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GNU Scientific Library
**********************

     GSL 2.6, Aug 16, 2019

     The GSL Team

     Copyright © 1996-2019 The GSL Team

* Menu:

* Introduction::
* Using the Library::
* Error Handling::
* Mathematical Functions::
* Complex Numbers::
* Polynomials::
* Special Functions::
* Vectors and Matrices::
* Permutations::
* Combinations::
* Multisets::
* Sorting::
* BLAS Support::
* Linear Algebra::
* Eigensystems::
* Fast Fourier Transforms (FFTs): Fast Fourier Transforms FFTs.
* Numerical Integration::
* Random Number Generation::
* Quasi-Random Sequences::
* Random Number Distributions::
* Statistics::
* Running Statistics::
* Moving Window Statistics::
* Digital Filtering::
* Histograms::
* N-tuples::
* Monte Carlo Integration::
* Simulated Annealing::
* Ordinary Differential Equations::
* Interpolation::
* Numerical Differentiation::
* Chebyshev Approximations::
* Series Acceleration::
* Wavelet Transforms::
* Discrete Hankel Transforms::
* One Dimensional Root-Finding::
* One Dimensional Minimization::
* Multidimensional Root-Finding::
* Multidimensional Minimization::
* Linear Least-Squares Fitting::
* Nonlinear Least-Squares Fitting::
* Basis Splines::
* Sparse Matrices::
* Sparse BLAS Support::
* Sparse Linear Algebra::
* Physical Constants::
* IEEE floating-point arithmetic::
* Debugging Numerical Programs::
* Contributors to GSL::
* Autoconf Macros::
* GSL CBLAS Library::
* GNU General Public License::
* GNU Free Documentation License::
* Index::

 — The Detailed Node Listing —

Introduction

* Routines available in GSL::
* GSL is Free Software::
* Obtaining GSL::
* No Warranty::
* Reporting Bugs::
* Further Information::
* Conventions used in this manual::

Using the Library

* An Example Program::
* Compiling and Linking::
* Shared Libraries::
* ANSI C Compliance::
* Inline functions::
* Long double::
* Portability functions::
* Alternative optimized functions::
* Support for different numeric types::
* Compatibility with C++::
* Aliasing of arrays::
* Thread-safety::
* Deprecated Functions::
* Code Reuse::

Compiling and Linking

* Linking programs with the library::
* Linking with an alternative BLAS library::

Error Handling

* Error Reporting::
* Error Codes::
* Error Handlers::
* Using GSL error reporting in your own functions::
* Examples::

Mathematical Functions

* Mathematical Constants::
* Infinities and Not-a-number::
* Elementary Functions::
* Small integer powers::
* Testing the Sign of Numbers::
* Testing for Odd and Even Numbers::
* Maximum and Minimum functions::
* Approximate Comparison of Floating Point Numbers::

Complex Numbers

* Representation of complex numbers::
* Properties of complex numbers::
* Complex arithmetic operators::
* Elementary Complex Functions::
* Complex Trigonometric Functions::
* Inverse Complex Trigonometric Functions::
* Complex Hyperbolic Functions::
* Inverse Complex Hyperbolic Functions::
* References and Further Reading::

Polynomials

* Polynomial Evaluation::
* Divided Difference Representation of Polynomials::
* Quadratic Equations::
* Cubic Equations::
* General Polynomial Equations::
* Examples: Examples<2>.
* References and Further Reading: References and Further Reading<2>.

Special Functions

* Usage::
* The gsl_sf_result struct::
* Modes::
* Airy Functions and Derivatives::
* Bessel Functions::
* Clausen Functions::
* Coulomb Functions::
* Coupling Coefficients::
* Dawson Function::
* Debye Functions::
* Dilogarithm::
* Elementary Operations::
* Elliptic Integrals::
* Elliptic Functions (Jacobi): Elliptic Functions Jacobi.
* Error Functions::
* Exponential Functions::
* Exponential Integrals::
* Fermi-Dirac Function::
* Gamma and Beta Functions::
* Gegenbauer Functions::
* Hermite Polynomials and Functions::
* Hypergeometric Functions::
* Laguerre Functions::
* Lambert W Functions::
* Legendre Functions and Spherical Harmonics::
* Logarithm and Related Functions::
* Mathieu Functions::
* Power Function::
* Psi (Digamma) Function: Psi Digamma Function.
* Synchrotron Functions::
* Transport Functions::
* Trigonometric Functions::
* Zeta Functions::
* Examples: Examples<3>.
* References and Further Reading: References and Further Reading<3>.

Airy Functions and Derivatives

* Airy Functions::
* Derivatives of Airy Functions::
* Zeros of Airy Functions::
* Zeros of Derivatives of Airy Functions::

Bessel Functions

* Regular Cylindrical Bessel Functions::
* Irregular Cylindrical Bessel Functions::
* Regular Modified Cylindrical Bessel Functions::
* Irregular Modified Cylindrical Bessel Functions::
* Regular Spherical Bessel Functions::
* Irregular Spherical Bessel Functions::
* Regular Modified Spherical Bessel Functions::
* Irregular Modified Spherical Bessel Functions::
* Regular Bessel Function—Fractional Order::
* Irregular Bessel Functions—Fractional Order::
* Regular Modified Bessel Functions—Fractional Order::
* Irregular Modified Bessel Functions—Fractional Order::
* Zeros of Regular Bessel Functions::

Coulomb Functions

* Normalized Hydrogenic Bound States::
* Coulomb Wave Functions::
* Coulomb Wave Function Normalization Constant::

Coupling Coefficients

* 3-j Symbols::
* 6-j Symbols::
* 9-j Symbols::

Dilogarithm

* Real Argument::
* Complex Argument::

Elliptic Integrals

* Definition of Legendre Forms::
* Definition of Carlson Forms::
* Legendre Form of Complete Elliptic Integrals::
* Legendre Form of Incomplete Elliptic Integrals::
* Carlson Forms::

Error Functions

* Error Function::
* Complementary Error Function::
* Log Complementary Error Function::
* Probability functions::

Exponential Functions

* Exponential Function::
* Relative Exponential Functions::
* Exponentiation With Error Estimate::

Exponential Integrals

* Exponential Integral::
* Ei(x): Ei x.
* Hyperbolic Integrals::
* Ei_3(x): Ei_3 x.
* Trigonometric Integrals::
* Arctangent Integral::

Fermi-Dirac Function

* Complete Fermi-Dirac Integrals::
* Incomplete Fermi-Dirac Integrals::

Gamma and Beta Functions

* Gamma Functions::
* Factorials::
* Pochhammer Symbol::
* Incomplete Gamma Functions::
* Beta Functions::
* Incomplete Beta Function::

Hermite Polynomials and Functions

* Hermite Polynomials::
* Derivatives of Hermite Polynomials::
* Hermite Functions::
* Derivatives of Hermite Functions::
* Zeros of Hermite Polynomials and Hermite Functions::

Legendre Functions and Spherical Harmonics

* Legendre Polynomials::
* Associated Legendre Polynomials and Spherical Harmonics::
* Conical Functions::
* Radial Functions for Hyperbolic Space::

Mathieu Functions

* Mathieu Function Workspace::
* Mathieu Function Characteristic Values::
* Angular Mathieu Functions::
* Radial Mathieu Functions::

Psi (Digamma) Function

* Digamma Function::
* Trigamma Function::
* Polygamma Function::

Trigonometric Functions

* Circular Trigonometric Functions::
* Trigonometric Functions for Complex Arguments::
* Hyperbolic Trigonometric Functions::
* Conversion Functions::
* Restriction Functions::
* Trigonometric Functions With Error Estimates::

Zeta Functions

* Riemann Zeta Function::
* Riemann Zeta Function Minus One::
* Hurwitz Zeta Function::
* Eta Function::

Vectors and Matrices

* Data types::
* Blocks::
* Vectors::
* Matrices::

Blocks

* Block allocation::
* Reading and writing blocks::
* Example programs for blocks::

Vectors

* Vector allocation::
* Accessing vector elements::
* Initializing vector elements::
* Reading and writing vectors::
* Vector views::
* Copying vectors::
* Exchanging elements::
* Vector operations::
* Finding maximum and minimum elements of vectors::
* Vector properties::
* Example programs for vectors::

Matrices

* Matrix allocation::
* Accessing matrix elements::
* Initializing matrix elements::
* Reading and writing matrices::
* Matrix views::
* Creating row and column views::
* Copying matrices::
* Copying rows and columns::
* Exchanging rows and columns::
* Matrix operations::
* Finding maximum and minimum elements of matrices::
* Matrix properties::
* Example programs for matrices::
* References and Further Reading: References and Further Reading<4>.

Permutations

* The Permutation struct::
* Permutation allocation::
* Accessing permutation elements::
* Permutation properties::
* Permutation functions::
* Applying Permutations::
* Reading and writing permutations::
* Permutations in cyclic form::
* Examples: Examples<4>.
* References and Further Reading: References and Further Reading<5>.

Combinations

* The Combination struct::
* Combination allocation::
* Accessing combination elements::
* Combination properties::
* Combination functions::
* Reading and writing combinations::
* Examples: Examples<5>.
* References and Further Reading: References and Further Reading<6>.

Multisets

* The Multiset struct::
* Multiset allocation::
* Accessing multiset elements::
* Multiset properties::
* Multiset functions::
* Reading and writing multisets::
* Examples: Examples<6>.

Sorting

* Sorting objects::
* Sorting vectors::
* Selecting the k smallest or largest elements::
* Computing the rank::
* Examples: Examples<7>.
* References and Further Reading: References and Further Reading<7>.

BLAS Support

* GSL BLAS Interface::
* Examples: Examples<8>.
* References and Further Reading: References and Further Reading<8>.

GSL BLAS Interface

* Level 1::
* Level 2::
* Level 3::

Linear Algebra

* LU Decomposition::
* QR Decomposition::
* QR Decomposition with Column Pivoting::
* LQ Decomposition::
* Complete Orthogonal Decomposition::
* Singular Value Decomposition::
* Cholesky Decomposition::
* Pivoted Cholesky Decomposition::
* Modified Cholesky Decomposition::
* LDLT Decomposition::
* Tridiagonal Decomposition of Real Symmetric Matrices::
* Tridiagonal Decomposition of Hermitian Matrices::
* Hessenberg Decomposition of Real Matrices::
* Hessenberg-Triangular Decomposition of Real Matrices::
* Bidiagonalization::
* Givens Rotations::
* Householder Transformations::
* Householder solver for linear systems::
* Tridiagonal Systems::
* Triangular Systems::
* Banded Systems::
* Balancing::
* Examples: Examples<9>.
* References and Further Reading: References and Further Reading<9>.

QR Decomposition

* Triangle on Top of Rectangle::

Banded Systems

* General Banded Format::
* Symmetric Banded Format::
* Banded Cholesky Decomposition::
* Banded LDLT Decomposition::

Eigensystems

* Real Symmetric Matrices::
* Complex Hermitian Matrices::
* Real Nonsymmetric Matrices::
* Real Generalized Symmetric-Definite Eigensystems::
* Complex Generalized Hermitian-Definite Eigensystems::
* Real Generalized Nonsymmetric Eigensystems::
* Sorting Eigenvalues and Eigenvectors::
* Examples: Examples<10>.
* References and Further Reading: References and Further Reading<10>.

Fast Fourier Transforms (FFTs)

* Mathematical Definitions::
* Overview of complex data FFTs::
* Radix-2 FFT routines for complex data::
* Mixed-radix FFT routines for complex data::
* Overview of real data FFTs::
* Radix-2 FFT routines for real data::
* Mixed-radix FFT routines for real data::
* References and Further Reading: References and Further Reading<11>.

Numerical Integration

* Introduction: Introduction<2>.
* QNG non-adaptive Gauss-Kronrod integration::
* QAG adaptive integration::
* QAGS adaptive integration with singularities::
* QAGP adaptive integration with known singular points::
* QAGI adaptive integration on infinite intervals::
* QAWC adaptive integration for Cauchy principal values::
* QAWS adaptive integration for singular functions::
* QAWO adaptive integration for oscillatory functions::
* QAWF adaptive integration for Fourier integrals::
* CQUAD doubly-adaptive integration::
* Romberg integration::
* Gauss-Legendre integration::
* Fixed point quadratures::
* Error codes::
* Examples: Examples<11>.
* References and Further Reading: References and Further Reading<12>.

Introduction

* Integrands without weight functions::
* Integrands with weight functions::
* Integrands with singular weight functions::

Examples

* Adaptive integration example::
* Fixed-point quadrature example::

Random Number Generation

* General comments on random numbers::
* The Random Number Generator Interface::
* Random number generator initialization::
* Sampling from a random number generator::
* Auxiliary random number generator functions::
* Random number environment variables::
* Copying random number generator state::
* Reading and writing random number generator state::
* Random number generator algorithms::
* Unix random number generators::
* Other random number generators::
* Performance::
* Examples: Examples<12>.
* References and Further Reading: References and Further Reading<13>.
* Acknowledgements::

Quasi-Random Sequences

* Quasi-random number generator initialization::
* Sampling from a quasi-random number generator::
* Auxiliary quasi-random number generator functions::
* Saving and restoring quasi-random number generator state::
* Quasi-random number generator algorithms::
* Examples: Examples<13>.
* References::

Random Number Distributions

* Introduction: Introduction<3>.
* The Gaussian Distribution::
* The Gaussian Tail Distribution::
* The Bivariate Gaussian Distribution::
* The Multivariate Gaussian Distribution::
* The Exponential Distribution::
* The Laplace Distribution::
* The Exponential Power Distribution::
* The Cauchy Distribution::
* The Rayleigh Distribution::
* The Rayleigh Tail Distribution::
* The Landau Distribution::
* The Levy alpha-Stable Distributions::
* The Levy skew alpha-Stable Distribution::
* The Gamma Distribution::
* The Flat (Uniform) Distribution: The Flat Uniform Distribution.
* The Lognormal Distribution::
* The Chi-squared Distribution::
* The F-distribution::
* The t-distribution::
* The Beta Distribution::
* The Logistic Distribution::
* The Pareto Distribution::
* Spherical Vector Distributions::
* The Weibull Distribution::
* The Type-1 Gumbel Distribution::
* The Type-2 Gumbel Distribution::
* The Dirichlet Distribution::
* General Discrete Distributions::
* The Poisson Distribution::
* The Bernoulli Distribution::
* The Binomial Distribution::
* The Multinomial Distribution::
* The Negative Binomial Distribution::
* The Pascal Distribution::
* The Geometric Distribution::
* The Hypergeometric Distribution::
* The Logarithmic Distribution::
* The Wishart Distribution::
* Shuffling and Sampling::
* Examples: Examples<14>.
* References and Further Reading: References and Further Reading<14>.

Statistics

* Mean, Standard Deviation and Variance: Mean Standard Deviation and Variance.
* Absolute deviation::
* Higher moments (skewness and kurtosis): Higher moments skewness and kurtosis.
* Autocorrelation::
* Covariance::
* Correlation::
* Weighted Samples::
* Maximum and Minimum values::
* Median and Percentiles::
* Order Statistics::
* Robust Location Estimates::
* Robust Scale Estimates::
* Examples: Examples<15>.
* References and Further Reading: References and Further Reading<15>.

Robust Location Estimates

* Trimmed Mean::
* Gastwirth Estimator::

Robust Scale Estimates

* Median Absolute Deviation (MAD): Median Absolute Deviation MAD.
* S_n Statistic::
* Q_n Statistic::

Running Statistics

* Initializing the Accumulator::
* Adding Data to the Accumulator::
* Current Statistics::
* Quantiles::
* Examples: Examples<16>.
* References and Further Reading: References and Further Reading<16>.

Moving Window Statistics

* Introduction: Introduction<4>.
* Handling Endpoints::
* Allocation for Moving Window Statistics::
* Moving Mean::
* Moving Variance and Standard Deviation::
* Moving Minimum and Maximum::
* Moving Sum::
* Moving Median::
* Robust Scale Estimation::
* User-defined Moving Statistics::
* Accumulators::
* Examples: Examples<17>.
* References and Further Reading: References and Further Reading<17>.

Robust Scale Estimation

* Moving MAD::
* Moving QQR::
* Moving S_n::
* Moving Q_n::

Examples

* Example 1::
* Example 2; Robust Scale: Example 2 Robust Scale.
* Example 3; User-defined Moving Window: Example 3 User-defined Moving Window.

Digital Filtering

* Introduction: Introduction<5>.
* Handling Endpoints: Handling Endpoints<2>.
* Linear Digital Filters::
* Nonlinear Digital Filters::
* Examples: Examples<18>.
* References and Further Reading: References and Further Reading<18>.

Linear Digital Filters

* Gaussian Filter::

Nonlinear Digital Filters

* Standard Median Filter::
* Recursive Median Filter::
* Impulse Detection Filter::

Examples

* Gaussian Example 1::
* Gaussian Example 2::
* Square Wave Signal Example::
* Impulse Detection Example::

Histograms

* The histogram struct::
* Histogram allocation::
* Copying Histograms::
* Updating and accessing histogram elements::
* Searching histogram ranges::
* Histogram Statistics::
* Histogram Operations::
* Reading and writing histograms::
* Resampling from histograms::
* The histogram probability distribution struct::
* Example programs for histograms::
* Two dimensional histograms::
* The 2D histogram struct::
* 2D Histogram allocation::
* Copying 2D Histograms::
* Updating and accessing 2D histogram elements::
* Searching 2D histogram ranges::
* 2D Histogram Statistics::
* 2D Histogram Operations::
* Reading and writing 2D histograms::
* Resampling from 2D histograms::
* Example programs for 2D histograms::

N-tuples

* The ntuple struct::
* Creating ntuples::
* Opening an existing ntuple file::
* Writing ntuples::
* Reading ntuples::
* Closing an ntuple file::
* Histogramming ntuple values::
* Examples: Examples<19>.
* References and Further Reading: References and Further Reading<19>.

Monte Carlo Integration

* Interface::
* PLAIN Monte Carlo::
* MISER::
* VEGAS::
* Examples: Examples<20>.
* References and Further Reading: References and Further Reading<20>.

Simulated Annealing

* Simulated Annealing algorithm::
* Simulated Annealing functions::
* Examples: Examples<21>.
* References and Further Reading: References and Further Reading<21>.

Examples

* Trivial example::
* Traveling Salesman Problem::

Ordinary Differential Equations

* Defining the ODE System::
* Stepping Functions::
* Adaptive Step-size Control::
* Evolution::
* Driver::
* Examples: Examples<22>.
* References and Further Reading: References and Further Reading<22>.

Interpolation

* Introduction to 1D Interpolation::
* 1D Interpolation Functions::
* 1D Interpolation Types::
* 1D Index Look-up and Acceleration::
* 1D Evaluation of Interpolating Functions::
* 1D Higher-level Interface::
* 1D Interpolation Example Programs::
* Introduction to 2D Interpolation::
* 2D Interpolation Functions::
* 2D Interpolation Grids::
* 2D Interpolation Types::
* 2D Evaluation of Interpolating Functions::
* 2D Higher-level Interface::
* 2D Interpolation Example programs::
* References and Further Reading: References and Further Reading<23>.

Numerical Differentiation

* Functions::
* Examples: Examples<23>.
* References and Further Reading: References and Further Reading<24>.

Chebyshev Approximations

* Definitions::
* Creation and Calculation of Chebyshev Series::
* Auxiliary Functions::
* Chebyshev Series Evaluation::
* Derivatives and Integrals::
* Examples: Examples<24>.
* References and Further Reading: References and Further Reading<25>.

Series Acceleration

* Acceleration functions::
* Acceleration functions without error estimation::
* Examples: Examples<25>.
* References and Further Reading: References and Further Reading<26>.

Wavelet Transforms

* Definitions: Definitions<2>.
* Initialization::
* Transform Functions::
* Examples: Examples<26>.
* References and Further Reading: References and Further Reading<27>.

Transform Functions

* Wavelet transforms in one dimension::
* Wavelet transforms in two dimension::

Discrete Hankel Transforms

* Definitions: Definitions<3>.
* Functions: Functions<2>.
* References and Further Reading: References and Further Reading<28>.

One Dimensional Root-Finding

* Overview::
* Caveats::
* Initializing the Solver::
* Providing the function to solve::
* Search Bounds and Guesses::
* Iteration::
* Search Stopping Parameters::
* Root Bracketing Algorithms::
* Root Finding Algorithms using Derivatives::
* Examples: Examples<27>.
* References and Further Reading: References and Further Reading<29>.

One Dimensional Minimization

* Overview: Overview<2>.
* Caveats: Caveats<2>.
* Initializing the Minimizer::
* Providing the function to minimize::
* Iteration: Iteration<2>.
* Stopping Parameters::
* Minimization Algorithms::
* Examples: Examples<28>.
* References and Further Reading: References and Further Reading<30>.

Multidimensional Root-Finding

* Overview: Overview<3>.
* Initializing the Solver: Initializing the Solver<2>.
* Providing the function to solve: Providing the function to solve<2>.
* Iteration: Iteration<3>.
* Search Stopping Parameters: Search Stopping Parameters<2>.
* Algorithms using Derivatives::
* Algorithms without Derivatives::
* Examples: Examples<29>.
* References and Further Reading: References and Further Reading<31>.

Multidimensional Minimization

* Overview: Overview<4>.
* Caveats: Caveats<3>.
* Initializing the Multidimensional Minimizer::
* Providing a function to minimize::
* Iteration: Iteration<4>.
* Stopping Criteria::
* Algorithms with Derivatives::
* Algorithms without Derivatives: Algorithms without Derivatives<2>.
* Examples: Examples<30>.
* References and Further Reading: References and Further Reading<32>.

Linear Least-Squares Fitting

* Overview: Overview<5>.
* Linear regression::
* Multi-parameter regression::
* Regularized regression::
* Robust linear regression::
* Large dense linear systems::
* Troubleshooting::
* Examples: Examples<31>.
* References and Further Reading: References and Further Reading<33>.

Linear regression

* Linear regression with a constant term::
* Linear regression without a constant term::

Large dense linear systems

* Normal Equations Approach::
* Tall Skinny QR (TSQR) Approach: Tall Skinny QR TSQR Approach.
* Large Dense Linear Systems Solution Steps::
* Large Dense Linear Least Squares Routines::

Examples

* Simple Linear Regression Example::
* Multi-parameter Linear Regression Example::
* Regularized Linear Regression Example 1::
* Regularized Linear Regression Example 2::
* Robust Linear Regression Example::
* Large Dense Linear Regression Example::

Nonlinear Least-Squares Fitting

* Overview: Overview<6>.
* Solving the Trust Region Subproblem (TRS): Solving the Trust Region Subproblem TRS.
* Weighted Nonlinear Least-Squares::
* Tunable Parameters::
* Initializing the Solver: Initializing the Solver<3>.
* Providing the Function to be Minimized::
* Iteration: Iteration<5>.
* Testing for Convergence::
* High Level Driver::
* Covariance matrix of best fit parameters::
* Troubleshooting: Troubleshooting<2>.
* Examples: Examples<32>.
* References and Further Reading: References and Further Reading<34>.

Solving the Trust Region Subproblem (TRS)

* Levenberg-Marquardt::
* Levenberg-Marquardt with Geodesic Acceleration::
* Dogleg::
* Double Dogleg::
* Two Dimensional Subspace::
* Steihaug-Toint Conjugate Gradient::

Examples

* Exponential Fitting Example::
* Geodesic Acceleration Example 1::
* Geodesic Acceleration Example 2::
* Comparing TRS Methods Example::
* Large Nonlinear Least Squares Example::

Basis Splines

* Overview: Overview<7>.
* Initializing the B-splines solver::
* Constructing the knots vector::
* Evaluation of B-splines::
* Evaluation of B-spline derivatives::
* Working with the Greville abscissae::
* Examples: Examples<33>.
* References and Further Reading: References and Further Reading<35>.

Sparse Matrices

* Data types: Data types<2>.
* Sparse Matrix Storage Formats::
* Overview: Overview<8>.
* Allocation::
* Accessing Matrix Elements::
* Initializing Matrix Elements::
* Reading and Writing Matrices::
* Copying Matrices::
* Exchanging Rows and Columns::
* Matrix Operations::
* Matrix Properties::
* Finding Maximum and Minimum Elements::
* Compressed Format::
* Conversion Between Sparse and Dense Matrices::
* Examples: Examples<34>.
* References and Further Reading: References and Further Reading<36>.

Sparse Matrix Storage Formats

* Coordinate Storage (COO): Coordinate Storage COO.
* Compressed Sparse Column (CSC): Compressed Sparse Column CSC.
* Compressed Sparse Row (CSR): Compressed Sparse Row CSR.

Sparse BLAS Support

* Sparse BLAS operations::
* References and Further Reading: References and Further Reading<37>.

Sparse Linear Algebra

* Overview: Overview<9>.
* Sparse Iterative Solvers::
* Examples: Examples<35>.
* References and Further Reading: References and Further Reading<38>.

Sparse Iterative Solvers

* Overview: Overview<10>.
* Types of Sparse Iterative Solvers::
* Iterating the Sparse Linear System::

Physical Constants

* Fundamental Constants::
* Astronomy and Astrophysics::
* Atomic and Nuclear Physics::
* Measurement of Time::
* Imperial Units::
* Speed and Nautical Units::
* Printers Units::
* Volume, Area and Length: Volume Area and Length.
* Mass and Weight::
* Thermal Energy and Power::
* Pressure::
* Viscosity::
* Light and Illumination::
* Radioactivity::
* Force and Energy::
* Prefixes::
* Examples: Examples<36>.
* References and Further Reading: References and Further Reading<39>.

IEEE floating-point arithmetic

* Representation of floating point numbers::
* Setting up your IEEE environment::
* References and Further Reading: References and Further Reading<40>.

Debugging Numerical Programs

* Using gdb::
* Examining floating point registers::
* Handling floating point exceptions::
* GCC warning options for numerical programs::
* References and Further Reading: References and Further Reading<41>.

GSL CBLAS Library

* Level 1: Level 1<2>.
* Level 2: Level 2<2>.
* Level 3: Level 3<2>.
* Examples: Examples<37>.



File: gsl-ref.info,  Node: Introduction,  Next: Using the Library,  Prev: Top,  Up: Top

1 Introduction
**************

The GNU Scientific Library (GSL) is a collection of routines for
numerical computing.  The routines have been written from scratch in C,
and present a modern Applications Programming Interface (API) for C
programmers, allowing wrappers to be written for very high level
languages.  The source code is distributed under the GNU General Public
License.

* Menu:

* Routines available in GSL::
* GSL is Free Software::
* Obtaining GSL::
* No Warranty::
* Reporting Bugs::
* Further Information::
* Conventions used in this manual::


File: gsl-ref.info,  Node: Routines available in GSL,  Next: GSL is Free Software,  Up: Introduction

1.1 Routines available in GSL
=============================

The library covers a wide range of topics in numerical computing.
Routines are available for the following areas,

Complex Numbers                 Roots of Polynomials            Special Functions
                                                                
                                                                
Vectors and Matrices            Permutations                    Combinations
                                                                
                                                                
Sorting                         BLAS Support                    Linear Algebra
                                                                
                                                                
CBLAS Library                   Fast Fourier Transforms         Eigensystems
                                                                
                                                                
Random Numbers                  Quadrature                      Random Distributions
                                                                
                                                                
Quasi-Random Sequences          Histograms                      Statistics
                                                                
                                                                
Monte Carlo Integration         N-Tuples                        Differential Equations
                                                                
                                                                
Simulated Annealing             Numerical Differentiation       Interpolation
                                                                
                                                                
Series Acceleration             Chebyshev Approximations        Root-Finding
                                                                
                                                                
Discrete Hankel Transforms      Least-Squares Fitting           Minimization
                                                                
                                                                
IEEE Floating-Point             Physical Constants              Basis Splines
                                                                
                                                                
Wavelets                        Sparse BLAS Support             Sparse Linear Algebra
                                                                

The use of these routines is described in this manual.  Each chapter
provides detailed definitions of the functions, followed by example
programs and references to the articles on which the algorithms are
based.

Where possible the routines have been based on reliable public-domain
packages such as FFTPACK and QUADPACK, which the developers of GSL have
reimplemented in C with modern coding conventions.


File: gsl-ref.info,  Node: GSL is Free Software,  Next: Obtaining GSL,  Prev: Routines available in GSL,  Up: Introduction

1.2 GSL is Free Software
========================

The subroutines in the GNU Scientific Library are “free software”; this
means that everyone is free to use them, and to redistribute them in
other free programs.  The library is not in the public domain; it is
copyrighted and there are conditions on its distribution.  These
conditions are designed to permit everything that a good cooperating
citizen would want to do.  What is not allowed is to try to prevent
others from further sharing any version of the software that they might
get from you.

Specifically, we want to make sure that you have the right to share
copies of programs that you are given which use the GNU Scientific
Library, that you receive their source code or else can get it if you
want it, that you can change these programs or use pieces of them in new
free programs, and that you know you can do these things.

To make sure that everyone has such rights, we have to forbid you to
deprive anyone else of these rights.  For example, if you distribute
copies of any code which uses the GNU Scientific Library, you must give
the recipients all the rights that you have received.  You must make
sure that they, too, receive or can get the source code, both to the
library and the code which uses it.  And you must tell them their
rights.  This means that the library should not be redistributed in
proprietary programs.

Also, for our own protection, we must make certain that everyone finds
out that there is no warranty for the GNU Scientific Library.  If these
programs are modified by someone else and passed on, we want their
recipients to know that what they have is not what we distributed, so
that any problems introduced by others will not reflect on our
reputation.

The precise conditions for the distribution of software related to the
GNU Scientific Library are found in the GNU General Public License(1).
Further information about this license is available from the GNU Project
webpage Frequently Asked Questions about the GNU GPL(2).

The Free Software Foundation also operates a license consulting service
for commercial users (contact details available from
‘http://www.fsf.org’.

   ---------- Footnotes ----------

   (1) 
https://www.gnu.org/software/gsl/manual/html_node/GNU-General-Public-License.html#GNU-General-Public-License

   (2) http://www.gnu.org/copyleft/gpl-faq.html


File: gsl-ref.info,  Node: Obtaining GSL,  Next: No Warranty,  Prev: GSL is Free Software,  Up: Introduction

1.3 Obtaining GSL
=================

The source code for the library can be obtained in different ways, by
copying it from a friend, purchasing it on CDROM or downloading it from
the internet.  A list of public ftp servers which carry the source code
can be found on the GNU website, ‘http://www.gnu.org/software/gsl/’.

The preferred platform for the library is a GNU system, which allows it
to take advantage of additional features in the GNU C compiler and GNU C
library.  However, the library is fully portable and should compile on
most systems with a C compiler.

Announcements of new releases, updates and other relevant events are
made on the <info-gsl@gnu.org> mailing list.  To subscribe to this
low-volume list, send an email of the following form:

     To: info-gsl-request@gnu.org
     Subject: subscribe

You will receive a response asking you to reply in order to confirm your
subscription.


File: gsl-ref.info,  Node: No Warranty,  Next: Reporting Bugs,  Prev: Obtaining GSL,  Up: Introduction

1.4 No Warranty
===============

The software described in this manual has no warranty, it is provided
“as is”.  It is your responsibility to validate the behavior of the
routines and their accuracy using the source code provided, or to
purchase support and warranties from commercial redistributors.  Consult
the GNU General Public License(1) for further details.

   ---------- Footnotes ----------

   (1) 
https://www.gnu.org/software/gsl/manual/html_node/GNU-General-Public-License.html#GNU-General-Public-License


File: gsl-ref.info,  Node: Reporting Bugs,  Next: Further Information,  Prev: No Warranty,  Up: Introduction

1.5 Reporting Bugs
==================

A list of known bugs can be found in the ‘BUGS’ file included in the GSL
distribution or online in the GSL bug tracker.  (1) Details of
compilation problems can be found in the ‘INSTALL’ file.

If you find a bug which is not listed in these files, please report it
to <bug-gsl@gnu.org>.

All bug reports should include:

   - The version number of GSL

   - The hardware and operating system

   - The compiler used, including version number and compilation options

   - A description of the bug behavior

   - A short program which exercises the bug

It is useful if you can check whether the same problem occurs when the
library is compiled without optimization.  Thank you.

Any errors or omissions in this manual can also be reported to the same
address.

   ---------- Footnotes ----------

   (1) ‘http://savannah.gnu.org/bugs/?group=gsl’


File: gsl-ref.info,  Node: Further Information,  Next: Conventions used in this manual,  Prev: Reporting Bugs,  Up: Introduction

1.6 Further Information
=======================

Additional information, including online copies of this manual, links to
related projects, and mailing list archives are available from the
website mentioned above.

Any questions about the use and installation of the library can be asked
on the mailing list <help-gsl@gnu.org>.  To subscribe to this list, send
an email of the following form:

     To: help-gsl-request@gnu.org
     Subject: subscribe

This mailing list can be used to ask questions not covered by this
manual, and to contact the developers of the library.

If you would like to refer to the GNU Scientific Library in a journal
article, the recommended way is to cite this reference manual, e.g.:

     M. Galassi et al, GNU Scientific Library Reference Manual (3rd Ed.), ISBN 0954612078.

If you want to give a url, use “‘http://www.gnu.org/software/gsl/’”.


File: gsl-ref.info,  Node: Conventions used in this manual,  Prev: Further Information,  Up: Introduction

1.7 Conventions used in this manual
===================================

This manual contains many examples which can be typed at the keyboard.
A command entered at the terminal is shown like this:

     $ command

The first character on the line is the terminal prompt, and should not
be typed.  The dollar sign $ is used as the standard prompt in this
manual, although some systems may use a different character.

The examples assume the use of the GNU operating system.  There may be
minor differences in the output on other systems.  The commands for
setting environment variables use the Bourne shell syntax of the
standard GNU shell (‘bash’).


File: gsl-ref.info,  Node: Using the Library,  Next: Error Handling,  Prev: Introduction,  Up: Top

2 Using the Library
*******************

This chapter describes how to compile programs that use GSL, and
introduces its conventions.

* Menu:

* An Example Program::
* Compiling and Linking::
* Shared Libraries::
* ANSI C Compliance::
* Inline functions::
* Long double::
* Portability functions::
* Alternative optimized functions::
* Support for different numeric types::
* Compatibility with C++::
* Aliasing of arrays::
* Thread-safety::
* Deprecated Functions::
* Code Reuse::


File: gsl-ref.info,  Node: An Example Program,  Next: Compiling and Linking,  Up: Using the Library

2.1 An Example Program
======================

The following short program demonstrates the use of the library by
computing the value of the Bessel function J_0(x) for x=5:

     #include <stdio.h>
     #include <gsl/gsl_sf_bessel.h>

     int
     main (void)
     {
       double x = 5.0;
       double y = gsl_sf_bessel_J0 (x);
       printf ("J0(%g) = %.18e\n", x, y);
       return 0;
     }

The output is shown below, and should be correct to double-precision
accuracy (1),

     J0(5) = -1.775967713143382642e-01

The steps needed to compile this program are described in the following
sections.

   ---------- Footnotes ----------

   (1) The last few digits may vary slightly depending on the compiler
and platform used—this is normal


File: gsl-ref.info,  Node: Compiling and Linking,  Next: Shared Libraries,  Prev: An Example Program,  Up: Using the Library

2.2 Compiling and Linking
=========================

The library header files are installed in their own ‘gsl’ directory.
You should write any preprocessor include statements with a ‘gsl/’
directory prefix thus:

     #include <gsl/gsl_math.h>

If the directory is not installed on the standard search path of your
compiler you will also need to provide its location to the preprocessor
as a command line flag.  The default location of the ‘gsl’ directory is
‘/usr/local/include/gsl’.  A typical compilation command for a source
file ‘example.c’ with the GNU C compiler ‘gcc’ is:

     $ gcc -Wall -I/usr/local/include -c example.c

This results in an object file ‘example.o’.  The default include path
for ‘gcc’ searches ‘/usr/local/include’ automatically so the ‘-I’ option
can actually be omitted when GSL is installed in its default location.

* Menu:

* Linking programs with the library::
* Linking with an alternative BLAS library::


File: gsl-ref.info,  Node: Linking programs with the library,  Next: Linking with an alternative BLAS library,  Up: Compiling and Linking

2.2.1 Linking programs with the library
---------------------------------------

The library is installed as a single file, ‘libgsl.a’.  A shared version
of the library ‘libgsl.so’ is also installed on systems that support
shared libraries.  The default location of these files is
‘/usr/local/lib’.  If this directory is not on the standard search path
of your linker you will also need to provide its location as a command
line flag.

To link against the library you need to specify both the main library
and a supporting CBLAS library, which provides standard basic linear
algebra subroutines.  A suitable CBLAS implementation is provided in the
library ‘libgslcblas.a’ if your system does not provide one.  The
following example shows how to link an application with the library:

     $ gcc -L/usr/local/lib example.o -lgsl -lgslcblas -lm

The default library path for ‘gcc’ searches ‘/usr/local/lib’
automatically so the ‘-L’ option can be omitted when GSL is installed in
its default location.

The option ‘-lm’ links with the system math library.  On some systems it
is not needed.  (1)

For a tutorial introduction to the GNU C Compiler and related programs,
see “An Introduction to GCC” (ISBN 0954161793).  (2)

   ---------- Footnotes ----------

   (1) It is not needed on MacOS X

   (2) ‘http://www.network-theory.co.uk/gcc/intro/’


File: gsl-ref.info,  Node: Linking with an alternative BLAS library,  Prev: Linking programs with the library,  Up: Compiling and Linking

2.2.2 Linking with an alternative BLAS library
----------------------------------------------

The following command line shows how you would link the same application
with an alternative CBLAS library ‘libcblas.a’:

     $ gcc example.o -lgsl -lcblas -lm

For the best performance an optimized platform-specific CBLAS library
should be used for ‘-lcblas’.  The library must conform to the CBLAS
standard.  The ATLAS package provides a portable high-performance BLAS
library with a CBLAS interface.  It is free software and should be
installed for any work requiring fast vector and matrix operations.  The
following command line will link with the ATLAS library and its CBLAS
interface:

     $ gcc example.o -lgsl -lcblas -latlas -lm

If the ATLAS library is installed in a non-standard directory use the
‘-L’ option to add it to the search path, as described above.

For more information about BLAS functions see *note BLAS Support: 11.


File: gsl-ref.info,  Node: Shared Libraries,  Next: ANSI C Compliance,  Prev: Compiling and Linking,  Up: Using the Library

2.3 Shared Libraries
====================

To run a program linked with the shared version of the library the
operating system must be able to locate the corresponding ‘.so’ file at
runtime.  If the library cannot be found, the following error will
occur:

     $ ./a.out
     ./a.out: error while loading shared libraries:
     libgsl.so.0: cannot open shared object file: No such file or directory

To avoid this error, either modify the system dynamic linker
configuration (1) or define the shell variable ‘LD_LIBRARY_PATH’ to
include the directory where the library is installed.

For example, in the Bourne shell (‘/bin/sh’ or ‘/bin/bash’), the library
search path can be set with the following commands:

     $ LD_LIBRARY_PATH=/usr/local/lib
     $ export LD_LIBRARY_PATH
     $ ./example

In the C-shell (‘/bin/csh’ or ‘/bin/tcsh’) the equivalent command is:

     % setenv LD_LIBRARY_PATH /usr/local/lib

The standard prompt for the C-shell in the example above is the percent
character %, and should not be typed as part of the command.

To save retyping these commands each session they can be placed in an
individual or system-wide login file.

To compile a statically linked version of the program, use the ‘-static’
flag in ‘gcc’:

     $ gcc -static example.o -lgsl -lgslcblas -lm

   ---------- Footnotes ----------

   (1) ‘/etc/ld.so.conf’ on GNU/Linux systems


File: gsl-ref.info,  Node: ANSI C Compliance,  Next: Inline functions,  Prev: Shared Libraries,  Up: Using the Library

2.4 ANSI C Compliance
=====================

The library is written in ANSI C and is intended to conform to the ANSI
C standard (C89).  It should be portable to any system with a working
ANSI C compiler.

The library does not rely on any non-ANSI extensions in the interface it
exports to the user.  Programs you write using GSL can be ANSI
compliant.  Extensions which can be used in a way compatible with pure
ANSI C are supported, however, via conditional compilation.  This allows
the library to take advantage of compiler extensions on those platforms
which support them.

When an ANSI C feature is known to be broken on a particular system the
library will exclude any related functions at compile-time.  This should
make it impossible to link a program that would use these functions and
give incorrect results.

To avoid namespace conflicts all exported function names and variables
have the prefix ‘gsl_’, while exported macros have the prefix ‘GSL_’.


File: gsl-ref.info,  Node: Inline functions,  Next: Long double,  Prev: ANSI C Compliance,  Up: Using the Library

2.5 Inline functions
====================

The ‘inline’ keyword is not part of the original ANSI C standard (C89)
so the library does not export any inline function definitions by
default.  Inline functions were introduced officially in the newer C99
standard but most C89 compilers have also included ‘inline’ as an
extension for a long time.

To allow the use of inline functions, the library provides optional
inline versions of performance-critical routines by conditional
compilation in the exported header files.  The inline versions of these
functions can be included by defining the macro ‘HAVE_INLINE’ when
compiling an application:

     $ gcc -Wall -c -DHAVE_INLINE example.c

If you use ‘autoconf’ this macro can be defined automatically.  If you
do not define the macro ‘HAVE_INLINE’ then the slower non-inlined
versions of the functions will be used instead.

By default, the actual form of the inline keyword is ‘extern inline’,
which is a ‘gcc’ extension that eliminates unnecessary function
definitions.  If the form ‘extern inline’ causes problems with other
compilers a stricter autoconf test can be used, see *note Autoconf
Macros: 16.

When compiling with ‘gcc’ in C99 mode (‘gcc -std=c99’) the header files
automatically switch to C99-compatible inline function declarations
instead of ‘extern inline’.  With other C99 compilers, define the macro
‘GSL_C99_INLINE’ to use these declarations.


File: gsl-ref.info,  Node: Long double,  Next: Portability functions,  Prev: Inline functions,  Up: Using the Library

2.6 Long double
===============

In general, the algorithms in the library are written for double
precision only.  The ‘long double’ type is not supported for actual
computation.

One reason for this choice is that the precision of ‘long double’ is
platform dependent.  The IEEE standard only specifies the minimum
precision of extended precision numbers, while the precision of ‘double’
is the same on all platforms.

However, it is sometimes necessary to interact with external data in
long-double format, so the vector and matrix datatypes include
long-double versions.

It should be noted that in some system libraries the ‘stdio.h’ formatted
input/output functions ‘printf’ and ‘scanf’ are not implemented
correctly for ‘long double’.  Undefined or incorrect results are avoided
by testing these functions during the ‘configure’ stage of library
compilation and eliminating certain GSL functions which depend on them
if necessary.  The corresponding line in the ‘configure’ output looks
like this:

     checking whether printf works with long double... no

Consequently when ‘long double’ formatted input/output does not work on
a given system it should be impossible to link a program which uses GSL
functions dependent on this.

If it is necessary to work on a system which does not support formatted
‘long double’ input/output then the options are to use binary formats or
to convert ‘long double’ results into ‘double’ for reading and writing.


File: gsl-ref.info,  Node: Portability functions,  Next: Alternative optimized functions,  Prev: Long double,  Up: Using the Library

2.7 Portability functions
=========================

To help in writing portable applications GSL provides some
implementations of functions that are found in other libraries, such as
the BSD math library.  You can write your application to use the native
versions of these functions, and substitute the GSL versions via a
preprocessor macro if they are unavailable on another platform.

For example, after determining whether the BSD function ‘hypot()’ is
available you can include the following macro definitions in a file
‘config.h’ with your application:

     /* Substitute gsl_hypot for missing system hypot */

     #ifndef HAVE_HYPOT
     #define hypot gsl_hypot
     #endif

The application source files can then use the include command ‘#include
<config.h>’ to replace each occurrence of ‘hypot()’ by *note
gsl_hypot(): 1a. when ‘hypot()’ is not available.  This substitution can
be made automatically if you use ‘autoconf’, see *note Autoconf Macros:
16.

In most circumstances the best strategy is to use the native versions of
these functions when available, and fall back to GSL versions otherwise,
since this allows your application to take advantage of any
platform-specific optimizations in the system library.  This is the
strategy used within GSL itself.


File: gsl-ref.info,  Node: Alternative optimized functions,  Next: Support for different numeric types,  Prev: Portability functions,  Up: Using the Library

2.8 Alternative optimized functions
===================================

The main implementation of some functions in the library will not be
optimal on all architectures.  For example, there are several ways to
compute a Gaussian random variate and their relative speeds are
platform-dependent.  In cases like this the library provides alternative
implementations of these functions with the same interface.  If you
write your application using calls to the standard implementation you
can select an alternative version later via a preprocessor definition.
It is also possible to introduce your own optimized functions this way
while retaining portability.  The following lines demonstrate the use of
a platform-dependent choice of methods for sampling from the Gaussian
distribution:

     #ifdef SPARC
     #define gsl_ran_gaussian gsl_ran_gaussian_ratio_method
     #endif
     #ifdef INTEL
     #define gsl_ran_gaussian my_gaussian
     #endif

These lines would be placed in the configuration header file ‘config.h’
of the application, which should then be included by all the source
files.  Note that the alternative implementations will not produce
bit-for-bit identical results, and in the case of random number
distributions will produce an entirely different stream of random
variates.


File: gsl-ref.info,  Node: Support for different numeric types,  Next: Compatibility with C++,  Prev: Alternative optimized functions,  Up: Using the Library

2.9 Support for different numeric types
=======================================

Many functions in the library are defined for different numeric types.
This feature is implemented by varying the name of the function with a
type-related modifier—a primitive form of C++ templates.  The modifier
is inserted into the function name after the initial module prefix.  The
following table shows the function names defined for all the numeric
types of an imaginary module ‘gsl_foo’ with function ‘fn()’:

     gsl_foo_fn               double
     gsl_foo_long_double_fn   long double
     gsl_foo_float_fn         float
     gsl_foo_long_fn          long
     gsl_foo_ulong_fn         unsigned long
     gsl_foo_int_fn           int
     gsl_foo_uint_fn          unsigned int
     gsl_foo_short_fn         short
     gsl_foo_ushort_fn        unsigned short
     gsl_foo_char_fn          char
     gsl_foo_uchar_fn         unsigned char

The normal numeric precision ‘double’ is considered the default and does
not require a suffix.  For example, the function *note gsl_stats_mean():
1d. computes the mean of double precision numbers, while the function
‘gsl_stats_int_mean()’ computes the mean of integers.

A corresponding scheme is used for library defined types, such as
‘gsl_vector’ and ‘gsl_matrix’.  In this case the modifier is appended to
the type name.  For example, if a module defines a new type-dependent
struct or typedef ‘gsl_foo’ it is modified for other types in the
following way:

     gsl_foo                  double
     gsl_foo_long_double      long double
     gsl_foo_float            float
     gsl_foo_long             long
     gsl_foo_ulong            unsigned long
     gsl_foo_int              int
     gsl_foo_uint             unsigned int
     gsl_foo_short            short
     gsl_foo_ushort           unsigned short
     gsl_foo_char             char
     gsl_foo_uchar            unsigned char

When a module contains type-dependent definitions the library provides
individual header files for each type.  The filenames are modified as
shown in the below.  For convenience the default header includes the
definitions for all the types.  To include only the double precision
header file, or any other specific type, use its individual filename:

     #include <gsl/gsl_foo.h>               All types
     #include <gsl/gsl_foo_double.h>        double
     #include <gsl/gsl_foo_long_double.h>   long double
     #include <gsl/gsl_foo_float.h>         float
     #include <gsl/gsl_foo_long.h>          long
     #include <gsl/gsl_foo_ulong.h>         unsigned long
     #include <gsl/gsl_foo_int.h>           int
     #include <gsl/gsl_foo_uint.h>          unsigned int
     #include <gsl/gsl_foo_short.h>         short
     #include <gsl/gsl_foo_ushort.h>        unsigned short
     #include <gsl/gsl_foo_char.h>          char
     #include <gsl/gsl_foo_uchar.h>         unsigned char


File: gsl-ref.info,  Node: Compatibility with C++,  Next: Aliasing of arrays,  Prev: Support for different numeric types,  Up: Using the Library

2.10 Compatibility with C++
===========================

The library header files automatically define functions to have ‘extern
"C"’ linkage when included in C++ programs.  This allows the functions
to be called directly from C++.

To use C++ exception handling within user-defined functions passed to
the library as parameters, the library must be built with the additional
‘CFLAGS’ compilation option ‘-fexceptions’.


File: gsl-ref.info,  Node: Aliasing of arrays,  Next: Thread-safety,  Prev: Compatibility with C++,  Up: Using the Library

2.11 Aliasing of arrays
=======================

The library assumes that arrays, vectors and matrices passed as
modifiable arguments are not aliased and do not overlap with each other.
This removes the need for the library to handle overlapping memory
regions as a special case, and allows additional optimizations to be
used.  If overlapping memory regions are passed as modifiable arguments
then the results of such functions will be undefined.  If the arguments
will not be modified (for example, if a function prototype declares them
as ‘const’ arguments) then overlapping or aliased memory regions can be
safely used.


File: gsl-ref.info,  Node: Thread-safety,  Next: Deprecated Functions,  Prev: Aliasing of arrays,  Up: Using the Library

2.12 Thread-safety
==================

The library can be used in multi-threaded programs.  All the functions
are thread-safe, in the sense that they do not use static variables.
Memory is always associated with objects and not with functions.  For
functions which use `workspace' objects as temporary storage the
workspaces should be allocated on a per-thread basis.  For functions
which use `table' objects as read-only memory the tables can be used by
multiple threads simultaneously.  Table arguments are always declared
‘const’ in function prototypes, to indicate that they may be safely
accessed by different threads.

There are a small number of static global variables which are used to
control the overall behavior of the library (e.g.  whether to use
range-checking, the function to call on fatal error, etc).  These
variables are set directly by the user, so they should be initialized
once at program startup and not modified by different threads.


File: gsl-ref.info,  Node: Deprecated Functions,  Next: Code Reuse,  Prev: Thread-safety,  Up: Using the Library

2.13 Deprecated Functions
=========================

From time to time, it may be necessary for the definitions of some
functions to be altered or removed from the library.  In these
circumstances the functions will first be declared `deprecated' and then
removed from subsequent versions of the library.  Functions that are
deprecated can be disabled in the current release by setting the
preprocessor definition ‘GSL_DISABLE_DEPRECATED’.  This allows existing
code to be tested for forwards compatibility.


File: gsl-ref.info,  Node: Code Reuse,  Prev: Deprecated Functions,  Up: Using the Library

2.14 Code Reuse
===============

Where possible the routines in the library have been written to avoid
dependencies between modules and files.  This should make it possible to
extract individual functions for use in your own applications, without
needing to have the whole library installed.  You may need to define
certain macros such as ‘GSL_ERROR’ and remove some ‘#include’ statements
in order to compile the files as standalone units.  Reuse of the library
code in this way is encouraged, subject to the terms of the GNU General
Public License.


File: gsl-ref.info,  Node: Error Handling,  Next: Mathematical Functions,  Prev: Using the Library,  Up: Top

3 Error Handling
****************

This chapter describes the way that GSL functions report and handle
errors.  By examining the status information returned by every function
you can determine whether it succeeded or failed, and if it failed you
can find out what the precise cause of failure was.  You can also define
your own error handling functions to modify the default behavior of the
library.

The functions described in this section are declared in the header file
‘gsl_errno.h’.

* Menu:

* Error Reporting::
* Error Codes::
* Error Handlers::
* Using GSL error reporting in your own functions::
* Examples::


File: gsl-ref.info,  Node: Error Reporting,  Next: Error Codes,  Up: Error Handling

3.1 Error Reporting
===================

The library follows the thread-safe error reporting conventions of the
POSIX Threads library.  Functions return a non-zero error code to
indicate an error and ‘0’ to indicate success:

     int status = gsl_function (...)

     if (status) { /* an error occurred */
       .....
       /* status value specifies the type of error */
     }

The routines report an error whenever they cannot perform the task
requested of them.  For example, a root-finding function would return a
non-zero error code if could not converge to the requested accuracy, or
exceeded a limit on the number of iterations.  Situations like this are
a normal occurrence when using any mathematical library and you should
check the return status of the functions that you call.

Whenever a routine reports an error the return value specifies the type
of error.  The return value is analogous to the value of the variable
‘errno’ in the C library.  The caller can examine the return code and
decide what action to take, including ignoring the error if it is not
considered serious.

In addition to reporting errors by return codes the library also has an
error handler function ‘gsl_error()’.  This function is called by other
library functions when they report an error, just before they return to
the caller.  The default behavior of the error handler is to print a
message and abort the program:

     gsl: file.c:67: ERROR: invalid argument supplied by user
     Default GSL error handler invoked.
     Aborted

The purpose of the ‘gsl_error()’ handler is to provide a function where
a breakpoint can be set that will catch library errors when running
under the debugger.  It is not intended for use in production programs,
which should handle any errors using the return codes.


File: gsl-ref.info,  Node: Error Codes,  Next: Error Handlers,  Prev: Error Reporting,  Up: Error Handling

3.2 Error Codes
===============

The error code numbers returned by library functions are defined in the
file ‘gsl_errno.h’.  They all have the prefix ‘GSL_’ and expand to
non-zero constant integer values.  Error codes above 1024 are reserved
for applications, and are not used by the library.  Many of the error
codes use the same base name as the corresponding error code in the C
library.  Here are some of the most common error codes,

 -- Variable: int GSL_EDOM

     Domain error; used by mathematical functions when an argument value
     does not fall into the domain over which the function is defined
     (like ‘EDOM’ in the C library)

 -- Variable: int GSL_ERANGE

     Range error; used by mathematical functions when the result value
     is not representable because of overflow or underflow (like
     ‘ERANGE’ in the C library)

 -- Variable: int GSL_ENOMEM

     No memory available.  The system cannot allocate more virtual
     memory because its capacity is full (like ‘ENOMEM’ in the C
     library).  This error is reported when a GSL routine encounters
     problems when trying to allocate memory with ‘malloc()’.

 -- Variable: int GSL_EINVAL

     Invalid argument.  This is used to indicate various kinds of
     problems with passing the wrong argument to a library function
     (like ‘EINVAL’ in the C library).

The error codes can be converted into an error message using the
function *note gsl_strerror(): 2c.

 -- Function: const char * gsl_strerror (const int gsl_errno)

     This function returns a pointer to a string describing the error
     code ‘gsl_errno’.  For example:

          printf ("error: %s\n", gsl_strerror (status));

     would print an error message like ‘error: output range error’ for a
     status value of *note GSL_ERANGE: 29.


File: gsl-ref.info,  Node: Error Handlers,  Next: Using GSL error reporting in your own functions,  Prev: Error Codes,  Up: Error Handling

3.3 Error Handlers
==================

The default behavior of the GSL error handler is to print a short
message and call ‘abort()’.  When this default is in use programs will
stop with a core-dump whenever a library routine reports an error.  This
is intended as a fail-safe default for programs which do not check the
return status of library routines (we don’t encourage you to write
programs this way).

If you turn off the default error handler it is your responsibility to
check the return values of routines and handle them yourself.  You can
also customize the error behavior by providing a new error handler.  For
example, an alternative error handler could log all errors to a file,
ignore certain error conditions (such as underflows), or start the
debugger and attach it to the current process when an error occurs.

All GSL error handlers have the type ‘gsl_error_handler_t’, which is
defined in ‘gsl_errno.h’,

 -- Type: gsl_error_handler_t

     This is the type of GSL error handler functions.  An error handler
     will be passed four arguments which specify the reason for the
     error (a string), the name of the source file in which it occurred
     (also a string), the line number in that file (an integer) and the
     error number (an integer).  The source file and line number are set
     at compile time using the ‘__FILE__’ and ‘__LINE__’ directives in
     the preprocessor.  An error handler function returns type ‘void’.
     Error handler functions should be defined like this:

          void handler (const char * reason,
                        const char * file,
                        int line,
                        int gsl_errno)

To request the use of your own error handler you need to call the
function *note gsl_set_error_handler(): 2f. which is also declared in
‘gsl_errno.h’,

 -- Function: gsl_error_handler_t * gsl_set_error_handler
          (gsl_error_handler_t * new_handler)

     This function sets a new error handler, ‘new_handler’, for the GSL
     library routines.  The previous handler is returned (so that you
     can restore it later).  Note that the pointer to a user defined
     error handler function is stored in a static variable, so there can
     be only one error handler per program.  This function should be not
     be used in multi-threaded programs except to set up a program-wide
     error handler from a master thread.  The following example shows
     how to set and restore a new error handler:

          /* save original handler, install new handler */
          old_handler = gsl_set_error_handler (&my_handler);

          /* code uses new handler */
          .....

          /* restore original handler */
          gsl_set_error_handler (old_handler);

     To use the default behavior (‘abort()’ on error) set the error
     handler to ‘NULL’:

          old_handler = gsl_set_error_handler (NULL);

 -- Function: gsl_error_handler_t * gsl_set_error_handler_off ()

     This function turns off the error handler by defining an error
     handler which does nothing.  This will cause the program to
     continue after any error, so the return values from any library
     routines must be checked.  This is the recommended behavior for
     production programs.  The previous handler is returned (so that you
     can restore it later).

The error behavior can be changed for specific applications by
recompiling the library with a customized definition of the ‘GSL_ERROR’
macro in the file ‘gsl_errno.h’.


File: gsl-ref.info,  Node: Using GSL error reporting in your own functions,  Next: Examples,  Prev: Error Handlers,  Up: Error Handling

3.4 Using GSL error reporting in your own functions
===================================================

If you are writing numerical functions in a program which also uses GSL
code you may find it convenient to adopt the same error reporting
conventions as in the library.

To report an error you need to call the function ‘gsl_error()’ with a
string describing the error and then return an appropriate error code
from ‘gsl_errno.h’, or a special value, such as ‘NaN’.  For convenience
the file ‘gsl_errno.h’ defines two macros which carry out these steps:

 -- Macro: GSL_ERROR (reason, gsl_errno)

     This macro reports an error using the GSL conventions and returns a
     status value of ‘gsl_errno’.  It expands to the following code
     fragment:

          gsl_error (reason, __FILE__, __LINE__, gsl_errno);
          return gsl_errno;

     The macro definition in ‘gsl_errno.h’ actually wraps the code in a
     ‘do { ... } while (0)’ block to prevent possible parsing problems.

Here is an example of how the macro could be used to report that a
routine did not achieve a requested tolerance.  To report the error the
routine needs to return the error code ‘GSL_ETOL’:

     if (residual > tolerance)
       {
         GSL_ERROR("residual exceeds tolerance", GSL_ETOL);
       }

 -- Macro: GSL_ERROR_VAL (reason, gsl_errno, value)

     This macro is the same as ‘GSL_ERROR’ but returns a user-defined
     value of ‘value’ instead of an error code.  It can be used for
     mathematical functions that return a floating point value.

The following example shows how to return a ‘NaN’ at a mathematical
singularity using the ‘GSL_ERROR_VAL’ macro:

     if (x == 0)
       {
         GSL_ERROR_VAL("argument lies on singularity", GSL_ERANGE, GSL_NAN);
       }


File: gsl-ref.info,  Node: Examples,  Prev: Using GSL error reporting in your own functions,  Up: Error Handling

3.5 Examples
============

Here is an example of some code which checks the return value of a
function where an error might be reported:

     #include <stdio.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_fft_complex.h>

     ...
       int status;
       size_t n = 37;

       gsl_set_error_handler_off();

       status = gsl_fft_complex_radix2_forward (data, stride, n);

       if (status) {
         if (status == GSL_EINVAL) {
            fprintf (stderr, "invalid argument, n=%d\n", n);
         } else {
            fprintf (stderr, "failed, gsl_errno=%d\n", status);
         }
         exit (-1);
       }
     ...

The function *note gsl_fft_complex_radix2_forward(): 35. only accepts
integer lengths which are a power of two.  If the variable ‘n’ is not a
power of two then the call to the library function will return
‘GSL_EINVAL’, indicating that the length argument is invalid.  The
function call to *note gsl_set_error_handler_off(): 30. stops the
default error handler from aborting the program.  The ‘else’ clause
catches any other possible errors.


File: gsl-ref.info,  Node: Mathematical Functions,  Next: Complex Numbers,  Prev: Error Handling,  Up: Top

4 Mathematical Functions
************************

This chapter describes basic mathematical functions.  Some of these
functions are present in system libraries, but the alternative versions
given here can be used as a substitute when the system functions are not
available.

The functions and macros described in this chapter are defined in the
header file ‘gsl_math.h’.

* Menu:

* Mathematical Constants::
* Infinities and Not-a-number::
* Elementary Functions::
* Small integer powers::
* Testing the Sign of Numbers::
* Testing for Odd and Even Numbers::
* Maximum and Minimum functions::
* Approximate Comparison of Floating Point Numbers::


File: gsl-ref.info,  Node: Mathematical Constants,  Next: Infinities and Not-a-number,  Up: Mathematical Functions

4.1 Mathematical Constants
==========================

The library ensures that the standard BSD mathematical constants are
defined.  For reference, here is a list of the constants:

‘M_E’                     The base of exponentials, e
                          
                          
‘M_LOG2E’                 The base-2 logarithm of e, \log_2 (e)
                          
                          
‘M_LOG10E’                The base-10 logarithm of e, \log_{10} (e)
                          
                          
‘M_SQRT2’                 The square root of two, \sqrt 2
                          
                          
‘M_SQRT1_2’               The square root of one-half, \sqrt{1/2}
                          
                          
‘M_SQRT3’                 The square root of three, \sqrt 3
                          
                          
‘M_PI’                    The constant pi, \pi
                          
                          
‘M_PI_2’                  Pi divided by two, \pi/2
                          
                          
‘M_PI_4’                  Pi divided by four, \pi/4
                          
                          
‘M_SQRTPI’                The square root of pi, \sqrt\pi
                          
                          
‘M_2_SQRTPI’              Two divided by the square root of pi, 2/\sqrt\pi
                          
                          
‘M_1_PI’                  The reciprocal of pi, 1/\pi
                          
                          
‘M_2_PI’                  Twice the reciprocal of pi, 2/\pi
                          
                          
‘M_LN10’                  The natural logarithm of ten, \ln(10)
                          
                          
‘M_LN2’                   The natural logarithm of two, \ln(2)
                          
                          
‘M_LNPI’                  The natural logarithm of pi, \ln(\pi)
                          
                          
‘M_EULER’                 Euler’s constant, \gamma
                          


File: gsl-ref.info,  Node: Infinities and Not-a-number,  Next: Elementary Functions,  Prev: Mathematical Constants,  Up: Mathematical Functions

4.2 Infinities and Not-a-number
===============================

 -- Macro: GSL_POSINF

     This macro contains the IEEE representation of positive infinity,
     +\infty.  It is computed from the expression ‘+1.0/0.0’.

 -- Macro: GSL_NEGINF

     This macro contains the IEEE representation of negative infinity,
     -\infty.  It is computed from the expression ‘-1.0/0.0’.

 -- Macro: GSL_NAN

     This macro contains the IEEE representation of the Not-a-Number
     symbol, ‘NaN’.  It is computed from the ratio ‘0.0/0.0’.

 -- Function: int gsl_isnan (const double x)

     This function returns 1 if ‘x’ is not-a-number.

 -- Function: int gsl_isinf (const double x)

     This function returns +1 if ‘x’ is positive infinity, -1 if ‘x’ is
     negative infinity and 0 otherwise.  (1)

 -- Function: int gsl_finite (const double x)

     This function returns 1 if ‘x’ is a real number, and 0 if it is
     infinite or not-a-number.

   ---------- Footnotes ----------

   (1) Note that the C99 standard only requires the system ‘isinf()’
function to return a non-zero value, without the sign of the infinity.
The implementation in some earlier versions of GSL used the system
‘isinf()’ function and may have this behavior on some platforms.
Therefore, it is advisable to test the sign of ‘x’ separately, if
needed, rather than relying the sign of the return value from *note
gsl_isinf(): 3e.


File: gsl-ref.info,  Node: Elementary Functions,  Next: Small integer powers,  Prev: Infinities and Not-a-number,  Up: Mathematical Functions

4.3 Elementary Functions
========================

The following routines provide portable implementations of functions
found in the BSD math library.  When native versions are not available
the functions described here can be used instead.  The substitution can
be made automatically if you use ‘autoconf’ to compile your application
(see *note Portability functions: 19.).

 -- Function: double gsl_log1p (const double x)

     This function computes the value of \log(1+x) in a way that is
     accurate for small ‘x’.  It provides an alternative to the BSD math
     function ‘log1p(x)’.

 -- Function: double gsl_expm1 (const double x)

     This function computes the value of \exp(x)-1 in a way that is
     accurate for small ‘x’.  It provides an alternative to the BSD math
     function ‘expm1(x)’.

 -- Function: double gsl_hypot (const double x, const double y)

     This function computes the value of \sqrt{x^2 + y^2} in a way that
     avoids overflow.  It provides an alternative to the BSD math
     function ‘hypot(x,y)’.

 -- Function: double gsl_hypot3 (const double x, const double y, const
          double z)

     This function computes the value of \sqrt{x^2 + y^2 + z^2} in a way
     that avoids overflow.

 -- Function: double gsl_acosh (const double x)

     This function computes the value of \arccosh{(x)}.  It provides an
     alternative to the standard math function ‘acosh(x)’.

 -- Function: double gsl_asinh (const double x)

     This function computes the value of \arcsinh{(x)}.  It provides an
     alternative to the standard math function ‘asinh(x)’.

 -- Function: double gsl_atanh (const double x)

     This function computes the value of \arctanh{(x)}.  It provides an
     alternative to the standard math function ‘atanh(x)’.

 -- Function: double gsl_ldexp (double x, int e)

     This function computes the value of x * 2^e.  It provides an
     alternative to the standard math function ‘ldexp(x,e)’.

 -- Function: double gsl_frexp (double x, int * e)

     This function splits the number ‘x’ into its normalized fraction f
     and exponent e, such that x = f * 2^e and 0.5 <= f < 1.  The
     function returns f and stores the exponent in e.  If x is zero,
     both f and e are set to zero.  This function provides an
     alternative to the standard math function ‘frexp(x, e)’.


File: gsl-ref.info,  Node: Small integer powers,  Next: Testing the Sign of Numbers,  Prev: Elementary Functions,  Up: Mathematical Functions

4.4 Small integer powers
========================

A common complaint about the standard C library is its lack of a
function for calculating (small) integer powers.  GSL provides some
simple functions to fill this gap.  For reasons of efficiency, these
functions do not check for overflow or underflow conditions.

 -- Function: double gsl_pow_int (double x, int n)
 -- Function: double gsl_pow_uint (double x, unsigned int n)

     These routines computes the power x^n for integer ‘n’.  The power
     is computed efficiently—for example, x^8 is computed as
     ((x^2)^2)^2, requiring only 3 multiplications.  A version of this
     function which also computes the numerical error in the result is
     available as *note gsl_sf_pow_int_e(): 4c.

 -- Function: double gsl_pow_2 (const double x)
 -- Function: double gsl_pow_3 (const double x)
 -- Function: double gsl_pow_4 (const double x)
 -- Function: double gsl_pow_5 (const double x)
 -- Function: double gsl_pow_6 (const double x)
 -- Function: double gsl_pow_7 (const double x)
 -- Function: double gsl_pow_8 (const double x)
 -- Function: double gsl_pow_9 (const double x)

     These functions can be used to compute small integer powers x^2,
     x^3, etc.  efficiently.  The functions will be inlined when
     ‘HAVE_INLINE’ is defined, so that use of these functions should be
     as efficient as explicitly writing the corresponding product
     expression:

          #include <gsl/gsl_math.h>
          double y = gsl_pow_4 (3.141)  /* compute 3.141**4 */


File: gsl-ref.info,  Node: Testing the Sign of Numbers,  Next: Testing for Odd and Even Numbers,  Prev: Small integer powers,  Up: Mathematical Functions

4.5 Testing the Sign of Numbers
===============================

 -- Macro: GSL_SIGN (x)

     This macro returns the sign of ‘x’.  It is defined as ‘((x) >= 0 ?
     1 : -1)’.  Note that with this definition the sign of zero is
     positive (regardless of its IEEE sign bit).


File: gsl-ref.info,  Node: Testing for Odd and Even Numbers,  Next: Maximum and Minimum functions,  Prev: Testing the Sign of Numbers,  Up: Mathematical Functions

4.6 Testing for Odd and Even Numbers
====================================

 -- Macro: GSL_IS_ODD (n)

     This macro evaluates to 1 if ‘n’ is odd and 0 if ‘n’ is even.  The
     argument ‘n’ must be of integer type.

 -- Macro: GSL_IS_EVEN (n)

     This macro is the opposite of *note GSL_IS_ODD: 58.  It evaluates
     to 1 if ‘n’ is even and 0 if ‘n’ is odd.  The argument ‘n’ must be
     of integer type.


File: gsl-ref.info,  Node: Maximum and Minimum functions,  Next: Approximate Comparison of Floating Point Numbers,  Prev: Testing for Odd and Even Numbers,  Up: Mathematical Functions

4.7 Maximum and Minimum functions
=================================

Note that the following macros perform multiple evaluations of their
arguments, so they should not be used with arguments that have side
effects (such as a call to a random number generator).

 -- Macro: GSL_MAX (a, b)

     This macro returns the maximum of ‘a’ and ‘b’.  It is defined as
     ‘((a) > (b) ? (a):(b))’.

 -- Macro: GSL_MIN (a, b)

     This macro returns the minimum of ‘a’ and ‘b’.  It is defined as
     ‘((a) < (b) ? (a):(b))’.

 -- Function: extern inline double GSL_MAX_DBL (double a, double b)

     This function returns the maximum of the double precision numbers
     ‘a’ and ‘b’ using an inline function.  The use of a function allows
     for type checking of the arguments as an extra safety feature.  On
     platforms where inline functions are not available the macro *note
     GSL_MAX: 5b. will be automatically substituted.

 -- Function: extern inline double GSL_MIN_DBL (double a, double b)

     This function returns the minimum of the double precision numbers
     ‘a’ and ‘b’ using an inline function.  The use of a function allows
     for type checking of the arguments as an extra safety feature.  On
     platforms where inline functions are not available the macro *note
     GSL_MIN: 5c. will be automatically substituted.

 -- Function: extern inline int GSL_MAX_INT (int a, int b)
 -- Function: extern inline int GSL_MIN_INT (int a, int b)

     These functions return the maximum or minimum of the integers ‘a’
     and ‘b’ using an inline function.  On platforms where inline
     functions are not available the macros *note GSL_MAX: 5b. or *note
     GSL_MIN: 5c. will be automatically substituted.

 -- Function: extern inline long double GSL_MAX_LDBL (long double a,
          long double b)
 -- Function: extern inline long double GSL_MIN_LDBL (long double a,
          long double b)

     These functions return the maximum or minimum of the long doubles
     ‘a’ and ‘b’ using an inline function.  On platforms where inline
     functions are not available the macros *note GSL_MAX: 5b. or *note
     GSL_MIN: 5c. will be automatically substituted.


File: gsl-ref.info,  Node: Approximate Comparison of Floating Point Numbers,  Prev: Maximum and Minimum functions,  Up: Mathematical Functions

4.8 Approximate Comparison of Floating Point Numbers
====================================================

It is sometimes useful to be able to compare two floating point numbers
approximately, to allow for rounding and truncation errors.  The
following function implements the approximate floating-point comparison
algorithm proposed by D.E. Knuth in Section 4.2.2 of “Seminumerical
Algorithms” (3rd edition).

 -- Function: int gsl_fcmp (double x, double y, double epsilon)

     This function determines whether ‘x’ and ‘y’ are approximately
     equal to a relative accuracy ‘epsilon’.

     The relative accuracy is measured using an interval of size 2
     \delta, where \delta = 2^k \epsilon and k is the maximum base-2
     exponent of x and y as computed by the function ‘frexp()’.

     If x and y lie within this interval, they are considered
     approximately equal and the function returns 0.  Otherwise if x <
     y, the function returns -1, or if x > y, the function returns +1.

     Note that x and y are compared to relative accuracy, so this
     function is not suitable for testing whether a value is
     approximately zero.

     The implementation is based on the package ‘fcmp’ by T.C. Belding.


File: gsl-ref.info,  Node: Complex Numbers,  Next: Polynomials,  Prev: Mathematical Functions,  Up: Top

5 Complex Numbers
*****************

The functions described in this chapter provide support for complex
numbers.  The algorithms take care to avoid unnecessary intermediate
underflows and overflows, allowing the functions to be evaluated over as
much of the complex plane as possible.

For multiple-valued functions the branch cuts have been chosen to follow
the conventions of Abramowitz and Stegun.  The functions return
principal values which are the same as those in GNU Calc, which in turn
are the same as those in “Common Lisp, The Language (Second Edition)”
(1) and the HP-28/48 series of calculators.

The complex types are defined in the header file ‘gsl_complex.h’, while
the corresponding complex functions and arithmetic operations are
defined in ‘gsl_complex_math.h’.

* Menu:

* Representation of complex numbers::
* Properties of complex numbers::
* Complex arithmetic operators::
* Elementary Complex Functions::
* Complex Trigonometric Functions::
* Inverse Complex Trigonometric Functions::
* Complex Hyperbolic Functions::
* Inverse Complex Hyperbolic Functions::
* References and Further Reading::

   ---------- Footnotes ----------

   (1) Note that the first edition uses different definitions.


File: gsl-ref.info,  Node: Representation of complex numbers,  Next: Properties of complex numbers,  Up: Complex Numbers

5.1 Representation of complex numbers
=====================================

Complex numbers are represented using the type ‘gsl_complex’.  The
internal representation of this type may vary across platforms and
should not be accessed directly.  The functions and macros described
below allow complex numbers to be manipulated in a portable way.

For reference, the default form of the ‘gsl_complex’ type is given by
the following struct:

     typedef struct
     {
       double dat[2];
     } gsl_complex;

The real and imaginary part are stored in contiguous elements of a two
element array.  This eliminates any padding between the real and
imaginary parts, ‘dat[0]’ and ‘dat[1]’, allowing the struct to be mapped
correctly onto packed complex arrays.

 -- Function: gsl_complex gsl_complex_rect (double x, double y)

     This function uses the rectangular Cartesian components (x,y) to
     return the complex number z = x + i y.  An inline version of this
     function is used when ‘HAVE_INLINE’ is defined.

 -- Function: gsl_complex gsl_complex_polar (double r, double theta)

     This function returns the complex number z = r \exp(i \theta) = r
     (\cos(\theta) + i \sin(\theta)) from the polar representation (‘r’,
     ‘theta’).

 -- Macro: GSL_REAL (z)
 -- Macro: GSL_IMAG (z)

     These macros return the real and imaginary parts of the complex
     number ‘z’.

 -- Macro: GSL_SET_COMPLEX (zp, x, y)

     This macro uses the Cartesian components (‘x’, ‘y’) to set the real
     and imaginary parts of the complex number pointed to by ‘zp’.  For
     example:

          GSL_SET_COMPLEX(&z, 3, 4)

     sets z to be 3 + 4i.

 -- Macro: GSL_SET_REAL (zp, x)
 -- Macro: GSL_SET_IMAG (zp, y)

     These macros allow the real and imaginary parts of the complex
     number pointed to by ‘zp’ to be set independently.


File: gsl-ref.info,  Node: Properties of complex numbers,  Next: Complex arithmetic operators,  Prev: Representation of complex numbers,  Up: Complex Numbers

5.2 Properties of complex numbers
=================================

 -- Function: double gsl_complex_arg (gsl_complex z)

     This function returns the argument of the complex number ‘z’,
     \arg(z), where -\pi < \arg(z) <= \pi.

 -- Function: double gsl_complex_abs (gsl_complex z)

     This function returns the magnitude of the complex number ‘z’, |z|.

 -- Function: double gsl_complex_abs2 (gsl_complex z)

     This function returns the squared magnitude of the complex number
     ‘z’, |z|^2.

 -- Function: double gsl_complex_logabs (gsl_complex z)

     This function returns the natural logarithm of the magnitude of the
     complex number ‘z’, \log|z|.  It allows an accurate evaluation of
     \log|z| when |z| is close to one.  The direct evaluation of
     ‘log(gsl_complex_abs(z))’ would lead to a loss of precision in this
     case.


File: gsl-ref.info,  Node: Complex arithmetic operators,  Next: Elementary Complex Functions,  Prev: Properties of complex numbers,  Up: Complex Numbers

5.3 Complex arithmetic operators
================================

 -- Function: gsl_complex gsl_complex_add (gsl_complex a, gsl_complex b)

     This function returns the sum of the complex numbers ‘a’ and ‘b’,
     z=a+b.

 -- Function: gsl_complex gsl_complex_sub (gsl_complex a, gsl_complex b)

     This function returns the difference of the complex numbers ‘a’ and
     ‘b’, z=a-b.

 -- Function: gsl_complex gsl_complex_mul (gsl_complex a, gsl_complex b)

     This function returns the product of the complex numbers ‘a’ and
     ‘b’, z=ab.

 -- Function: gsl_complex gsl_complex_div (gsl_complex a, gsl_complex b)

     This function returns the quotient of the complex numbers ‘a’ and
     ‘b’, z=a/b.

 -- Function: gsl_complex gsl_complex_add_real (gsl_complex a, double x)

     This function returns the sum of the complex number ‘a’ and the
     real number ‘x’, z=a+x.

 -- Function: gsl_complex gsl_complex_sub_real (gsl_complex a, double x)

     This function returns the difference of the complex number ‘a’ and
     the real number ‘x’, z=a-x.

 -- Function: gsl_complex gsl_complex_mul_real (gsl_complex a, double x)

     This function returns the product of the complex number ‘a’ and the
     real number ‘x’, z=ax.

 -- Function: gsl_complex gsl_complex_div_real (gsl_complex a, double x)

     This function returns the quotient of the complex number ‘a’ and
     the real number ‘x’, z=a/x.

 -- Function: gsl_complex gsl_complex_add_imag (gsl_complex a, double y)

     This function returns the sum of the complex number ‘a’ and the
     imaginary number iy, z=a+iy.

 -- Function: gsl_complex gsl_complex_sub_imag (gsl_complex a, double y)

     This function returns the difference of the complex number ‘a’ and
     the imaginary number iy, z=a-iy.

 -- Function: gsl_complex gsl_complex_mul_imag (gsl_complex a, double y)

     This function returns the product of the complex number ‘a’ and the
     imaginary number iy, z=a*(iy).

 -- Function: gsl_complex gsl_complex_div_imag (gsl_complex a, double y)

     This function returns the quotient of the complex number ‘a’ and
     the imaginary number iy, z=a/(iy).

 -- Function: gsl_complex gsl_complex_conjugate (gsl_complex z)

     This function returns the complex conjugate of the complex number
     ‘z’, z^* = x - i y.

 -- Function: gsl_complex gsl_complex_inverse (gsl_complex z)

     This function returns the inverse, or reciprocal, of the complex
     number ‘z’, 1/z = (x - i y)/(x^2 + y^2).

 -- Function: gsl_complex gsl_complex_negative (gsl_complex z)

     This function returns the negative of the complex number ‘z’, -z =
     (-x) + i(-y).


File: gsl-ref.info,  Node: Elementary Complex Functions,  Next: Complex Trigonometric Functions,  Prev: Complex arithmetic operators,  Up: Complex Numbers

5.4 Elementary Complex Functions
================================

 -- Function: gsl_complex gsl_complex_sqrt (gsl_complex z)

     This function returns the square root of the complex number ‘z’,
     \sqrt z.  The branch cut is the negative real axis.  The result
     always lies in the right half of the complex plane.

 -- Function: gsl_complex gsl_complex_sqrt_real (double x)

     This function returns the complex square root of the real number
     ‘x’, where ‘x’ may be negative.

 -- Function: gsl_complex gsl_complex_pow (gsl_complex z, gsl_complex a)

     The function returns the complex number ‘z’ raised to the complex
     power ‘a’, z^a.  This is computed as \exp(\log(z)*a) using complex
     logarithms and complex exponentials.

 -- Function: gsl_complex gsl_complex_pow_real (gsl_complex z, double x)

     This function returns the complex number ‘z’ raised to the real
     power ‘x’, z^x.

 -- Function: gsl_complex gsl_complex_exp (gsl_complex z)

     This function returns the complex exponential of the complex number
     ‘z’, \exp(z).

 -- Function: gsl_complex gsl_complex_log (gsl_complex z)

     This function returns the complex natural logarithm (base e) of the
     complex number ‘z’, \log(z).  The branch cut is the negative real
     axis.

 -- Function: gsl_complex gsl_complex_log10 (gsl_complex z)

     This function returns the complex base-10 logarithm of the complex
     number ‘z’, \log_{10} (z).

 -- Function: gsl_complex gsl_complex_log_b (gsl_complex z,
          gsl_complex b)

     This function returns the complex base-‘b’ logarithm of the complex
     number ‘z’, \log_b(z).  This quantity is computed as the ratio
     \log(z)/\log(b).


File: gsl-ref.info,  Node: Complex Trigonometric Functions,  Next: Inverse Complex Trigonometric Functions,  Prev: Elementary Complex Functions,  Up: Complex Numbers

5.5 Complex Trigonometric Functions
===================================

 -- Function: gsl_complex gsl_complex_sin (gsl_complex z)

     This function returns the complex sine of the complex number ‘z’,
     \sin(z) = (\exp(iz) - \exp(-iz))/(2i).

 -- Function: gsl_complex gsl_complex_cos (gsl_complex z)

     This function returns the complex cosine of the complex number ‘z’,
     \cos(z) = (\exp(iz) + \exp(-iz))/2.

 -- Function: gsl_complex gsl_complex_tan (gsl_complex z)

     This function returns the complex tangent of the complex number
     ‘z’, \tan(z) = \sin(z)/\cos(z).

 -- Function: gsl_complex gsl_complex_sec (gsl_complex z)

     This function returns the complex secant of the complex number ‘z’,
     \sec(z) = 1/\cos(z).

 -- Function: gsl_complex gsl_complex_csc (gsl_complex z)

     This function returns the complex cosecant of the complex number
     ‘z’, \csc(z) = 1/\sin(z).

 -- Function: gsl_complex gsl_complex_cot (gsl_complex z)

     This function returns the complex cotangent of the complex number
     ‘z’, \cot(z) = 1/\tan(z).


File: gsl-ref.info,  Node: Inverse Complex Trigonometric Functions,  Next: Complex Hyperbolic Functions,  Prev: Complex Trigonometric Functions,  Up: Complex Numbers

5.6 Inverse Complex Trigonometric Functions
===========================================

 -- Function: gsl_complex gsl_complex_arcsin (gsl_complex z)

     This function returns the complex arcsine of the complex number
     ‘z’, \arcsin(z).  The branch cuts are on the real axis, less than
     -1 and greater than 1.

 -- Function: gsl_complex gsl_complex_arcsin_real (double z)

     This function returns the complex arcsine of the real number ‘z’,
     \arcsin(z).  For z between -1 and 1, the function returns a real
     value in the range [-\pi/2,\pi/2].  For z less than -1 the result
     has a real part of -\pi/2 and a positive imaginary part.  For z
     greater than 1 the result has a real part of \pi/2 and a negative
     imaginary part.

 -- Function: gsl_complex gsl_complex_arccos (gsl_complex z)

     This function returns the complex arccosine of the complex number
     ‘z’, \arccos(z).  The branch cuts are on the real axis, less than
     -1 and greater than 1.

 -- Function: gsl_complex gsl_complex_arccos_real (double z)

     This function returns the complex arccosine of the real number ‘z’,
     \arccos(z).  For z between -1 and 1, the function returns a real
     value in the range [0,\pi].  For z less than -1 the result has a
     real part of \pi and a negative imaginary part.  For z greater than
     1 the result is purely imaginary and positive.

 -- Function: gsl_complex gsl_complex_arctan (gsl_complex z)

     This function returns the complex arctangent of the complex number
     ‘z’, \arctan(z).  The branch cuts are on the imaginary axis, below
     -i and above i.

 -- Function: gsl_complex gsl_complex_arcsec (gsl_complex z)

     This function returns the complex arcsecant of the complex number
     ‘z’, \arcsec(z) = \arccos(1/z).

 -- Function: gsl_complex gsl_complex_arcsec_real (double z)

     This function returns the complex arcsecant of the real number ‘z’,
     \arcsec(z) = \arccos(1/z).

 -- Function: gsl_complex gsl_complex_arccsc (gsl_complex z)

     This function returns the complex arccosecant of the complex number
     ‘z’, \arccsc(z) = \arcsin(1/z).

 -- Function: gsl_complex gsl_complex_arccsc_real (double z)

     This function returns the complex arccosecant of the real number
     ‘z’, \arccsc(z) = \arcsin(1/z).

 -- Function: gsl_complex gsl_complex_arccot (gsl_complex z)

     This function returns the complex arccotangent of the complex
     number ‘z’, \arccot(z) = \arctan(1/z).


File: gsl-ref.info,  Node: Complex Hyperbolic Functions,  Next: Inverse Complex Hyperbolic Functions,  Prev: Inverse Complex Trigonometric Functions,  Up: Complex Numbers

5.7 Complex Hyperbolic Functions
================================

 -- Function: gsl_complex gsl_complex_sinh (gsl_complex z)

     This function returns the complex hyperbolic sine of the complex
     number ‘z’, \sinh(z) = (\exp(z) - \exp(-z))/2.

 -- Function: gsl_complex gsl_complex_cosh (gsl_complex z)

     This function returns the complex hyperbolic cosine of the complex
     number ‘z’, \cosh(z) = (\exp(z) + \exp(-z))/2.

 -- Function: gsl_complex gsl_complex_tanh (gsl_complex z)

     This function returns the complex hyperbolic tangent of the complex
     number ‘z’, \tanh(z) = \sinh(z)/\cosh(z).

 -- Function: gsl_complex gsl_complex_sech (gsl_complex z)

     This function returns the complex hyperbolic secant of the complex
     number ‘z’, \sech(z) = 1/\cosh(z).

 -- Function: gsl_complex gsl_complex_csch (gsl_complex z)

     This function returns the complex hyperbolic cosecant of the
     complex number ‘z’, \csch(z) = 1/\sinh(z).

 -- Function: gsl_complex gsl_complex_coth (gsl_complex z)

     This function returns the complex hyperbolic cotangent of the
     complex number ‘z’, \coth(z) = 1/\tanh(z).


File: gsl-ref.info,  Node: Inverse Complex Hyperbolic Functions,  Next: References and Further Reading,  Prev: Complex Hyperbolic Functions,  Up: Complex Numbers

5.8 Inverse Complex Hyperbolic Functions
========================================

 -- Function: gsl_complex gsl_complex_arcsinh (gsl_complex z)

     This function returns the complex hyperbolic arcsine of the complex
     number ‘z’, \arcsinh(z).  The branch cuts are on the imaginary
     axis, below -i and above i.

 -- Function: gsl_complex gsl_complex_arccosh (gsl_complex z)

     This function returns the complex hyperbolic arccosine of the
     complex number ‘z’, \arccosh(z).  The branch cut is on the real
     axis, less than 1.  Note that in this case we use the negative
     square root in formula 4.6.21 of Abramowitz & Stegun giving
     \arccosh(z)=\log(z-\sqrt{z^2-1}).

 -- Function: gsl_complex gsl_complex_arccosh_real (double z)

     This function returns the complex hyperbolic arccosine of the real
     number ‘z’, \arccosh(z).

 -- Function: gsl_complex gsl_complex_arctanh (gsl_complex z)

     This function returns the complex hyperbolic arctangent of the
     complex number ‘z’, \arctanh(z).  The branch cuts are on the real
     axis, less than -1 and greater than 1.

 -- Function: gsl_complex gsl_complex_arctanh_real (double z)

     This function returns the complex hyperbolic arctangent of the real
     number ‘z’, \arctanh(z).

 -- Function: gsl_complex gsl_complex_arcsech (gsl_complex z)

     This function returns the complex hyperbolic arcsecant of the
     complex number ‘z’, \arcsech(z) = \arccosh(1/z).

 -- Function: gsl_complex gsl_complex_arccsch (gsl_complex z)

     This function returns the complex hyperbolic arccosecant of the
     complex number ‘z’, \arccsch(z) = \arcsinh(1/z).

 -- Function: gsl_complex gsl_complex_arccoth (gsl_complex z)

     This function returns the complex hyperbolic arccotangent of the
     complex number ‘z’, \arccoth(z) = \arctanh(1/z).


File: gsl-ref.info,  Node: References and Further Reading,  Prev: Inverse Complex Hyperbolic Functions,  Up: Complex Numbers

5.9 References and Further Reading
==================================

The implementations of the elementary and trigonometric functions are
based on the following papers,

   * T. E. Hull, Thomas F. Fairgrieve, Ping Tak Peter Tang,
     “Implementing Complex Elementary Functions Using Exception
     Handling”, ACM Transactions on Mathematical Software, Volume 20
     (1994), pp 215–244, Corrigenda, p553

   * T. E. Hull, Thomas F. Fairgrieve, Ping Tak Peter Tang,
     “Implementing the complex arcsin and arccosine functions using
     exception handling”, ACM Transactions on Mathematical Software,
     Volume 23 (1997) pp 299–335

The general formulas and details of branch cuts can be found in the
following books,

   * Abramowitz and Stegun, Handbook of Mathematical Functions,
     “Circular Functions in Terms of Real and Imaginary Parts”, Formulas
     4.3.55–58, “Inverse Circular Functions in Terms of Real and
     Imaginary Parts”, Formulas 4.4.37–39, “Hyperbolic Functions in
     Terms of Real and Imaginary Parts”, Formulas 4.5.49–52, “Inverse
     Hyperbolic Functions—relation to Inverse Circular Functions”,
     Formulas 4.6.14–19.

   * Dave Gillespie, Calc Manual, Free Software Foundation, ISBN
     1-882114-18-3


File: gsl-ref.info,  Node: Polynomials,  Next: Special Functions,  Prev: Complex Numbers,  Up: Top

6 Polynomials
*************

This chapter describes functions for evaluating and solving polynomials.
There are routines for finding real and complex roots of quadratic and
cubic equations using analytic methods.  An iterative polynomial solver
is also available for finding the roots of general polynomials with real
coefficients (of any order).  The functions are declared in the header
file ‘gsl_poly.h’.

* Menu:

* Polynomial Evaluation::
* Divided Difference Representation of Polynomials::
* Quadratic Equations::
* Cubic Equations::
* General Polynomial Equations::
* Examples: Examples<2>.
* References and Further Reading: References and Further Reading<2>.


File: gsl-ref.info,  Node: Polynomial Evaluation,  Next: Divided Difference Representation of Polynomials,  Up: Polynomials

6.1 Polynomial Evaluation
=========================

The functions described here evaluate the polynomial

P(x) = c[0] + c[1] x + c[2] x^2 + … + c[len-1] x^{len-1}

using Horner’s method for stability.  Inline versions of these functions
are used when ‘HAVE_INLINE’ is defined.

 -- Function: double gsl_poly_eval (const double c[], const int len,
          const double x)

     This function evaluates a polynomial with real coefficients for the
     real variable ‘x’.

 -- Function: gsl_complex gsl_poly_complex_eval (const double c[], const
          int len, const gsl_complex z)

     This function evaluates a polynomial with real coefficients for the
     complex variable ‘z’.

 -- Function: gsl_complex gsl_complex_poly_complex_eval (const
          gsl_complex c[], const int len, const gsl_complex z)

     This function evaluates a polynomial with complex coefficients for
     the complex variable ‘z’.

 -- Function: int gsl_poly_eval_derivs (const double c[], const
          size_t lenc, const double x, double res[], const
          size_t lenres)

     This function evaluates a polynomial and its derivatives storing
     the results in the array ‘res’ of size ‘lenres’.  The output array
     contains the values of d^k P(x)/d x^k for the specified value of
     ‘x’ starting with k = 0.


File: gsl-ref.info,  Node: Divided Difference Representation of Polynomials,  Next: Quadratic Equations,  Prev: Polynomial Evaluation,  Up: Polynomials

6.2 Divided Difference Representation of Polynomials
====================================================

The functions described here manipulate polynomials stored in Newton’s
divided-difference representation.  The use of divided-differences is
described in Abramowitz & Stegun sections 25.1.4 and 25.2.26, and Burden
and Faires, chapter 3, and discussed briefly below.

Given a function f(x), an nth degree interpolating polynomial P_{n}(x)
can be constructed which agrees with f at n+1 distinct points
x_0,x_1,...,x_{n}.  This polynomial can be written in a form known as
Newton’s divided-difference representation

P_{n}(x) = f(x_0) + sum_{k=1}^n [x_0,x_1,…,x_k] (x-x_0)(x-x_1) …
(x-x_{k-1})

where the divided differences [x_0,x_1,...,x_k] are defined in section
25.1.4 of Abramowitz and Stegun.  Additionally, it is possible to
construct an interpolating polynomial of degree 2n+1 which also matches
the first derivatives of f at the points x_0,x_1,...,x_n.  This is
called the Hermite interpolating polynomial and is defined as

H_{2n+1}(x) = f(z_0) + sum_{k=1}^{2n+1} [z_0,z_1,…,z_k] (x-z_0)(x-z_1) …
(x-z_{k-1})

where the elements of z = \{x_0,x_0,x_1,x_1,...,x_n,x_n\} are defined by
z_{2k} = z_{2k+1} = x_k.  The divided-differences [z_0,z_1,...,z_k] are
discussed in Burden and Faires, section 3.4.

 -- Function: int gsl_poly_dd_init (double dd[], const double xa[],
          const double ya[], size_t size)

     This function computes a divided-difference representation of the
     interpolating polynomial for the points (x, y) stored in the arrays
     ‘xa’ and ‘ya’ of length ‘size’.  On output the divided-differences
     of (‘xa’, ‘ya’) are stored in the array ‘dd’, also of length
     ‘size’.  Using the notation above, dd[k] = [x_0,x_1,...,x_k].

 -- Function: double gsl_poly_dd_eval (const double dd[], const
          double xa[], const size_t size, const double x)

     This function evaluates the polynomial stored in divided-difference
     form in the arrays ‘dd’ and ‘xa’ of length ‘size’ at the point ‘x’.
     An inline version of this function is used when ‘HAVE_INLINE’ is
     defined.

 -- Function: int gsl_poly_dd_taylor (double c[], double xp, const
          double dd[], const double xa[], size_t size, double w[])

     This function converts the divided-difference representation of a
     polynomial to a Taylor expansion.  The divided-difference
     representation is supplied in the arrays ‘dd’ and ‘xa’ of length
     ‘size’.  On output the Taylor coefficients of the polynomial
     expanded about the point ‘xp’ are stored in the array ‘c’ also of
     length ‘size’.  A workspace of length ‘size’ must be provided in
     the array ‘w’.

 -- Function: int gsl_poly_dd_hermite_init (double dd[], double za[],
          const double xa[], const double ya[], const double dya[],
          const size_t size)

     This function computes a divided-difference representation of the
     interpolating Hermite polynomial for the points (x,y) stored in the
     arrays ‘xa’ and ‘ya’ of length ‘size’.  Hermite interpolation
     constructs polynomials which also match first derivatives dy/dx
     which are provided in the array ‘dya’ also of length ‘size’.  The
     first derivatives can be incorported into the usual
     divided-difference algorithm by forming a new dataset z =
     \{x_0,x_0,x_1,x_1,...\}, which is stored in the array ‘za’ of
     length 2*‘size’ on output.  On output the divided-differences of
     the Hermite representation are stored in the array ‘dd’, also of
     length 2*‘size’.  Using the notation above, dd[k] =
     [z_0,z_1,...,z_k].  The resulting Hermite polynomial can be
     evaluated by calling *note gsl_poly_dd_eval(): b9. and using ‘za’
     for the input argument ‘xa’.


File: gsl-ref.info,  Node: Quadratic Equations,  Next: Cubic Equations,  Prev: Divided Difference Representation of Polynomials,  Up: Polynomials

6.3 Quadratic Equations
=======================

 -- Function: int gsl_poly_solve_quadratic (double a, double b,
          double c, double * x0, double * x1)

     This function finds the real roots of the quadratic equation,

          a x^2 + b x + c = 0

     The number of real roots (either zero, one or two) is returned, and
     their locations are stored in ‘x0’ and ‘x1’.  If no real roots are
     found then ‘x0’ and ‘x1’ are not modified.  If one real root is
     found (i.e.  if a=0) then it is stored in ‘x0’.  When two real
     roots are found they are stored in ‘x0’ and ‘x1’ in ascending
     order.  The case of coincident roots is not considered special.
     For example (x-1)^2=0 will have two roots, which happen to have
     exactly equal values.

     The number of roots found depends on the sign of the discriminant
     b^2 - 4 a c.  This will be subject to rounding and cancellation
     errors when computed in double precision, and will also be subject
     to errors if the coefficients of the polynomial are inexact.  These
     errors may cause a discrete change in the number of roots.
     However, for polynomials with small integer coefficients the
     discriminant can always be computed exactly.

 -- Function: int gsl_poly_complex_solve_quadratic (double a, double b,
          double c, gsl_complex * z0, gsl_complex * z1)

     This function finds the complex roots of the quadratic equation,

          a z^2 + b z + c = 0

     The number of complex roots is returned (either one or two) and the
     locations of the roots are stored in ‘z0’ and ‘z1’.  The roots are
     returned in ascending order, sorted first by their real components
     and then by their imaginary components.  If only one real root is
     found (i.e.  if a=0) then it is stored in ‘z0’.


File: gsl-ref.info,  Node: Cubic Equations,  Next: General Polynomial Equations,  Prev: Quadratic Equations,  Up: Polynomials

6.4 Cubic Equations
===================

 -- Function: int gsl_poly_solve_cubic (double a, double b, double c,
          double * x0, double * x1, double * x2)

     This function finds the real roots of the cubic equation,

          x^3 + a x^2 + b x + c = 0

     with a leading coefficient of unity.  The number of real roots
     (either one or three) is returned, and their locations are stored
     in ‘x0’, ‘x1’ and ‘x2’.  If one real root is found then only ‘x0’
     is modified.  When three real roots are found they are stored in
     ‘x0’, ‘x1’ and ‘x2’ in ascending order.  The case of coincident
     roots is not considered special.  For example, the equation
     (x-1)^3=0 will have three roots with exactly equal values.  As in
     the quadratic case, finite precision may cause equal or
     closely-spaced real roots to move off the real axis into the
     complex plane, leading to a discrete change in the number of real
     roots.

 -- Function: int gsl_poly_complex_solve_cubic (double a, double b,
          double c, gsl_complex * z0, gsl_complex * z1, gsl_complex
          * z2)

     This function finds the complex roots of the cubic equation,

          z^3 + a z^2 + b z + c = 0

     The number of complex roots is returned (always three) and the
     locations of the roots are stored in ‘z0’, ‘z1’ and ‘z2’.  The
     roots are returned in ascending order, sorted first by their real
     components and then by their imaginary components.


File: gsl-ref.info,  Node: General Polynomial Equations,  Next: Examples<2>,  Prev: Cubic Equations,  Up: Polynomials

6.5 General Polynomial Equations
================================

The roots of polynomial equations cannot be found analytically beyond
the special cases of the quadratic, cubic and quartic equation.  The
algorithm described in this section uses an iterative method to find the
approximate locations of roots of higher order polynomials.

 -- Type: gsl_poly_complex_workspace

     This workspace contains parameters used for finding roots of
     general polynomials

 -- Function: gsl_poly_complex_workspace *
          gsl_poly_complex_workspace_alloc (size_t n)

     This function allocates space for a *note
     gsl_poly_complex_workspace: c3. struct and a workspace suitable for
     solving a polynomial with ‘n’ coefficients using the routine *note
     gsl_poly_complex_solve(): c5.

     The function returns a pointer to the newly allocated *note
     gsl_poly_complex_workspace: c3. if no errors were detected, and a
     null pointer in the case of error.

 -- Function: void gsl_poly_complex_workspace_free
          (gsl_poly_complex_workspace * w)

     This function frees all the memory associated with the workspace
     ‘w’.

 -- Function: int gsl_poly_complex_solve (const double * a, size_t n,
          gsl_poly_complex_workspace * w, gsl_complex_packed_ptr z)

     This function computes the roots of the general polynomial

     P(x) = a_0 + a_1 x + a_2 x^2 + … + a_{n-1} x^{n-1}

     using balanced-QR reduction of the companion matrix.  The parameter
     ‘n’ specifies the length of the coefficient array.  The coefficient
     of the highest order term must be non-zero.  The function requires
     a workspace ‘w’ of the appropriate size.  The n-1 roots are
     returned in the packed complex array ‘z’ of length 2(n-1),
     alternating real and imaginary parts.

     The function returns ‘GSL_SUCCESS’ if all the roots are found.  If
     the QR reduction does not converge, the error handler is invoked
     with an error code of ‘GSL_EFAILED’.  Note that due to finite
     precision, roots of higher multiplicity are returned as a cluster
     of simple roots with reduced accuracy.  The solution of polynomials
     with higher-order roots requires specialized algorithms that take
     the multiplicity structure into account (see e.g.  Z. Zeng,
     Algorithm 835, ACM Transactions on Mathematical Software, Volume
     30, Issue 2 (2004), pp 218–236).


File: gsl-ref.info,  Node: Examples<2>,  Next: References and Further Reading<2>,  Prev: General Polynomial Equations,  Up: Polynomials

6.6 Examples
============

To demonstrate the use of the general polynomial solver we will take the
polynomial P(x) = x^5 - 1 which has these roots:

1, e^{2*pi i / 5}, e^{4*pi i / 5}, e^{6*pi i / 5}, e^{8*pi i / 5}

The following program will find these roots.

     #include <stdio.h>
     #include <gsl/gsl_poly.h>

     int
     main (void)
     {
       int i;
       /* coefficients of P(x) =  -1 + x^5  */
       double a[6] = { -1, 0, 0, 0, 0, 1 };
       double z[10];

       gsl_poly_complex_workspace * w
           = gsl_poly_complex_workspace_alloc (6);

       gsl_poly_complex_solve (a, 6, w, z);

       gsl_poly_complex_workspace_free (w);

       for (i = 0; i < 5; i++)
         {
           printf ("z%d = %+.18f %+.18f\n",
                   i, z[2*i], z[2*i+1]);
         }

       return 0;
     }

The output of the program is

     z0 = -0.809016994374947673 +0.587785252292473359
     z1 = -0.809016994374947673 -0.587785252292473359
     z2 = +0.309016994374947507 +0.951056516295152976
     z3 = +0.309016994374947507 -0.951056516295152976
     z4 = +0.999999999999999889 +0.000000000000000000

which agrees with the analytic result, z_n = \exp(2 \pi n i/5).


File: gsl-ref.info,  Node: References and Further Reading<2>,  Prev: Examples<2>,  Up: Polynomials

6.7 References and Further Reading
==================================

The balanced-QR method and its error analysis are described in the
following papers,

   * R.S. Martin, G. Peters and J.H. Wilkinson, “The QR Algorithm for
     Real Hessenberg Matrices”, Numerische Mathematik, 14 (1970),
     219–231.

   * B.N. Parlett and C. Reinsch, “Balancing a Matrix for Calculation of
     Eigenvalues and Eigenvectors”, Numerische Mathematik, 13 (1969),
     293–304.

   * A. Edelman and H. Murakami, “Polynomial roots from companion matrix
     eigenvalues”, Mathematics of Computation, Vol.: 64, No.: 210
     (1995), 763–776.

The formulas for divided differences are given in the following texts,

   * Abramowitz and Stegun, Handbook of Mathematical Functions, Sections
     25.1.4 and 25.2.26.

   * R. L. Burden and J. D. Faires, Numerical Analysis, 9th edition,
     ISBN 0-538-73351-9, 2011.


File: gsl-ref.info,  Node: Special Functions,  Next: Vectors and Matrices,  Prev: Polynomials,  Up: Top

7 Special Functions
*******************

This chapter describes the GSL special function library.  The library
includes routines for calculating the values of Airy functions, Bessel
functions, Clausen functions, Coulomb wave functions, Coupling
coefficients, the Dawson function, Debye functions, Dilogarithms,
Elliptic integrals, Jacobi elliptic functions, Error functions,
Exponential integrals, Fermi-Dirac functions, Gamma functions,
Gegenbauer functions, Hermite polynomials and functions, Hypergeometric
functions, Laguerre functions, Legendre functions and Spherical
Harmonics, the Psi (Digamma) Function, Synchrotron functions, Transport
functions, Trigonometric functions and Zeta functions.  Each routine
also computes an estimate of the numerical error in the calculated value
of the function.

The functions in this chapter are declared in individual header files,
such as ‘gsl_sf_airy.h’, ‘gsl_sf_bessel.h’, etc.  The complete set of
header files can be included using the file ‘gsl_sf.h’.

* Menu:

* Usage::
* The gsl_sf_result struct::
* Modes::
* Airy Functions and Derivatives::
* Bessel Functions::
* Clausen Functions::
* Coulomb Functions::
* Coupling Coefficients::
* Dawson Function::
* Debye Functions::
* Dilogarithm::
* Elementary Operations::
* Elliptic Integrals::
* Elliptic Functions (Jacobi): Elliptic Functions Jacobi.
* Error Functions::
* Exponential Functions::
* Exponential Integrals::
* Fermi-Dirac Function::
* Gamma and Beta Functions::
* Gegenbauer Functions::
* Hermite Polynomials and Functions::
* Hypergeometric Functions::
* Laguerre Functions::
* Lambert W Functions::
* Legendre Functions and Spherical Harmonics::
* Logarithm and Related Functions::
* Mathieu Functions::
* Power Function::
* Psi (Digamma) Function: Psi Digamma Function.
* Synchrotron Functions::
* Transport Functions::
* Trigonometric Functions::
* Zeta Functions::
* Examples: Examples<3>.
* References and Further Reading: References and Further Reading<3>.


File: gsl-ref.info,  Node: Usage,  Next: The gsl_sf_result struct,  Up: Special Functions

7.1 Usage
=========

The special functions are available in two calling conventions, a
`natural form' which returns the numerical value of the function and an
`error-handling form' which returns an error code.  The two types of
function provide alternative ways of accessing the same underlying code.

The `natural form' returns only the value of the function and can be
used directly in mathematical expressions.  For example, the following
function call will compute the value of the Bessel function J_0(x):

     double y = gsl_sf_bessel_J0 (x);

There is no way to access an error code or to estimate the error using
this method.  To allow access to this information the alternative
error-handling form stores the value and error in a modifiable argument:

     gsl_sf_result result;
     int status = gsl_sf_bessel_J0_e (x, &result);

The error-handling functions have the suffix ‘_e’.  The returned status
value indicates error conditions such as overflow, underflow or loss of
precision.  If there are no errors the error-handling functions return
‘GSL_SUCCESS’.


File: gsl-ref.info,  Node: The gsl_sf_result struct,  Next: Modes,  Prev: Usage,  Up: Special Functions

7.2 The gsl_sf_result struct
============================

The error handling form of the special functions always calculate an
error estimate along with the value of the result.  Therefore,
structures are provided for amalgamating a value and error estimate.
These structures are declared in the header file ‘gsl_sf_result.h’.

The following struct contains value and error fields.

 -- Type: gsl_sf_result

          typedef struct
          {
            double val;
            double err;
          } gsl_sf_result;

     The field ‘val’ contains the value and the field ‘err’ contains an
     estimate of the absolute error in the value.

In some cases, an overflow or underflow can be detected and handled by a
function.  In this case, it may be possible to return a scaling exponent
as well as an error/value pair in order to save the result from
exceeding the dynamic range of the built-in types.  The following struct
contains value and error fields as well as an exponent field such that
the actual result is obtained as ‘result * 10^(e10)’.

 -- Type: gsl_sf_result_e10

          typedef struct
          {
            double val;
            double err;
            int    e10;
          } gsl_sf_result_e10;


File: gsl-ref.info,  Node: Modes,  Next: Airy Functions and Derivatives,  Prev: The gsl_sf_result struct,  Up: Special Functions

7.3 Modes
=========

The goal of the library is to achieve double precision accuracy wherever
possible.  However the cost of evaluating some special functions to
double precision can be significant, particularly where very high order
terms are required.  In these cases a ‘mode’ argument, of type *note
gsl_mode_t: d0. allows the accuracy of the function to be reduced in
order to improve performance.  The following precision levels are
available for the mode argument,

 -- Type: gsl_mode_t

      -- Macro: GSL_PREC_DOUBLE

          Double-precision, a relative accuracy of approximately 2 *
          10^{-16}.

      -- Macro: GSL_PREC_SINGLE

          Single-precision, a relative accuracy of approximately
          10^{-7}.

      -- Macro: GSL_PREC_APPROX

          Approximate values, a relative accuracy of approximately 5 *
          10^{-4}.

The approximate mode provides the fastest evaluation at the lowest
accuracy.


File: gsl-ref.info,  Node: Airy Functions and Derivatives,  Next: Bessel Functions,  Prev: Modes,  Up: Special Functions

7.4 Airy Functions and Derivatives
==================================

The Airy functions Ai(x) and Bi(x) are defined by the integral
representations,

     Ai(x) = (1/pi) int_0^infty cos((1/3) t^3 + xt) dt 
     Bi(x) = (1/pi) int_0^infty (e^(-(1/3) t^3 + xt) + sin((1/3) t^3 + xt)) dt 

For further information see Abramowitz & Stegun, Section 10.4.  The Airy
functions are defined in the header file ‘gsl_sf_airy.h’.

* Menu:

* Airy Functions::
* Derivatives of Airy Functions::
* Zeros of Airy Functions::
* Zeros of Derivatives of Airy Functions::


File: gsl-ref.info,  Node: Airy Functions,  Next: Derivatives of Airy Functions,  Up: Airy Functions and Derivatives

7.4.1 Airy Functions
--------------------

 -- Function: double gsl_sf_airy_Ai (double x, gsl_mode_t mode)
 -- Function: int gsl_sf_airy_Ai_e (double x, gsl_mode_t mode,
          gsl_sf_result * result)

     These routines compute the Airy function Ai(x) with an accuracy
     specified by *note mode: d8.

 -- Function: double gsl_sf_airy_Bi (double x, gsl_mode_t mode)
 -- Function: int gsl_sf_airy_Bi_e (double x, gsl_mode_t mode,
          gsl_sf_result * result)

     These routines compute the Airy function Bi(x) with an accuracy
     specified by *note mode: d8.

 -- Function: double gsl_sf_airy_Ai_scaled (double x, gsl_mode_t mode)
 -- Function: int gsl_sf_airy_Ai_scaled_e (double x, gsl_mode_t mode,
          gsl_sf_result * result)

     These routines compute a scaled version of the Airy function S_A(x)
     Ai(x).  For x > 0 the scaling factor S_A(x) is \exp(+(2/3)
     x^{3/2}), and is 1 for x < 0.

 -- Function: double gsl_sf_airy_Bi_scaled (double x, gsl_mode_t mode)
 -- Function: int gsl_sf_airy_Bi_scaled_e (double x, gsl_mode_t mode,
          gsl_sf_result * result)

     These routines compute a scaled version of the Airy function S_B(x)
     Bi(x).  For x > 0 the scaling factor S_B(x) is exp(-(2/3) x^{3/2}),
     and is 1 for x < 0.


File: gsl-ref.info,  Node: Derivatives of Airy Functions,  Next: Zeros of Airy Functions,  Prev: Airy Functions,  Up: Airy Functions and Derivatives

7.4.2 Derivatives of Airy Functions
-----------------------------------

 -- Function: double gsl_sf_airy_Ai_deriv (double x, gsl_mode_t mode)
 -- Function: int gsl_sf_airy_Ai_deriv_e (double x, gsl_mode_t mode,
          gsl_sf_result * result)

     These routines compute the Airy function derivative Ai'(x) with an
     accuracy specified by *note mode: d8.

 -- Function: double gsl_sf_airy_Bi_deriv (double x, gsl_mode_t mode)
 -- Function: int gsl_sf_airy_Bi_deriv_e (double x, gsl_mode_t mode,
          gsl_sf_result * result)

     These routines compute the Airy function derivative Bi'(x) with an
     accuracy specified by *note mode: d8.

 -- Function: double gsl_sf_airy_Ai_deriv_scaled (double x,
          gsl_mode_t mode)
 -- Function: int gsl_sf_airy_Ai_deriv_scaled_e (double x,
          gsl_mode_t mode, gsl_sf_result * result)

     These routines compute the scaled Airy function derivative S_A(x)
     Ai'(x).  For x > 0 the scaling factor S_A(x) is \exp(+(2/3)
     x^{3/2}), and is 1 for x < 0.

 -- Function: double gsl_sf_airy_Bi_deriv_scaled (double x,
          gsl_mode_t mode)
 -- Function: int gsl_sf_airy_Bi_deriv_scaled_e (double x,
          gsl_mode_t mode, gsl_sf_result * result)

     These routines compute the scaled Airy function derivative S_B(x)
     Bi'(x).  For x > 0 the scaling factor S_B(x) is exp(-(2/3)
     x^{3/2}), and is 1 for x < 0.


File: gsl-ref.info,  Node: Zeros of Airy Functions,  Next: Zeros of Derivatives of Airy Functions,  Prev: Derivatives of Airy Functions,  Up: Airy Functions and Derivatives

7.4.3 Zeros of Airy Functions
-----------------------------

 -- Function: double gsl_sf_airy_zero_Ai (unsigned int s)
 -- Function: int gsl_sf_airy_zero_Ai_e (unsigned int s, gsl_sf_result
          * result)

     These routines compute the location of the ‘s’-th zero of the Airy
     function Ai(x).

 -- Function: double gsl_sf_airy_zero_Bi (unsigned int s)
 -- Function: int gsl_sf_airy_zero_Bi_e (unsigned int s, gsl_sf_result
          * result)

     These routines compute the location of the ‘s’-th zero of the Airy
     function Bi(x).


File: gsl-ref.info,  Node: Zeros of Derivatives of Airy Functions,  Prev: Zeros of Airy Functions,  Up: Airy Functions and Derivatives

7.4.4 Zeros of Derivatives of Airy Functions
--------------------------------------------

 -- Function: double gsl_sf_airy_zero_Ai_deriv (unsigned int s)
 -- Function: int gsl_sf_airy_zero_Ai_deriv_e (unsigned int s,
          gsl_sf_result * result)

     These routines compute the location of the ‘s’-th zero of the Airy
     function derivative Ai'(x).

 -- Function: double gsl_sf_airy_zero_Bi_deriv (unsigned int s)
 -- Function: int gsl_sf_airy_zero_Bi_deriv_e (unsigned int s,
          gsl_sf_result * result)

     These routines compute the location of the ‘s’-th zero of the Airy
     function derivative Bi'(x).


File: gsl-ref.info,  Node: Bessel Functions,  Next: Clausen Functions,  Prev: Airy Functions and Derivatives,  Up: Special Functions

7.5 Bessel Functions
====================

The routines described in this section compute the Cylindrical Bessel
functions J_n(x), Y_n(x), Modified cylindrical Bessel functions I_n(x),
K_n(x), Spherical Bessel functions j_l(x), y_l(x), and Modified
Spherical Bessel functions i_l(x), k_l(x).  For more information see
Abramowitz & Stegun, Chapters 9 and 10.  The Bessel functions are
defined in the header file ‘gsl_sf_bessel.h’.

* Menu:

* Regular Cylindrical Bessel Functions::
* Irregular Cylindrical Bessel Functions::
* Regular Modified Cylindrical Bessel Functions::
* Irregular Modified Cylindrical Bessel Functions::
* Regular Spherical Bessel Functions::
* Irregular Spherical Bessel Functions::
* Regular Modified Spherical Bessel Functions::
* Irregular Modified Spherical Bessel Functions::
* Regular Bessel Function—Fractional Order::
* Irregular Bessel Functions—Fractional Order::
* Regular Modified Bessel Functions—Fractional Order::
* Irregular Modified Bessel Functions—Fractional Order::
* Zeros of Regular Bessel Functions::


File: gsl-ref.info,  Node: Regular Cylindrical Bessel Functions,  Next: Irregular Cylindrical Bessel Functions,  Up: Bessel Functions

7.5.1 Regular Cylindrical Bessel Functions
------------------------------------------

 -- Function: double gsl_sf_bessel_J0 (double x)
 -- Function: int gsl_sf_bessel_J0_e (double x, gsl_sf_result * result)

     These routines compute the regular cylindrical Bessel function of
     zeroth order, J_0(x).

 -- Function: double gsl_sf_bessel_J1 (double x)
 -- Function: int gsl_sf_bessel_J1_e (double x, gsl_sf_result * result)

     These routines compute the regular cylindrical Bessel function of
     first order, J_1(x).

 -- Function: double gsl_sf_bessel_Jn (int n, double x)
 -- Function: int gsl_sf_bessel_Jn_e (int n, double x, gsl_sf_result
          * result)

     These routines compute the regular cylindrical Bessel function of
     order ‘n’, J_n(x).

 -- Function: int gsl_sf_bessel_Jn_array (int nmin, int nmax, double x,
          double result_array[])

     This routine computes the values of the regular cylindrical Bessel
     functions J_n(x) for n from ‘nmin’ to ‘nmax’ inclusive, storing the
     results in the array ‘result_array’.  The values are computed using
     recurrence relations for efficiency, and therefore may differ
     slightly from the exact values.


File: gsl-ref.info,  Node: Irregular Cylindrical Bessel Functions,  Next: Regular Modified Cylindrical Bessel Functions,  Prev: Regular Cylindrical Bessel Functions,  Up: Bessel Functions

7.5.2 Irregular Cylindrical Bessel Functions
--------------------------------------------

 -- Function: double gsl_sf_bessel_Y0 (double x)
 -- Function: int gsl_sf_bessel_Y0_e (double x, gsl_sf_result * result)

     These routines compute the irregular cylindrical Bessel function of
     zeroth order, Y_0(x), for x>0.

 -- Function: double gsl_sf_bessel_Y1 (double x)
 -- Function: int gsl_sf_bessel_Y1_e (double x, gsl_sf_result * result)

     These routines compute the irregular cylindrical Bessel function of
     first order, Y_1(x), for x>0.

 -- Function: double gsl_sf_bessel_Yn (int n, double x)
 -- Function: int gsl_sf_bessel_Yn_e (int n, double x, gsl_sf_result
          * result)

     These routines compute the irregular cylindrical Bessel function of
     order ‘n’, Y_n(x), for x>0.

 -- Function: int gsl_sf_bessel_Yn_array (int nmin, int nmax, double x,
          double result_array[])

     This routine computes the values of the irregular cylindrical
     Bessel functions Y_n(x) for n from ‘nmin’ to ‘nmax’ inclusive,
     storing the results in the array ‘result_array’.  The domain of the
     function is x>0.  The values are computed using recurrence
     relations for efficiency, and therefore may differ slightly from
     the exact values.


File: gsl-ref.info,  Node: Regular Modified Cylindrical Bessel Functions,  Next: Irregular Modified Cylindrical Bessel Functions,  Prev: Irregular Cylindrical Bessel Functions,  Up: Bessel Functions

7.5.3 Regular Modified Cylindrical Bessel Functions
---------------------------------------------------

 -- Function: double gsl_sf_bessel_I0 (double x)
 -- Function: int gsl_sf_bessel_I0_e (double x, gsl_sf_result * result)

     These routines compute the regular modified cylindrical Bessel
     function of zeroth order, I_0(x).

 -- Function: double gsl_sf_bessel_I1 (double x)
 -- Function: int gsl_sf_bessel_I1_e (double x, gsl_sf_result * result)

     These routines compute the regular modified cylindrical Bessel
     function of first order, I_1(x).

 -- Function: double gsl_sf_bessel_In (int n, double x)
 -- Function: int gsl_sf_bessel_In_e (int n, double x, gsl_sf_result
          * result)

     These routines compute the regular modified cylindrical Bessel
     function of order ‘n’, I_n(x).

 -- Function: int gsl_sf_bessel_In_array (int nmin, int nmax, double x,
          double result_array[])

     This routine computes the values of the regular modified
     cylindrical Bessel functions I_n(x) for n from ‘nmin’ to ‘nmax’
     inclusive, storing the results in the array ‘result_array’.  The
     start of the range ‘nmin’ must be positive or zero.  The values are
     computed using recurrence relations for efficiency, and therefore
     may differ slightly from the exact values.

 -- Function: double gsl_sf_bessel_I0_scaled (double x)
 -- Function: int gsl_sf_bessel_I0_scaled_e (double x, gsl_sf_result
          * result)

     These routines compute the scaled regular modified cylindrical
     Bessel function of zeroth order \exp(-|x|) I_0(x).

 -- Function: double gsl_sf_bessel_I1_scaled (double x)
 -- Function: int gsl_sf_bessel_I1_scaled_e (double x, gsl_sf_result
          * result)

     These routines compute the scaled regular modified cylindrical
     Bessel function of first order \exp(-|x|) I_1(x).

 -- Function: double gsl_sf_bessel_In_scaled (int n, double x)
 -- Function: int gsl_sf_bessel_In_scaled_e (int n, double x,
          gsl_sf_result * result)

     These routines compute the scaled regular modified cylindrical
     Bessel function of order ‘n’, \exp(-|x|) I_n(x)

 -- Function: int gsl_sf_bessel_In_scaled_array (int nmin, int nmax,
          double x, double result_array[])

     This routine computes the values of the scaled regular cylindrical
     Bessel functions \exp(-|x|) I_n(x) for n from ‘nmin’ to ‘nmax’
     inclusive, storing the results in the array ‘result_array’.  The
     start of the range ‘nmin’ must be positive or zero.  The values are
     computed using recurrence relations for efficiency, and therefore
     may differ slightly from the exact values.


File: gsl-ref.info,  Node: Irregular Modified Cylindrical Bessel Functions,  Next: Regular Spherical Bessel Functions,  Prev: Regular Modified Cylindrical Bessel Functions,  Up: Bessel Functions

7.5.4 Irregular Modified Cylindrical Bessel Functions
-----------------------------------------------------

 -- Function: double gsl_sf_bessel_K0 (double x)
 -- Function: int gsl_sf_bessel_K0_e (double x, gsl_sf_result * result)

     These routines compute the irregular modified cylindrical Bessel
     function of zeroth order, K_0(x), for x > 0.

 -- Function: double gsl_sf_bessel_K1 (double x)
 -- Function: int gsl_sf_bessel_K1_e (double x, gsl_sf_result * result)

     These routines compute the irregular modified cylindrical Bessel
     function of first order, K_1(x), for x > 0.

 -- Function: double gsl_sf_bessel_Kn (int n, double x)
 -- Function: int gsl_sf_bessel_Kn_e (int n, double x, gsl_sf_result
          * result)

     These routines compute the irregular modified cylindrical Bessel
     function of order ‘n’, K_n(x), for x > 0.

 -- Function: int gsl_sf_bessel_Kn_array (int nmin, int nmax, double x,
          double result_array[])

     This routine computes the values of the irregular modified
     cylindrical Bessel functions K_n(x) for n from ‘nmin’ to ‘nmax’
     inclusive, storing the results in the array ‘result_array’.  The
     start of the range ‘nmin’ must be positive or zero.  The domain of
     the function is x>0.  The values are computed using recurrence
     relations for efficiency, and therefore may differ slightly from
     the exact values.

 -- Function: double gsl_sf_bessel_K0_scaled (double x)
 -- Function: int gsl_sf_bessel_K0_scaled_e (double x, gsl_sf_result
          * result)

     These routines compute the scaled irregular modified cylindrical
     Bessel function of zeroth order \exp(x) K_0(x) for x>0.

 -- Function: double gsl_sf_bessel_K1_scaled (double x)
 -- Function: int gsl_sf_bessel_K1_scaled_e (double x, gsl_sf_result
          * result)

     These routines compute the scaled irregular modified cylindrical
     Bessel function of first order \exp(x) K_1(x) for x>0.

 -- Function: double gsl_sf_bessel_Kn_scaled (int n, double x)
 -- Function: int gsl_sf_bessel_Kn_scaled_e (int n, double x,
          gsl_sf_result * result)

     These routines compute the scaled irregular modified cylindrical
     Bessel function of order ‘n’, \exp(x) K_n(x), for x>0.

 -- Function: int gsl_sf_bessel_Kn_scaled_array (int nmin, int nmax,
          double x, double result_array[])

     This routine computes the values of the scaled irregular
     cylindrical Bessel functions \exp(x) K_n(x) for n from ‘nmin’ to
     ‘nmax’ inclusive, storing the results in the array ‘result_array’.
     The start of the range ‘nmin’ must be positive or zero.  The domain
     of the function is x>0.  The values are computed using recurrence
     relations for efficiency, and therefore may differ slightly from
     the exact values.


File: gsl-ref.info,  Node: Regular Spherical Bessel Functions,  Next: Irregular Spherical Bessel Functions,  Prev: Irregular Modified Cylindrical Bessel Functions,  Up: Bessel Functions

7.5.5 Regular Spherical Bessel Functions
----------------------------------------

 -- Function: double gsl_sf_bessel_j0 (double x)
 -- Function: int gsl_sf_bessel_j0_e (double x, gsl_sf_result * result)

     These routines compute the regular spherical Bessel function of
     zeroth order, j_0(x) = \sin(x)/x.

 -- Function: double gsl_sf_bessel_j1 (double x)
 -- Function: int gsl_sf_bessel_j1_e (double x, gsl_sf_result * result)

     These routines compute the regular spherical Bessel function of
     first order, j_1(x) = (\sin(x)/x - \cos(x))/x.

 -- Function: double gsl_sf_bessel_j2 (double x)
 -- Function: int gsl_sf_bessel_j2_e (double x, gsl_sf_result * result)

     These routines compute the regular spherical Bessel function of
     second order, j_2(x) = ((3/x^2 - 1)\sin(x) - 3\cos(x)/x)/x.

 -- Function: double gsl_sf_bessel_jl (int l, double x)
 -- Function: int gsl_sf_bessel_jl_e (int l, double x, gsl_sf_result
          * result)

     These routines compute the regular spherical Bessel function of
     order ‘l’, j_l(x), for l \geq 0 and x \geq 0.

 -- Function: int gsl_sf_bessel_jl_array (int lmax, double x,
          double result_array[])

     This routine computes the values of the regular spherical Bessel
     functions j_l(x) for l from 0 to ‘lmax’ inclusive for lmax \geq 0
     and x \geq 0, storing the results in the array ‘result_array’.  The
     values are computed using recurrence relations for efficiency, and
     therefore may differ slightly from the exact values.

 -- Function: int gsl_sf_bessel_jl_steed_array (int lmax, double x,
          double * result_array)

     This routine uses Steed’s method to compute the values of the
     regular spherical Bessel functions j_l(x) for l from 0 to ‘lmax’
     inclusive for lmax \geq 0 and x \geq 0, storing the results in the
     array ‘result_array’.  The Steed/Barnett algorithm is described in
     Comp.  Phys.  Comm.  21, 297 (1981).  Steed’s method is more stable
     than the recurrence used in the other functions but is also slower.


File: gsl-ref.info,  Node: Irregular Spherical Bessel Functions,  Next: Regular Modified Spherical Bessel Functions,  Prev: Regular Spherical Bessel Functions,  Up: Bessel Functions

7.5.6 Irregular Spherical Bessel Functions
------------------------------------------

 -- Function: double gsl_sf_bessel_y0 (double x)
 -- Function: int gsl_sf_bessel_y0_e (double x, gsl_sf_result * result)

     These routines compute the irregular spherical Bessel function of
     zeroth order, y_0(x) = -\cos(x)/x.

 -- Function: double gsl_sf_bessel_y1 (double x)
 -- Function: int gsl_sf_bessel_y1_e (double x, gsl_sf_result * result)

     These routines compute the irregular spherical Bessel function of
     first order, y_1(x) = -(\cos(x)/x + \sin(x))/x.

 -- Function: double gsl_sf_bessel_y2 (double x)
 -- Function: int gsl_sf_bessel_y2_e (double x, gsl_sf_result * result)

     These routines compute the irregular spherical Bessel function of
     second order, y_2(x) = (-3/x^3 + 1/x)\cos(x) - (3/x^2)\sin(x).

 -- Function: double gsl_sf_bessel_yl (int l, double x)
 -- Function: int gsl_sf_bessel_yl_e (int l, double x, gsl_sf_result
          * result)

     These routines compute the irregular spherical Bessel function of
     order ‘l’, y_l(x), for l \geq 0.

 -- Function: int gsl_sf_bessel_yl_array (int lmax, double x,
          double result_array[])

     This routine computes the values of the irregular spherical Bessel
     functions y_l(x) for l from 0 to ‘lmax’ inclusive for lmax \geq 0,
     storing the results in the array ‘result_array’.  The values are
     computed using recurrence relations for efficiency, and therefore
     may differ slightly from the exact values.


File: gsl-ref.info,  Node: Regular Modified Spherical Bessel Functions,  Next: Irregular Modified Spherical Bessel Functions,  Prev: Irregular Spherical Bessel Functions,  Up: Bessel Functions

7.5.7 Regular Modified Spherical Bessel Functions
-------------------------------------------------

The regular modified spherical Bessel functions i_l(x) are related to
the modified Bessel functions of fractional order, i_l(x) =
\sqrt{\pi/(2x)} I_{l+1/2}(x)

 -- Function: double gsl_sf_bessel_i0_scaled (double x)
 -- Function: int gsl_sf_bessel_i0_scaled_e (double x, gsl_sf_result
          * result)

     These routines compute the scaled regular modified spherical Bessel
     function of zeroth order, \exp(-|x|) i_0(x).

 -- Function: double gsl_sf_bessel_i1_scaled (double x)
 -- Function: int gsl_sf_bessel_i1_scaled_e (double x, gsl_sf_result
          * result)

     These routines compute the scaled regular modified spherical Bessel
     function of first order, \exp(-|x|) i_1(x).

 -- Function: double gsl_sf_bessel_i2_scaled (double x)
 -- Function: int gsl_sf_bessel_i2_scaled_e (double x, gsl_sf_result
          * result)

     These routines compute the scaled regular modified spherical Bessel
     function of second order, \exp(-|x|) i_2(x)

 -- Function: double gsl_sf_bessel_il_scaled (int l, double x)
 -- Function: int gsl_sf_bessel_il_scaled_e (int l, double x,
          gsl_sf_result * result)

     These routines compute the scaled regular modified spherical Bessel
     function of order ‘l’, \exp(-|x|) i_l(x)

 -- Function: int gsl_sf_bessel_il_scaled_array (int lmax, double x,
          double result_array[])

     This routine computes the values of the scaled regular modified
     spherical Bessel functions \exp(-|x|) i_l(x) for l from 0 to ‘lmax’
     inclusive for lmax \geq 0, storing the results in the array
     ‘result_array’.  The values are computed using recurrence relations
     for efficiency, and therefore may differ slightly from the exact
     values.


File: gsl-ref.info,  Node: Irregular Modified Spherical Bessel Functions,  Next: Regular Bessel Function—Fractional Order,  Prev: Regular Modified Spherical Bessel Functions,  Up: Bessel Functions

7.5.8 Irregular Modified Spherical Bessel Functions
---------------------------------------------------

The irregular modified spherical Bessel functions k_l(x) are related to
the irregular modified Bessel functions of fractional order, k_l(x) =
\sqrt{\pi/(2x)} K_{l+1/2}(x).

 -- Function: double gsl_sf_bessel_k0_scaled (double x)
 -- Function: int gsl_sf_bessel_k0_scaled_e (double x, gsl_sf_result
          * result)

     These routines compute the scaled irregular modified spherical
     Bessel function of zeroth order, \exp(x) k_0(x), for x>0.

 -- Function: double gsl_sf_bessel_k1_scaled (double x)
 -- Function: int gsl_sf_bessel_k1_scaled_e (double x, gsl_sf_result
          * result)

     These routines compute the scaled irregular modified spherical
     Bessel function of first order, \exp(x) k_1(x), for x>0.

 -- Function: double gsl_sf_bessel_k2_scaled (double x)
 -- Function: int gsl_sf_bessel_k2_scaled_e (double x, gsl_sf_result
          * result)

     These routines compute the scaled irregular modified spherical
     Bessel function of second order, \exp(x) k_2(x), for x>0.

 -- Function: double gsl_sf_bessel_kl_scaled (int l, double x)
 -- Function: int gsl_sf_bessel_kl_scaled_e (int l, double x,
          gsl_sf_result * result)

     These routines compute the scaled irregular modified spherical
     Bessel function of order ‘l’, \exp(x) k_l(x), for x>0.

 -- Function: int gsl_sf_bessel_kl_scaled_array (int lmax, double x,
          double result_array[])

     This routine computes the values of the scaled irregular modified
     spherical Bessel functions \exp(x) k_l(x) for l from 0 to ‘lmax’
     inclusive for lmax \geq 0 and x>0, storing the results in the array
     ‘result_array’.  The values are computed using recurrence relations
     for efficiency, and therefore may differ slightly from the exact
     values.


File: gsl-ref.info,  Node: Regular Bessel Function—Fractional Order,  Next: Irregular Bessel Functions—Fractional Order,  Prev: Irregular Modified Spherical Bessel Functions,  Up: Bessel Functions

7.5.9 Regular Bessel Function—Fractional Order
----------------------------------------------

 -- Function: double gsl_sf_bessel_Jnu (double nu, double x)
 -- Function: int gsl_sf_bessel_Jnu_e (double nu, double x,
          gsl_sf_result * result)

     These routines compute the regular cylindrical Bessel function of
     fractional order \nu, J_\nu(x).

 -- Function: int gsl_sf_bessel_sequence_Jnu_e (double nu,
          gsl_mode_t mode, size_t size, double v[])

     This function computes the regular cylindrical Bessel function of
     fractional order \nu, J_\nu(x), evaluated at a series of x values.
     The array ‘v’ of length ‘size’ contains the x values.  They are
     assumed to be strictly ordered and positive.  The array is
     over-written with the values of J_\nu(x_i).


File: gsl-ref.info,  Node: Irregular Bessel Functions—Fractional Order,  Next: Regular Modified Bessel Functions—Fractional Order,  Prev: Regular Bessel Function—Fractional Order,  Up: Bessel Functions

7.5.10 Irregular Bessel Functions—Fractional Order
--------------------------------------------------

 -- Function: double gsl_sf_bessel_Ynu (double nu, double x)
 -- Function: int gsl_sf_bessel_Ynu_e (double nu, double x,
          gsl_sf_result * result)

     These routines compute the irregular cylindrical Bessel function of
     fractional order \nu, Y_\nu(x).


File: gsl-ref.info,  Node: Regular Modified Bessel Functions—Fractional Order,  Next: Irregular Modified Bessel Functions—Fractional Order,  Prev: Irregular Bessel Functions—Fractional Order,  Up: Bessel Functions

7.5.11 Regular Modified Bessel Functions—Fractional Order
---------------------------------------------------------

 -- Function: double gsl_sf_bessel_Inu (double nu, double x)
 -- Function: int gsl_sf_bessel_Inu_e (double nu, double x,
          gsl_sf_result * result)

     These routines compute the regular modified Bessel function of
     fractional order \nu, I_\nu(x) for x>0, \nu>0.

 -- Function: double gsl_sf_bessel_Inu_scaled (double nu, double x)
 -- Function: int gsl_sf_bessel_Inu_scaled_e (double nu, double x,
          gsl_sf_result * result)

     These routines compute the scaled regular modified Bessel function
     of fractional order \nu, \exp(-|x|)I_\nu(x) for x>0, \nu>0.


File: gsl-ref.info,  Node: Irregular Modified Bessel Functions—Fractional Order,  Next: Zeros of Regular Bessel Functions,  Prev: Regular Modified Bessel Functions—Fractional Order,  Up: Bessel Functions

7.5.12 Irregular Modified Bessel Functions—Fractional Order
-----------------------------------------------------------

 -- Function: double gsl_sf_bessel_Knu (double nu, double x)
 -- Function: int gsl_sf_bessel_Knu_e (double nu, double x,
          gsl_sf_result * result)

     These routines compute the irregular modified Bessel function of
     fractional order \nu, K_\nu(x) for x>0, \nu>0.

 -- Function: double gsl_sf_bessel_lnKnu (double nu, double x)
 -- Function: int gsl_sf_bessel_lnKnu_e (double nu, double x,
          gsl_sf_result * result)

     These routines compute the logarithm of the irregular modified
     Bessel function of fractional order \nu, \ln(K_\nu(x)) for x>0,
     \nu>0.

 -- Function: double gsl_sf_bessel_Knu_scaled (double nu, double x)
 -- Function: int gsl_sf_bessel_Knu_scaled_e (double nu, double x,
          gsl_sf_result * result)

     These routines compute the scaled irregular modified Bessel
     function of fractional order \nu, \exp(+|x|) K_\nu(x) for x>0,
     \nu>0.


File: gsl-ref.info,  Node: Zeros of Regular Bessel Functions,  Prev: Irregular Modified Bessel Functions—Fractional Order,  Up: Bessel Functions

7.5.13 Zeros of Regular Bessel Functions
----------------------------------------

 -- Function: double gsl_sf_bessel_zero_J0 (unsigned int s)
 -- Function: int gsl_sf_bessel_zero_J0_e (unsigned int s, gsl_sf_result
          * result)

     These routines compute the location of the ‘s’-th positive zero of
     the Bessel function J_0(x).

 -- Function: double gsl_sf_bessel_zero_J1 (unsigned int s)
 -- Function: int gsl_sf_bessel_zero_J1_e (unsigned int s, gsl_sf_result
          * result)

     These routines compute the location of the ‘s’-th positive zero of
     the Bessel function J_1(x).

 -- Function: double gsl_sf_bessel_zero_Jnu (double nu, unsigned int s)
 -- Function: int gsl_sf_bessel_zero_Jnu_e (double nu, unsigned int s,
          gsl_sf_result * result)

     These routines compute the location of the ‘s’-th positive zero of
     the Bessel function J_\nu(x).  The current implementation does not
     support negative values of ‘nu’.


File: gsl-ref.info,  Node: Clausen Functions,  Next: Coulomb Functions,  Prev: Bessel Functions,  Up: Special Functions

7.6 Clausen Functions
=====================

The Clausen function is defined by the following integral,

Cl_2(x) = - int_0^x dt log( 2 sin(t/2) )

It is related to the *note dilogarithm: 165. by Cl_2(\theta) = \Im
Li_2(\exp(i\theta)).  The Clausen functions are declared in the header
file ‘gsl_sf_clausen.h’.

 -- Function: double gsl_sf_clausen (double x)
 -- Function: int gsl_sf_clausen_e (double x, gsl_sf_result * result)

     These routines compute the Clausen integral Cl_2(x).


File: gsl-ref.info,  Node: Coulomb Functions,  Next: Coupling Coefficients,  Prev: Clausen Functions,  Up: Special Functions

7.7 Coulomb Functions
=====================

The prototypes of the Coulomb functions are declared in the header file
‘gsl_sf_coulomb.h’.  Both bound state and scattering solutions are
available.

* Menu:

* Normalized Hydrogenic Bound States::
* Coulomb Wave Functions::
* Coulomb Wave Function Normalization Constant::


File: gsl-ref.info,  Node: Normalized Hydrogenic Bound States,  Next: Coulomb Wave Functions,  Up: Coulomb Functions

7.7.1 Normalized Hydrogenic Bound States
----------------------------------------

 -- Function: double gsl_sf_hydrogenicR_1 (double Z, double r)
 -- Function: int gsl_sf_hydrogenicR_1_e (double Z, double r,
          gsl_sf_result * result)

     These routines compute the lowest-order normalized hydrogenic bound
     state radial wavefunction R_1 := 2Z \sqrt{Z} \exp(-Z r).

 -- Function: double gsl_sf_hydrogenicR (int n, int l, double Z,
          double r)
 -- Function: int gsl_sf_hydrogenicR_e (int n, int l, double Z,
          double r, gsl_sf_result * result)

     These routines compute the ‘n’-th normalized hydrogenic bound state
     radial wavefunction,

          R_n := 2 (Z^{3/2}/n^2) sqrt{(n-l-1)!/(n+l)!} exp(-Z r/n) (2Zr/n)^l 
               L^{2l+1}_{n-l-1}(2Zr/n). 

     where L^a_b(x) is the *note generalized Laguerre polynomial: 16e.
     The normalization is chosen such that the wavefunction \psi is
     given by \psi(n,l,r) = R_n Y_{lm}.


File: gsl-ref.info,  Node: Coulomb Wave Functions,  Next: Coulomb Wave Function Normalization Constant,  Prev: Normalized Hydrogenic Bound States,  Up: Coulomb Functions

7.7.2 Coulomb Wave Functions
----------------------------

The Coulomb wave functions F_L(\eta,x), G_L(\eta,x) are described in
Abramowitz & Stegun, Chapter 14.  Because there can be a large dynamic
range of values for these functions, overflows are handled gracefully.
If an overflow occurs, ‘GSL_EOVRFLW’ is signalled and exponent(s) are
returned through the modifiable parameters ‘exp_F’, ‘exp_G’.  The full
solution can be reconstructed from the following relations,

     F_L(eta,x) = fc[k_L] * exp(exp_F) 
     G_L(eta,x) = gc[k_L] * exp(exp_G) 
      
     F_L’(eta,x) = fcp[k_L] * exp(exp_F) 
     G_L’(eta,x) = gcp[k_L] * exp(exp_G) 

 -- Function: int gsl_sf_coulomb_wave_FG_e (double eta, double x,
          double L_F, int k, gsl_sf_result * F, gsl_sf_result * Fp,
          gsl_sf_result * G, gsl_sf_result * Gp, double * exp_F, double
          * exp_G)

     This function computes the Coulomb wave functions F_L(\eta,x),
     G_{L-k}(\eta,x) and their derivatives F'_L(\eta,x),
     G'_{L-k}(\eta,x) with respect to x.  The parameters are restricted
     to L, L-k > -1/2, x > 0 and integer k.  Note that L itself is not
     restricted to being an integer.  The results are stored in the
     parameters F, G for the function values and ‘Fp’, ‘Gp’ for the
     derivative values.  If an overflow occurs, ‘GSL_EOVRFLW’ is
     returned and scaling exponents are stored in the modifiable
     parameters ‘exp_F’, ‘exp_G’.

 -- Function: int gsl_sf_coulomb_wave_F_array (double L_min, int kmax,
          double eta, double x, double fc_array[], double * F_exponent)

     This function computes the Coulomb wave function F_L(\eta,x) for L
     = Lmin \dots Lmin + kmax, storing the results in ‘fc_array’.  In
     the case of overflow the exponent is stored in ‘F_exponent’.

 -- Function: int gsl_sf_coulomb_wave_FG_array (double L_min, int kmax,
          double eta, double x, double fc_array[], double gc_array[],
          double * F_exponent, double * G_exponent)

     This function computes the functions F_L(\eta,x), G_L(\eta,x) for L
     = Lmin \dots Lmin + kmax storing the results in ‘fc_array’ and
     ‘gc_array’.  In the case of overflow the exponents are stored in
     ‘F_exponent’ and ‘G_exponent’.

 -- Function: int gsl_sf_coulomb_wave_FGp_array (double L_min, int kmax,
          double eta, double x, double fc_array[], double fcp_array[],
          double gc_array[], double gcp_array[], double * F_exponent,
          double * G_exponent)

     This function computes the functions F_L(\eta,x), G_L(\eta,x) and
     their derivatives F'_L(\eta,x), G'_L(\eta,x) for L = Lmin \dots
     Lmin + kmax storing the results in ‘fc_array’, ‘gc_array’,
     ‘fcp_array’ and ‘gcp_array’.  In the case of overflow the exponents
     are stored in ‘F_exponent’ and ‘G_exponent’.

 -- Function: int gsl_sf_coulomb_wave_sphF_array (double L_min,
          int kmax, double eta, double x, double fc_array[],
          double F_exponent[])

     This function computes the Coulomb wave function divided by the
     argument F_L(\eta, x)/x for L = Lmin \dots Lmin + kmax, storing the
     results in ‘fc_array’.  In the case of overflow the exponent is
     stored in ‘F_exponent’.  This function reduces to spherical Bessel
     functions in the limit \eta \to 0.


File: gsl-ref.info,  Node: Coulomb Wave Function Normalization Constant,  Prev: Coulomb Wave Functions,  Up: Coulomb Functions

7.7.3 Coulomb Wave Function Normalization Constant
--------------------------------------------------

The Coulomb wave function normalization constant is defined in
Abramowitz 14.1.7.

 -- Function: int gsl_sf_coulomb_CL_e (double L, double eta,
          gsl_sf_result * result)

     This function computes the Coulomb wave function normalization
     constant C_L(\eta) for L > -1.

 -- Function: int gsl_sf_coulomb_CL_array (double Lmin, int kmax,
          double eta, double cl[])

     This function computes the Coulomb wave function normalization
     constant C_L(\eta) for L = Lmin \dots Lmin + kmax, Lmin > -1.


File: gsl-ref.info,  Node: Coupling Coefficients,  Next: Dawson Function,  Prev: Coulomb Functions,  Up: Special Functions

7.8 Coupling Coefficients
=========================

The Wigner 3-j, 6-j and 9-j symbols give the coupling coefficients for
combined angular momentum vectors.  Since the arguments of the standard
coupling coefficient functions are integer or half-integer, the
arguments of the following functions are, by convention, integers equal
to twice the actual spin value.  For information on the 3-j coefficients
see Abramowitz & Stegun, Section 27.9.  The functions described in this
section are declared in the header file ‘gsl_sf_coupling.h’.

* Menu:

* 3-j Symbols::
* 6-j Symbols::
* 9-j Symbols::


File: gsl-ref.info,  Node: 3-j Symbols,  Next: 6-j Symbols,  Up: Coupling Coefficients

7.8.1 3-j Symbols
-----------------

 -- Function: double gsl_sf_coupling_3j (int two_ja, int two_jb,
          int two_jc, int two_ma, int two_mb, int two_mc)
 -- Function: int gsl_sf_coupling_3j_e (int two_ja, int two_jb,
          int two_jc, int two_ma, int two_mb, int two_mc, gsl_sf_result
          * result)

     These routines compute the Wigner 3-j coefficient,

          ( ja jb jc ) 
          ( ma mb mc ) 

     where the arguments are given in half-integer units, ja =
     ‘two_ja’/2, ma = ‘two_ma’/2, etc.


File: gsl-ref.info,  Node: 6-j Symbols,  Next: 9-j Symbols,  Prev: 3-j Symbols,  Up: Coupling Coefficients

7.8.2 6-j Symbols
-----------------

 -- Function: double gsl_sf_coupling_6j (int two_ja, int two_jb,
          int two_jc, int two_jd, int two_je, int two_jf)
 -- Function: int gsl_sf_coupling_6j_e (int two_ja, int two_jb,
          int two_jc, int two_jd, int two_je, int two_jf, gsl_sf_result
          * result)

     These routines compute the Wigner 6-j coefficient,

          { ja jb jc } 
          { jd je jf } 

     where the arguments are given in half-integer units, ja =
     ‘two_ja’/2, ma = ‘two_ma’/2, etc.


File: gsl-ref.info,  Node: 9-j Symbols,  Prev: 6-j Symbols,  Up: Coupling Coefficients

7.8.3 9-j Symbols
-----------------

 -- Function: double gsl_sf_coupling_9j (int two_ja, int two_jb,
          int two_jc, int two_jd, int two_je, int two_jf, int two_jg,
          int two_jh, int two_ji)
 -- Function: int gsl_sf_coupling_9j_e (int two_ja, int two_jb,
          int two_jc, int two_jd, int two_je, int two_jf, int two_jg,
          int two_jh, int two_ji, gsl_sf_result * result)

     These routines compute the Wigner 9-j coefficient,

          { ja jb jc } 
          { jd je jf } 
          { jg jh ji } 

     where the arguments are given in half-integer units, ja =
     ‘two_ja’/2, ma = ‘two_ma’/2, etc.


File: gsl-ref.info,  Node: Dawson Function,  Next: Debye Functions,  Prev: Coupling Coefficients,  Up: Special Functions

7.9 Dawson Function
===================

The Dawson integral is defined by

     \exp(-x^2) \int_0^x dt \exp(t^2)

A table of Dawson’s integral can be found in Abramowitz & Stegun, Table
7.5.  The Dawson functions are declared in the header file
‘gsl_sf_dawson.h’.

 -- Function: double gsl_sf_dawson (double x)
 -- Function: int gsl_sf_dawson_e (double x, gsl_sf_result * result)

     These routines compute the value of Dawson’s integral for ‘x’.


File: gsl-ref.info,  Node: Debye Functions,  Next: Dilogarithm,  Prev: Dawson Function,  Up: Special Functions

7.10 Debye Functions
====================

The Debye functions D_n(x) are defined by the following integral,

     D_n(x) = n/x^n \int_0^x dt (t^n/(e^t - 1))

For further information see Abramowitz & Stegun, Section 27.1.  The
Debye functions are declared in the header file ‘gsl_sf_debye.h’.

 -- Function: double gsl_sf_debye_1 (double x)
 -- Function: int gsl_sf_debye_1_e (double x, gsl_sf_result * result)

     These routines compute the first-order Debye function D_1(x).

 -- Function: double gsl_sf_debye_2 (double x)
 -- Function: int gsl_sf_debye_2_e (double x, gsl_sf_result * result)

     These routines compute the second-order Debye function D_2(x).

 -- Function: double gsl_sf_debye_3 (double x)
 -- Function: int gsl_sf_debye_3_e (double x, gsl_sf_result * result)

     These routines compute the third-order Debye function D_3(x).

 -- Function: double gsl_sf_debye_4 (double x)
 -- Function: int gsl_sf_debye_4_e (double x, gsl_sf_result * result)

     These routines compute the fourth-order Debye function D_4(x).

 -- Function: double gsl_sf_debye_5 (double x)
 -- Function: int gsl_sf_debye_5_e (double x, gsl_sf_result * result)

     These routines compute the fifth-order Debye function D_5(x).

 -- Function: double gsl_sf_debye_6 (double x)
 -- Function: int gsl_sf_debye_6_e (double x, gsl_sf_result * result)

     These routines compute the sixth-order Debye function D_6(x).


File: gsl-ref.info,  Node: Dilogarithm,  Next: Elementary Operations,  Prev: Debye Functions,  Up: Special Functions

7.11 Dilogarithm
================

The dilogarithm is defined as

     Li_2(z) = - \int_0^z ds log(1-s) / s

The functions described in this section are declared in the header file
‘gsl_sf_dilog.h’.

* Menu:

* Real Argument::
* Complex Argument::


File: gsl-ref.info,  Node: Real Argument,  Next: Complex Argument,  Up: Dilogarithm

7.11.1 Real Argument
--------------------

 -- Function: double gsl_sf_dilog (double x)
 -- Function: int gsl_sf_dilog_e (double x, gsl_sf_result * result)

     These routines compute the dilogarithm for a real argument.  In
     Lewin’s notation this is Li_2(x), the real part of the dilogarithm
     of a real x.  It is defined by the integral representation

          Li_2(x) = - \Re \int_0^x ds \log(1-s) / s

     Note that \Im(Li_2(x)) = 0 for x \le 1, and -\pi\log(x) for x > 1.

     Note that Abramowitz & Stegun refer to the Spence integral S(x) =
     Li_2(1 - x) as the dilogarithm rather than Li_2(x).


File: gsl-ref.info,  Node: Complex Argument,  Prev: Real Argument,  Up: Dilogarithm

7.11.2 Complex Argument
-----------------------

 -- Function: int gsl_sf_complex_dilog_e (double r, double theta,
          gsl_sf_result * result_re, gsl_sf_result * result_im)

     This function computes the full complex-valued dilogarithm for the
     complex argument z = r \exp(i \theta).  The real and imaginary
     parts of the result are returned in ‘result_re’, ‘result_im’.


File: gsl-ref.info,  Node: Elementary Operations,  Next: Elliptic Integrals,  Prev: Dilogarithm,  Up: Special Functions

7.12 Elementary Operations
==========================

The following functions allow for the propagation of errors when
combining quantities by multiplication.  The functions are declared in
the header file ‘gsl_sf_elementary.h’.

 -- Function: double gsl_sf_multiply (double x, double y)
 -- Function: int gsl_sf_multiply_e (double x, double y, gsl_sf_result
          * result)

     This function multiplies ‘x’ and ‘y’ storing the product and its
     associated error in ‘result’.

 -- Function: int gsl_sf_multiply_err_e (double x, double dx, double y,
          double dy, gsl_sf_result * result)

     This function multiplies ‘x’ and ‘y’ with associated absolute
     errors ‘dx’ and ‘dy’.  The product xy \pm xy \sqrt{(dx/x)^2
     +(dy/y)^2} is stored in ‘result’.


File: gsl-ref.info,  Node: Elliptic Integrals,  Next: Elliptic Functions Jacobi,  Prev: Elementary Operations,  Up: Special Functions

7.13 Elliptic Integrals
=======================

The functions described in this section are declared in the header file
‘gsl_sf_ellint.h’.  Further information about the elliptic integrals can
be found in Abramowitz & Stegun, Chapter 17.

* Menu:

* Definition of Legendre Forms::
* Definition of Carlson Forms::
* Legendre Form of Complete Elliptic Integrals::
* Legendre Form of Incomplete Elliptic Integrals::
* Carlson Forms::


File: gsl-ref.info,  Node: Definition of Legendre Forms,  Next: Definition of Carlson Forms,  Up: Elliptic Integrals

7.13.1 Definition of Legendre Forms
-----------------------------------

The Legendre forms of elliptic integrals F(\phi,k), E(\phi,k) and
\Pi(\phi,k,n) are defined by,

     F(\phi,k) = \int_0^\phi dt 1/\sqrt((1 - k^2 \sin^2(t)))
     E(\phi,k) = \int_0^\phi dt   \sqrt((1 - k^2 \sin^2(t)))
     Pi(\phi,k,n) = \int_0^\phi dt 1/((1 + n \sin^2(t))\sqrt(1 - k^2 \sin^2(t)))

The complete Legendre forms are denoted by K(k) = F(\pi/2, k) and E(k) =
E(\pi/2, k).

The notation used here is based on Carlson, “Numerische Mathematik” 33
(1979) 1 and differs slightly from that used by Abramowitz & Stegun,
where the functions are given in terms of the parameter m = k^2 and n is
replaced by -n.


File: gsl-ref.info,  Node: Definition of Carlson Forms,  Next: Legendre Form of Complete Elliptic Integrals,  Prev: Definition of Legendre Forms,  Up: Elliptic Integrals

7.13.2 Definition of Carlson Forms
----------------------------------

The Carlson symmetric forms of elliptical integrals RC(x,y), RD(x,y,z),
RF(x,y,z) and RJ(x,y,z,p) are defined by,

     RC(x,y) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1)
     RD(x,y,z) = 3/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-3/2)
     RF(x,y,z) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2)
     RJ(x,y,z,p) = 3/2 \int_0^\infty dt
                      (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2) (t+p)^(-1)


File: gsl-ref.info,  Node: Legendre Form of Complete Elliptic Integrals,  Next: Legendre Form of Incomplete Elliptic Integrals,  Prev: Definition of Carlson Forms,  Up: Elliptic Integrals

7.13.3 Legendre Form of Complete Elliptic Integrals
---------------------------------------------------

 -- Function: double gsl_sf_ellint_Kcomp (double k, gsl_mode_t mode)
 -- Function: int gsl_sf_ellint_Kcomp_e (double k, gsl_mode_t mode,
          gsl_sf_result * result)

     These routines compute the complete elliptic integral K(k) to the
     accuracy specified by the mode variable *note mode: d8.  Note that
     Abramowitz & Stegun define this function in terms of the parameter
     m = k^2.

 -- Function: double gsl_sf_ellint_Ecomp (double k, gsl_mode_t mode)
 -- Function: int gsl_sf_ellint_Ecomp_e (double k, gsl_mode_t mode,
          gsl_sf_result * result)

     These routines compute the complete elliptic integral E(k) to the
     accuracy specified by the mode variable *note mode: d8.  Note that
     Abramowitz & Stegun define this function in terms of the parameter
     m = k^2.

 -- Function: double gsl_sf_ellint_Pcomp (double k, double n,
          gsl_mode_t mode)
 -- Function: int gsl_sf_ellint_Pcomp_e (double k, double n,
          gsl_mode_t mode, gsl_sf_result * result)

     These routines compute the complete elliptic integral \Pi(k,n) to
     the accuracy specified by the mode variable *note mode: d8.  Note
     that Abramowitz & Stegun define this function in terms of the
     parameters m = k^2 and \sin^2(\alpha) = k^2, with the change of
     sign n \to -n.


File: gsl-ref.info,  Node: Legendre Form of Incomplete Elliptic Integrals,  Next: Carlson Forms,  Prev: Legendre Form of Complete Elliptic Integrals,  Up: Elliptic Integrals

7.13.4 Legendre Form of Incomplete Elliptic Integrals
-----------------------------------------------------

 -- Function: double gsl_sf_ellint_F (double phi, double k,
          gsl_mode_t mode)
 -- Function: int gsl_sf_ellint_F_e (double phi, double k,
          gsl_mode_t mode, gsl_sf_result * result)

     These routines compute the incomplete elliptic integral F(\phi,k)
     to the accuracy specified by the mode variable *note mode: d8.
     Note that Abramowitz & Stegun define this function in terms of the
     parameter m = k^2.

 -- Function: double gsl_sf_ellint_E (double phi, double k,
          gsl_mode_t mode)
 -- Function: int gsl_sf_ellint_E_e (double phi, double k,
          gsl_mode_t mode, gsl_sf_result * result)

     These routines compute the incomplete elliptic integral E(\phi,k)
     to the accuracy specified by the mode variable *note mode: d8.
     Note that Abramowitz & Stegun define this function in terms of the
     parameter m = k^2.

 -- Function: double gsl_sf_ellint_P (double phi, double k, double n,
          gsl_mode_t mode)
 -- Function: int gsl_sf_ellint_P_e (double phi, double k, double n,
          gsl_mode_t mode, gsl_sf_result * result)

     These routines compute the incomplete elliptic integral
     \Pi(\phi,k,n) to the accuracy specified by the mode variable *note
     mode: d8.  Note that Abramowitz & Stegun define this function in
     terms of the parameters m = k^2 and \sin^2(\alpha) = k^2, with the
     change of sign n \to -n.

 -- Function: double gsl_sf_ellint_D (double phi, double k,
          gsl_mode_t mode)
 -- Function: int gsl_sf_ellint_D_e (double phi, double k,
          gsl_mode_t mode, gsl_sf_result * result)

     These functions compute the incomplete elliptic integral D(\phi,k)
     which is defined through the Carlson form RD(x,y,z) by the
     following relation,

          D(\phi,k) = (1/3)(\sin(\phi))^3 RD (1-\sin^2(\phi), 1-k^2 \sin^2(\phi), 1).


File: gsl-ref.info,  Node: Carlson Forms,  Prev: Legendre Form of Incomplete Elliptic Integrals,  Up: Elliptic Integrals

7.13.5 Carlson Forms
--------------------

 -- Function: double gsl_sf_ellint_RC (double x, double y,
          gsl_mode_t mode)
 -- Function: int gsl_sf_ellint_RC_e (double x, double y,
          gsl_mode_t mode, gsl_sf_result * result)

     These routines compute the incomplete elliptic integral RC(x,y) to
     the accuracy specified by the mode variable *note mode: d8.

 -- Function: double gsl_sf_ellint_RD (double x, double y, double z,
          gsl_mode_t mode)
 -- Function: int gsl_sf_ellint_RD_e (double x, double y, double z,
          gsl_mode_t mode, gsl_sf_result * result)

     These routines compute the incomplete elliptic integral RD(x,y,z)
     to the accuracy specified by the mode variable *note mode: d8.

 -- Function: double gsl_sf_ellint_RF (double x, double y, double z,
          gsl_mode_t mode)
 -- Function: int gsl_sf_ellint_RF_e (double x, double y, double z,
          gsl_mode_t mode, gsl_sf_result * result)

     These routines compute the incomplete elliptic integral RF(x,y,z)
     to the accuracy specified by the mode variable *note mode: d8.

 -- Function: double gsl_sf_ellint_RJ (double x, double y, double z,
          double p, gsl_mode_t mode)
 -- Function: int gsl_sf_ellint_RJ_e (double x, double y, double z,
          double p, gsl_mode_t mode, gsl_sf_result * result)

     These routines compute the incomplete elliptic integral RJ(x,y,z,p)
     to the accuracy specified by the mode variable *note mode: d8.


File: gsl-ref.info,  Node: Elliptic Functions Jacobi,  Next: Error Functions,  Prev: Elliptic Integrals,  Up: Special Functions

7.14 Elliptic Functions (Jacobi)
================================

The Jacobian Elliptic functions are defined in Abramowitz & Stegun,
Chapter 16.  The functions are declared in the header file
‘gsl_sf_elljac.h’.

 -- Function: int gsl_sf_elljac_e (double u, double m, double * sn,
          double * cn, double * dn)

     This function computes the Jacobian elliptic functions sn(u|m),
     cn(u|m), dn(u|m) by descending Landen transformations.


File: gsl-ref.info,  Node: Error Functions,  Next: Exponential Functions,  Prev: Elliptic Functions Jacobi,  Up: Special Functions

7.15 Error Functions
====================

The error function is described in Abramowitz & Stegun, Chapter 7.  The
functions in this section are declared in the header file
‘gsl_sf_erf.h’.

* Menu:

* Error Function::
* Complementary Error Function::
* Log Complementary Error Function::
* Probability functions::


File: gsl-ref.info,  Node: Error Function,  Next: Complementary Error Function,  Up: Error Functions

7.15.1 Error Function
---------------------

 -- Function: double gsl_sf_erf (double x)
 -- Function: int gsl_sf_erf_e (double x, gsl_sf_result * result)

     These routines compute the error function \erf(x), where \erf(x) =
     (2/\sqrt{\pi}) \int_0^x dt \exp(-t^2).


File: gsl-ref.info,  Node: Complementary Error Function,  Next: Log Complementary Error Function,  Prev: Error Function,  Up: Error Functions

7.15.2 Complementary Error Function
-----------------------------------

 -- Function: double gsl_sf_erfc (double x)
 -- Function: int gsl_sf_erfc_e (double x, gsl_sf_result * result)

     These routines compute the complementary error function \erfc(x) =
     1 - \erf(x) = (2/\sqrt{\pi}) \int_x^\infty \exp(-t^2)


File: gsl-ref.info,  Node: Log Complementary Error Function,  Next: Probability functions,  Prev: Complementary Error Function,  Up: Error Functions

7.15.3 Log Complementary Error Function
---------------------------------------

 -- Function: double gsl_sf_log_erfc (double x)
 -- Function: int gsl_sf_log_erfc_e (double x, gsl_sf_result * result)

     These routines compute the logarithm of the complementary error
     function \log(\erfc(x)).


File: gsl-ref.info,  Node: Probability functions,  Prev: Log Complementary Error Function,  Up: Error Functions

7.15.4 Probability functions
----------------------------

The probability functions for the Normal or Gaussian distribution are
described in Abramowitz & Stegun, Section 26.2.

 -- Function: double gsl_sf_erf_Z (double x)
 -- Function: int gsl_sf_erf_Z_e (double x, gsl_sf_result * result)

     These routines compute the Gaussian probability density function
     Z(x) = (1/\sqrt{2\pi}) \exp(-x^2/2)

 -- Function: double gsl_sf_erf_Q (double x)
 -- Function: int gsl_sf_erf_Q_e (double x, gsl_sf_result * result)

     These routines compute the upper tail of the Gaussian probability
     function Q(x) = (1/\sqrt{2\pi}) \int_x^\infty dt \exp(-t^2/2)

The `hazard function' for the normal distribution, also known as the
inverse Mills’ ratio, is defined as,

     h(x) = Z(x)/Q(x) = \sqrt{2/\pi} \exp(-x^2 / 2) / \erfc(x/\sqrt 2)

It decreases rapidly as x approaches -\infty and asymptotes to h(x) \sim
x as x approaches +\infty.

 -- Function: double gsl_sf_hazard (double x)
 -- Function: int gsl_sf_hazard_e (double x, gsl_sf_result * result)

     These routines compute the hazard function for the normal
     distribution.


File: gsl-ref.info,  Node: Exponential Functions,  Next: Exponential Integrals,  Prev: Error Functions,  Up: Special Functions

7.16 Exponential Functions
==========================

The functions described in this section are declared in the header file
‘gsl_sf_exp.h’.

* Menu:

* Exponential Function::
* Relative Exponential Functions::
* Exponentiation With Error Estimate::


File: gsl-ref.info,  Node: Exponential Function,  Next: Relative Exponential Functions,  Up: Exponential Functions

7.16.1 Exponential Function
---------------------------

 -- Function: double gsl_sf_exp (double x)
 -- Function: int gsl_sf_exp_e (double x, gsl_sf_result * result)

     These routines provide an exponential function \exp(x) using GSL
     semantics and error checking.

 -- Function: int gsl_sf_exp_e10_e (double x, gsl_sf_result_e10
          * result)

     This function computes the exponential \exp(x) using the *note
     gsl_sf_result_e10: ce. type to return a result with extended range.
     This function may be useful if the value of \exp(x) would overflow
     the numeric range of ‘double’.

 -- Function: double gsl_sf_exp_mult (double x, double y)
 -- Function: int gsl_sf_exp_mult_e (double x, double y, gsl_sf_result
          * result)

     These routines exponentiate ‘x’ and multiply by the factor ‘y’ to
     return the product y \exp(x).

 -- Function: int gsl_sf_exp_mult_e10_e (const double x, const double y,
          gsl_sf_result_e10 * result)

     This function computes the product y \exp(x) using the *note
     gsl_sf_result_e10: ce. type to return a result with extended
     numeric range.


File: gsl-ref.info,  Node: Relative Exponential Functions,  Next: Exponentiation With Error Estimate,  Prev: Exponential Function,  Up: Exponential Functions

7.16.2 Relative Exponential Functions
-------------------------------------

 -- Function: double gsl_sf_expm1 (double x)
 -- Function: int gsl_sf_expm1_e (double x, gsl_sf_result * result)

     These routines compute the quantity \exp(x)-1 using an algorithm
     that is accurate for small x.

 -- Function: double gsl_sf_exprel (double x)
 -- Function: int gsl_sf_exprel_e (double x, gsl_sf_result * result)

     These routines compute the quantity (\exp(x)-1)/x using an
     algorithm that is accurate for small ‘x’.  For small ‘x’ the
     algorithm is based on the expansion (\exp(x)-1)/x = 1 + x/2 +
     x^2/(2*3) + x^3/(2*3*4) + \dots.

 -- Function: double gsl_sf_exprel_2 (double x)
 -- Function: int gsl_sf_exprel_2_e (double x, gsl_sf_result * result)

     These routines compute the quantity 2(\exp(x)-1-x)/x^2 using an
     algorithm that is accurate for small ‘x’.  For small ‘x’ the
     algorithm is based on the expansion 2(\exp(x)-1-x)/x^2 = 1 + x/3 +
     x^2/(3*4) + x^3/(3*4*5) + \dots.

 -- Function: double gsl_sf_exprel_n (int n, double x)
 -- Function: int gsl_sf_exprel_n_e (int n, double x, gsl_sf_result
          * result)

     These routines compute the N-relative exponential, which is the
     ‘n’-th generalization of the functions *note gsl_sf_exprel(): 1d6.
     and *note gsl_sf_exprel_2(): 1d8.  The N-relative exponential is
     given by,

          exprel_N(x) = N!/x^N (\exp(x) - \sum_{k=0}^{N-1} x^k/k!)
                      = 1 + x/(N+1) + x^2/((N+1)(N+2)) + ...
                      = 1F1 (1,1+N,x)


File: gsl-ref.info,  Node: Exponentiation With Error Estimate,  Prev: Relative Exponential Functions,  Up: Exponential Functions

7.16.3 Exponentiation With Error Estimate
-----------------------------------------

 -- Function: int gsl_sf_exp_err_e (double x, double dx, gsl_sf_result
          * result)

     This function exponentiates ‘x’ with an associated absolute error
     ‘dx’.

 -- Function: int gsl_sf_exp_err_e10_e (double x, double dx,
          gsl_sf_result_e10 * result)

     This function exponentiates a quantity ‘x’ with an associated
     absolute error ‘dx’ using the *note gsl_sf_result_e10: ce. type to
     return a result with extended range.

 -- Function: int gsl_sf_exp_mult_err_e (double x, double dx, double y,
          double dy, gsl_sf_result * result)

     This routine computes the product y \exp(x) for the quantities ‘x’,
     ‘y’ with associated absolute errors ‘dx’, ‘dy’.

 -- Function: int gsl_sf_exp_mult_err_e10_e (double x, double dx,
          double y, double dy, gsl_sf_result_e10 * result)

     This routine computes the product y \exp(x) for the quantities ‘x’,
     ‘y’ with associated absolute errors ‘dx’, ‘dy’ using the *note
     gsl_sf_result_e10: ce. type to return a result with extended range.


File: gsl-ref.info,  Node: Exponential Integrals,  Next: Fermi-Dirac Function,  Prev: Exponential Functions,  Up: Special Functions

7.17 Exponential Integrals
==========================

Information on the exponential integrals can be found in Abramowitz &
Stegun, Chapter 5.  These functions are declared in the header file
‘gsl_sf_expint.h’.

* Menu:

* Exponential Integral::
* Ei(x): Ei x.
* Hyperbolic Integrals::
* Ei_3(x): Ei_3 x.
* Trigonometric Integrals::
* Arctangent Integral::


File: gsl-ref.info,  Node: Exponential Integral,  Next: Ei x,  Up: Exponential Integrals

7.17.1 Exponential Integral
---------------------------

 -- Function: double gsl_sf_expint_E1 (double x)
 -- Function: int gsl_sf_expint_E1_e (double x, gsl_sf_result * result)

     These routines compute the exponential integral E_1(x),

          E_1(x) := \Re \int_1^\infty dt \exp(-xt)/t.

 -- Function: double gsl_sf_expint_E2 (double x)
 -- Function: int gsl_sf_expint_E2_e (double x, gsl_sf_result * result)

     These routines compute the second-order exponential integral
     E_2(x),

          E_2(x) := \Re \int_1^\infty dt \exp(-xt)/t^2

 -- Function: double gsl_sf_expint_En (int n, double x)
 -- Function: int gsl_sf_expint_En_e (int n, double x, gsl_sf_result
          * result)

     These routines compute the exponential integral E_n(x) of order
     ‘n’,

          E_n(x) := \Re \int_1^\infty dt \exp(-xt)/t^n.


File: gsl-ref.info,  Node: Ei x,  Next: Hyperbolic Integrals,  Prev: Exponential Integral,  Up: Exponential Integrals

7.17.2 Ei(x)
------------

 -- Function: double gsl_sf_expint_Ei (double x)
 -- Function: int gsl_sf_expint_Ei_e (double x, gsl_sf_result * result)

     These routines compute the exponential integral Ei(x),

          Ei(x) = - PV(\int_{-x}^\infty dt \exp(-t)/t)

     where PV denotes the principal value of the integral.


File: gsl-ref.info,  Node: Hyperbolic Integrals,  Next: Ei_3 x,  Prev: Ei x,  Up: Exponential Integrals

7.17.3 Hyperbolic Integrals
---------------------------

 -- Function: double gsl_sf_Shi (double x)
 -- Function: int gsl_sf_Shi_e (double x, gsl_sf_result * result)

     These routines compute the integral

          Shi(x) = \int_0^x dt \sinh(t)/t

 -- Function: double gsl_sf_Chi (double x)
 -- Function: int gsl_sf_Chi_e (double x, gsl_sf_result * result)

     These routines compute the integral

          Chi(x) := \Re[ \gamma_E + \log(x) + \int_0^x dt (\cosh(t)-1)/t ]

     where \gamma_E is the Euler constant (available as the macro
     ‘M_EULER’).


File: gsl-ref.info,  Node: Ei_3 x,  Next: Trigonometric Integrals,  Prev: Hyperbolic Integrals,  Up: Exponential Integrals

7.17.4 Ei_3(x)
--------------

 -- Function: double gsl_sf_expint_3 (double x)
 -- Function: int gsl_sf_expint_3_e (double x, gsl_sf_result * result)

     These routines compute the third-order exponential integral

          Ei_3(x) = \int_0^x dt \exp(-t^3)`

     for x \ge 0.


File: gsl-ref.info,  Node: Trigonometric Integrals,  Next: Arctangent Integral,  Prev: Ei_3 x,  Up: Exponential Integrals

7.17.5 Trigonometric Integrals
------------------------------

 -- Function: double gsl_sf_Si (const double x)
 -- Function: int gsl_sf_Si_e (double x, gsl_sf_result * result)

     These routines compute the Sine integral

          Si(x) = \int_0^x dt \sin(t)/t

 -- Function: double gsl_sf_Ci (const double x)
 -- Function: int gsl_sf_Ci_e (double x, gsl_sf_result * result)

     These routines compute the Cosine integral

          Ci(x) = -\int_x^\infty dt \cos(t)/t}

     for x > 0


File: gsl-ref.info,  Node: Arctangent Integral,  Prev: Trigonometric Integrals,  Up: Exponential Integrals

7.17.6 Arctangent Integral
--------------------------

 -- Function: double gsl_sf_atanint (double x)
 -- Function: int gsl_sf_atanint_e (double x, gsl_sf_result * result)

     These routines compute the Arctangent integral, which is defined as

          AtanInt(x) = \int_0^x dt \arctan(t)/t


File: gsl-ref.info,  Node: Fermi-Dirac Function,  Next: Gamma and Beta Functions,  Prev: Exponential Integrals,  Up: Special Functions

7.18 Fermi-Dirac Function
=========================

The functions described in this section are declared in the header file
‘gsl_sf_fermi_dirac.h’.

* Menu:

* Complete Fermi-Dirac Integrals::
* Incomplete Fermi-Dirac Integrals::


File: gsl-ref.info,  Node: Complete Fermi-Dirac Integrals,  Next: Incomplete Fermi-Dirac Integrals,  Up: Fermi-Dirac Function

7.18.1 Complete Fermi-Dirac Integrals
-------------------------------------

The complete Fermi-Dirac integral F_j(x) is given by,

     F_j(x) := (1/\Gamma(j+1)) \int_0^\infty dt (t^j / (\exp(t-x) + 1))

Note that the Fermi-Dirac integral is sometimes defined without the
normalisation factor in other texts.

 -- Function: double gsl_sf_fermi_dirac_m1 (double x)
 -- Function: int gsl_sf_fermi_dirac_m1_e (double x, gsl_sf_result
          * result)

     These routines compute the complete Fermi-Dirac integral with an
     index of -1.  This integral is given by F_{-1}(x) = e^x / (1 +
     e^x).

 -- Function: double gsl_sf_fermi_dirac_0 (double x)
 -- Function: int gsl_sf_fermi_dirac_0_e (double x, gsl_sf_result
          * result)

     These routines compute the complete Fermi-Dirac integral with an
     index of 0.  This integral is given by F_0(x) = \ln(1 + e^x).

 -- Function: double gsl_sf_fermi_dirac_1 (double x)
 -- Function: int gsl_sf_fermi_dirac_1_e (double x, gsl_sf_result
          * result)

     These routines compute the complete Fermi-Dirac integral with an
     index of 1, F_1(x) = \int_0^\infty dt (t /(\exp(t-x)+1)).

 -- Function: double gsl_sf_fermi_dirac_2 (double x)
 -- Function: int gsl_sf_fermi_dirac_2_e (double x, gsl_sf_result
          * result)

     These routines compute the complete Fermi-Dirac integral with an
     index of 2, F_2(x) = (1/2) \int_0^\infty dt (t^2 /(\exp(t-x)+1)).

 -- Function: double gsl_sf_fermi_dirac_int (int j, double x)
 -- Function: int gsl_sf_fermi_dirac_int_e (int j, double x,
          gsl_sf_result * result)

     These routines compute the complete Fermi-Dirac integral with an
     integer index of j, F_j(x) = (1/\Gamma(j+1)) \int_0^\infty dt (t^j
     /(\exp(t-x)+1)).

 -- Function: double gsl_sf_fermi_dirac_mhalf (double x)
 -- Function: int gsl_sf_fermi_dirac_mhalf_e (double x, gsl_sf_result
          * result)

     These routines compute the complete Fermi-Dirac integral
     F_{-1/2}(x).

 -- Function: double gsl_sf_fermi_dirac_half (double x)
 -- Function: int gsl_sf_fermi_dirac_half_e (double x, gsl_sf_result
          * result)

     These routines compute the complete Fermi-Dirac integral
     F_{1/2}(x).

 -- Function: double gsl_sf_fermi_dirac_3half (double x)
 -- Function: int gsl_sf_fermi_dirac_3half_e (double x, gsl_sf_result
          * result)

     These routines compute the complete Fermi-Dirac integral
     F_{3/2}(x).


File: gsl-ref.info,  Node: Incomplete Fermi-Dirac Integrals,  Prev: Complete Fermi-Dirac Integrals,  Up: Fermi-Dirac Function

7.18.2 Incomplete Fermi-Dirac Integrals
---------------------------------------

The incomplete Fermi-Dirac integral F_j(x,b) is given by,

     F_j(x,b) := (1/\Gamma(j+1)) \int_b^\infty dt (t^j / (\Exp(t-x) + 1))

 -- Function: double gsl_sf_fermi_dirac_inc_0 (double x, double b)
 -- Function: int gsl_sf_fermi_dirac_inc_0_e (double x, double b,
          gsl_sf_result * result)

     These routines compute the incomplete Fermi-Dirac integral with an
     index of zero, F_0(x,b) = \ln(1 + e^{b-x}) - (b-x)


File: gsl-ref.info,  Node: Gamma and Beta Functions,  Next: Gegenbauer Functions,  Prev: Fermi-Dirac Function,  Up: Special Functions

7.19 Gamma and Beta Functions
=============================

The following routines compute the gamma and beta functions in their
full and incomplete forms, as well as various kinds of factorials.  The
functions described in this section are declared in the header file
‘gsl_sf_gamma.h’.

* Menu:

* Gamma Functions::
* Factorials::
* Pochhammer Symbol::
* Incomplete Gamma Functions::
* Beta Functions::
* Incomplete Beta Function::


File: gsl-ref.info,  Node: Gamma Functions,  Next: Factorials,  Up: Gamma and Beta Functions

7.19.1 Gamma Functions
----------------------

The Gamma function is defined by the following integral,

     \Gamma(x) = \int_0^{\infty} dt t^{x-1} \exp(-t)

It is related to the factorial function by \Gamma(n) = (n-1)! for
positive integer n.  Further information on the Gamma function can be
found in Abramowitz & Stegun, Chapter 6.

 -- Function: double gsl_sf_gamma (double x)
 -- Function: int gsl_sf_gamma_e (double x, gsl_sf_result * result)

     These routines compute the Gamma function \Gamma(x), subject to x
     not being a negative integer or zero.  The function is computed
     using the real Lanczos method.  The maximum value of x such that
     \Gamma(x) is not considered an overflow is given by the macro
     ‘GSL_SF_GAMMA_XMAX’ and is 171.0.

 -- Function: double gsl_sf_lngamma (double x)
 -- Function: int gsl_sf_lngamma_e (double x, gsl_sf_result * result)

     These routines compute the logarithm of the Gamma function,
     \log(\Gamma(x)), subject to x not being a negative integer or zero.
     For x < 0 the real part of \log(\Gamma(x)) is returned, which is
     equivalent to \log(|\Gamma(x)|).  The function is computed using
     the real Lanczos method.

 -- Function: int gsl_sf_lngamma_sgn_e (double x, gsl_sf_result
          * result_lg, double * sgn)

     This routine computes the sign of the gamma function and the
     logarithm of its magnitude, subject to x not being a negative
     integer or zero.  The function is computed using the real Lanczos
     method.  The value of the gamma function and its error can be
     reconstructed using the relation \Gamma(x) = sgn *
     \exp(result\_lg), taking into account the two components of
     ‘result_lg’.

 -- Function: double gsl_sf_gammastar (double x)
 -- Function: int gsl_sf_gammastar_e (double x, gsl_sf_result * result)

     These routines compute the regulated Gamma Function \Gamma^*(x) for
     x > 0.  The regulated gamma function is given by,

          \Gamma^*(x) = \Gamma(x)/(\sqrt{2\pi} x^{(x-1/2)} \exp(-x))
                      = (1 + (1/12x) + ...)  for x \to \infty

     and is a useful suggestion of Temme.

 -- Function: double gsl_sf_gammainv (double x)
 -- Function: int gsl_sf_gammainv_e (double x, gsl_sf_result * result)

     These routines compute the reciprocal of the gamma function,
     1/\Gamma(x) using the real Lanczos method.

 -- Function: int gsl_sf_lngamma_complex_e (double zr, double zi,
          gsl_sf_result * lnr, gsl_sf_result * arg)

     This routine computes \log(\Gamma(z)) for complex z = z_r + i z_i
     and z not a negative integer or zero, using the complex Lanczos
     method.  The returned parameters are lnr = \log|\Gamma(z)| and arg
     = \arg(\Gamma(z)) in (-\pi,\pi].  Note that the phase part (‘arg’)
     is not well-determined when |z| is very large, due to inevitable
     roundoff in restricting to (-\pi,\pi].  This will result in a
     ‘GSL_ELOSS’ error when it occurs.  The absolute value part (‘lnr’),
     however, never suffers from loss of precision.


File: gsl-ref.info,  Node: Factorials,  Next: Pochhammer Symbol,  Prev: Gamma Functions,  Up: Gamma and Beta Functions

7.19.2 Factorials
-----------------

Although factorials can be computed from the Gamma function, using the
relation n! = \Gamma(n+1) for non-negative integer n, it is usually more
efficient to call the functions in this section, particularly for small
values of n, whose factorial values are maintained in hardcoded tables.

 -- Function: double gsl_sf_fact (unsigned int n)
 -- Function: int gsl_sf_fact_e (unsigned int n, gsl_sf_result * result)

     These routines compute the factorial n!.  The factorial is related
     to the Gamma function by n! = \Gamma(n+1).  The maximum value of n
     such that n! is not considered an overflow is given by the macro
     ‘GSL_SF_FACT_NMAX’ and is 170.

 -- Function: double gsl_sf_doublefact (unsigned int n)
 -- Function: int gsl_sf_doublefact_e (unsigned int n, gsl_sf_result
          * result)

     These routines compute the double factorial n!! = n(n-2)(n-4)
     \dots.  The maximum value of n such that n!! is not considered an
     overflow is given by the macro ‘GSL_SF_DOUBLEFACT_NMAX’ and is 297.

 -- Function: double gsl_sf_lnfact (unsigned int n)
 -- Function: int gsl_sf_lnfact_e (unsigned int n, gsl_sf_result
          * result)

     These routines compute the logarithm of the factorial of ‘n’,
     \log(n!).  The algorithm is faster than computing \ln(\Gamma(n+1))
     via *note gsl_sf_lngamma(): 215. for n < 170, but defers for larger
     ‘n’.

 -- Function: double gsl_sf_lndoublefact (unsigned int n)
 -- Function: int gsl_sf_lndoublefact_e (unsigned int n, gsl_sf_result
          * result)

     These routines compute the logarithm of the double factorial of
     ‘n’, \log(n!!).

 -- Function: double gsl_sf_choose (unsigned int n, unsigned int m)
 -- Function: int gsl_sf_choose_e (unsigned int n, unsigned int m,
          gsl_sf_result * result)

     These routines compute the combinatorial factor ‘n choose m’ =
     n!/(m!(n-m)!)

 -- Function: double gsl_sf_lnchoose (unsigned int n, unsigned int m)
 -- Function: int gsl_sf_lnchoose_e (unsigned int n, unsigned int m,
          gsl_sf_result * result)

     These routines compute the logarithm of ‘n choose m’.  This is
     equivalent to the sum \log(n!) - \log(m!) - \log((n-m)!).

 -- Function: double gsl_sf_taylorcoeff (int n, double x)
 -- Function: int gsl_sf_taylorcoeff_e (int n, double x, gsl_sf_result
          * result)

     These routines compute the Taylor coefficient x^n / n! for x \ge 0,
     n \ge 0


File: gsl-ref.info,  Node: Pochhammer Symbol,  Next: Incomplete Gamma Functions,  Prev: Factorials,  Up: Gamma and Beta Functions

7.19.3 Pochhammer Symbol
------------------------

 -- Function: double gsl_sf_poch (double a, double x)
 -- Function: int gsl_sf_poch_e (double a, double x, gsl_sf_result
          * result)

     These routines compute the Pochhammer symbol (a)_x = \Gamma(a +
     x)/\Gamma(a).  The Pochhammer symbol is also known as the Apell
     symbol and sometimes written as (a,x).  When a and a + x are
     negative integers or zero, the limiting value of the ratio is
     returned.

 -- Function: double gsl_sf_lnpoch (double a, double x)
 -- Function: int gsl_sf_lnpoch_e (double a, double x, gsl_sf_result
          * result)

     These routines compute the logarithm of the Pochhammer symbol,
     \log((a)_x) = \log(\Gamma(a + x)/\Gamma(a)).

 -- Function: int gsl_sf_lnpoch_sgn_e (double a, double x, gsl_sf_result
          * result, double * sgn)

     These routines compute the sign of the Pochhammer symbol and the
     logarithm of its magnitude.  The computed parameters are result =
     \log(|(a)_x|) with a corresponding error term, and sgn =
     \sgn((a)_x) where (a)_x = \Gamma(a + x)/\Gamma(a).

 -- Function: double gsl_sf_pochrel (double a, double x)
 -- Function: int gsl_sf_pochrel_e (double a, double x, gsl_sf_result
          * result)

     These routines compute the relative Pochhammer symbol ((a)_x - 1)/x
     where (a)_x = \Gamma(a + x)/\Gamma(a).


File: gsl-ref.info,  Node: Incomplete Gamma Functions,  Next: Beta Functions,  Prev: Pochhammer Symbol,  Up: Gamma and Beta Functions

7.19.4 Incomplete Gamma Functions
---------------------------------

 -- Function: double gsl_sf_gamma_inc (double a, double x)
 -- Function: int gsl_sf_gamma_inc_e (double a, double x, gsl_sf_result
          * result)

     These functions compute the unnormalized incomplete Gamma Function
     \Gamma(a,x) = \int_x^\infty dt t^{(a-1)} \exp(-t) for a real and x
     \ge 0.

 -- Function: double gsl_sf_gamma_inc_Q (double a, double x)
 -- Function: int gsl_sf_gamma_inc_Q_e (double a, double x,
          gsl_sf_result * result)

     These routines compute the normalized incomplete Gamma Function
     Q(a,x) = 1/\Gamma(a) \int_x^\infty dt t^{(a-1)} \exp(-t) for a > 0,
     x \ge 0.

 -- Function: double gsl_sf_gamma_inc_P (double a, double x)
 -- Function: int gsl_sf_gamma_inc_P_e (double a, double x,
          gsl_sf_result * result)

     These routines compute the complementary normalized incomplete
     Gamma Function P(a,x) = 1 - Q(a,x) = 1/\Gamma(a) \int_0^x dt
     t^{(a-1)} \exp(-t) for a > 0, x \ge 0.

     Note that Abramowitz & Stegun call P(a,x) the incomplete gamma
     function (section 6.5).


File: gsl-ref.info,  Node: Beta Functions,  Next: Incomplete Beta Function,  Prev: Incomplete Gamma Functions,  Up: Gamma and Beta Functions

7.19.5 Beta Functions
---------------------

 -- Function: double gsl_sf_beta (double a, double b)
 -- Function: int gsl_sf_beta_e (double a, double b, gsl_sf_result
          * result)

     These routines compute the Beta Function, B(a,b) =
     \Gamma(a)\Gamma(b)/\Gamma(a+b) subject to a and b not being
     negative integers.

 -- Function: double gsl_sf_lnbeta (double a, double b)
 -- Function: int gsl_sf_lnbeta_e (double a, double b, gsl_sf_result
          * result)

     These routines compute the logarithm of the Beta Function,
     \log(B(a,b)) subject to a and b not being negative integers.


File: gsl-ref.info,  Node: Incomplete Beta Function,  Prev: Beta Functions,  Up: Gamma and Beta Functions

7.19.6 Incomplete Beta Function
-------------------------------

 -- Function: double gsl_sf_beta_inc (double a, double b, double x)
 -- Function: int gsl_sf_beta_inc_e (double a, double b, double x,
          gsl_sf_result * result)

     These routines compute the normalized incomplete Beta function
     I_x(a,b) = B_x(a,b) / B(a,b) where

          B_x(a,b) = \int_0^x t^{a-1} (1-t)^{b-1} dt

     for 0 \le x \le 1.  For a > 0, b > 0 the value is computed using a
     continued fraction expansion.  For all other values it is computed
     using the relation

          I_x(a,b,x) = (1/a) x^a 2F1(a,1-b,a+1,x) / B(a,b)


File: gsl-ref.info,  Node: Gegenbauer Functions,  Next: Hermite Polynomials and Functions,  Prev: Gamma and Beta Functions,  Up: Special Functions

7.20 Gegenbauer Functions
=========================

The Gegenbauer polynomials are defined in Abramowitz & Stegun, Chapter
22, where they are known as Ultraspherical polynomials.  The functions
described in this section are declared in the header file
‘gsl_sf_gegenbauer.h’.

 -- Function: double gsl_sf_gegenpoly_1 (double lambda, double x)
 -- Function: double gsl_sf_gegenpoly_2 (double lambda, double x)
 -- Function: double gsl_sf_gegenpoly_3 (double lambda, double x)
 -- Function: int gsl_sf_gegenpoly_1_e (double lambda, double x,
          gsl_sf_result * result)
 -- Function: int gsl_sf_gegenpoly_2_e (double lambda, double x,
          gsl_sf_result * result)
 -- Function: int gsl_sf_gegenpoly_3_e (double lambda, double x,
          gsl_sf_result * result)

     These functions evaluate the Gegenbauer polynomials
     C^{(\lambda)}_n(x) using explicit representations for n = 1, 2, 3.

 -- Function: double gsl_sf_gegenpoly_n (int n, double lambda, double x)
 -- Function: int gsl_sf_gegenpoly_n_e (int n, double lambda, double x,
          gsl_sf_result * result)

     These functions evaluate the Gegenbauer polynomial
     C^{(\lambda)}_n(x) for a specific value of ‘n’, ‘lambda’, ‘x’
     subject to \lambda > -1/2, n \ge 0.

 -- Function: int gsl_sf_gegenpoly_array (int nmax, double lambda,
          double x, double result_array[])

     This function computes an array of Gegenbauer polynomials
     C^{(\lambda)}_n(x) for n = 0, 1, 2, \dots, nmax, subject to \lambda
     > -1/2, nmax \ge 0.


File: gsl-ref.info,  Node: Hermite Polynomials and Functions,  Next: Hypergeometric Functions,  Prev: Gegenbauer Functions,  Up: Special Functions

7.21 Hermite Polynomials and Functions
======================================

Hermite polynomials and functions are discussed in Abramowitz & Stegun,
Chapter 22 and Szego, Gabor (1939, 1958, 1967), Orthogonal Polynomials,
American Mathematical Society.  The Hermite polynomials and functions
are defined in the header file ‘gsl_sf_hermite.h’.

* Menu:

* Hermite Polynomials::
* Derivatives of Hermite Polynomials::
* Hermite Functions::
* Derivatives of Hermite Functions::
* Zeros of Hermite Polynomials and Hermite Functions::


File: gsl-ref.info,  Node: Hermite Polynomials,  Next: Derivatives of Hermite Polynomials,  Up: Hermite Polynomials and Functions

7.21.1 Hermite Polynomials
--------------------------

The Hermite polynomials exist in two variants: the physicist version
H_n(x) and the probabilist version He_n(x).  They are defined by the
derivatives

     H_n(x) = (-1)^n e^{x^2} (d / dx)^n e^{-x^2}
     He_n(x) = (-1)^n e^{x^2/2} (d / dx)^n e^{-x^2/2}

They are connected via

     H_n(x) = 2^{n/2} He_n(\sqrt{2} x)
     He_n(x) = 2^{-n/2} H_n(x / \sqrt{2})

and satisfy the ordinary differential equations

     H_n^{''}(x) - 2x H_n^{'}(x) + 2n H_n(x) = 0
     He_n^{''}(x) - x He_n^{'}(x) + n He_n(x) = 0

 -- Function: double gsl_sf_hermite (const int n, const double x)
 -- Function: int gsl_sf_hermite_e (const int n, const double x,
          gsl_sf_result * result)

     These routines evaluate the physicist Hermite polynomial H_n(x) of
     order ‘n’ at position ‘x’.  If an overflow is detected,
     ‘GSL_EOVRFLW’ is returned without calling the error handler.

 -- Function: int gsl_sf_hermite_array (const int nmax, const double x,
          double * result_array)

     This routine evaluates all physicist Hermite polynomials H_n up to
     order ‘nmax’ at position ‘x’.  The results are stored in
     ‘result_array’.

 -- Function: double gsl_sf_hermite_series (const int n, const double x,
          const double * a)
 -- Function: int gsl_sf_hermite_series_e (const int n, const double x,
          const double * a, gsl_sf_result * result)

     These routines evaluate the series \sum_{j=0}^n a_j H_j(x) with H_j
     being the j-th physicist Hermite polynomial using the Clenshaw
     algorithm.

 -- Function: double gsl_sf_hermite_prob (const int n, const double x)
 -- Function: int gsl_sf_hermite_prob_e (const int n, const double x,
          gsl_sf_result * result)

     These routines evaluate the probabilist Hermite polynomial He_n(x)
     of order ‘n’ at position ‘x’.  If an overflow is detected,
     ‘GSL_EOVRFLW’ is returned without calling the error handler.

 -- Function: int gsl_sf_hermite_prob_array (const int nmax, const
          double x, double * result_array)

     This routine evaluates all probabilist Hermite polynomials He_n(x)
     up to order ‘nmax’ at position ‘x’.  The results are stored in
     ‘result_array’.

 -- Function: double gsl_sf_hermite_prob_series (const int n, const
          double x, const double * a)
 -- Function: int gsl_sf_hermite_prob_series_e (const int n, const
          double x, const double * a, gsl_sf_result * result)

     These routines evaluate the series \sum_{j=0}^n a_j He_j(x) with
     He_j being the j-th probabilist Hermite polynomial using the
     Clenshaw algorithm.


File: gsl-ref.info,  Node: Derivatives of Hermite Polynomials,  Next: Hermite Functions,  Prev: Hermite Polynomials,  Up: Hermite Polynomials and Functions

7.21.2 Derivatives of Hermite Polynomials
-----------------------------------------

 -- Function: double gsl_sf_hermite_deriv (const int m, const int n,
          const double x)
 -- Function: int gsl_sf_hermite_deriv_e (const int m, const int n,
          const double x, gsl_sf_result * result)

     These routines evaluate the ‘m’-th derivative of the physicist
     Hermite polynomial H_n(x) of order ‘n’ at position ‘x’.

 -- Function: int gsl_sf_hermite_array_deriv (const int m, const
          int nmax, const double x, double * result_array)

     This routine evaluates the ‘m’-th derivative of all physicist
     Hermite polynomials H_n(x) from orders 0, \dots, \text{nmax} at
     position ‘x’.  The result d^m/dx^m H_n(x) is stored in
     ‘result_array[n]’.  The output ‘result_array’ must have length at
     least ‘nmax + 1’.

 -- Function: int gsl_sf_hermite_deriv_array (const int mmax, const
          int n, const double x, double * result_array)

     This routine evaluates all derivative orders from 0, \dots,
     \text{mmax} of the physicist Hermite polynomial of order ‘n’, H_n,
     at position ‘x’.  The result d^m/dx^m H_n(x) is stored in
     ‘result_array[m]’.  The output ‘result_array’ must have length at
     least ‘mmax + 1’.

 -- Function: double gsl_sf_hermite_prob_deriv (const int m, const
          int n, const double x)
 -- Function: int gsl_sf_hermite_prob_deriv_e (const int m, const int n,
          const double x, gsl_sf_result * result)

     These routines evaluate the ‘m’-th derivative of the probabilist
     Hermite polynomial He_n(x) of order ‘n’ at position ‘x’.

 -- Function: int gsl_sf_hermite_prob_array_deriv (const int m, const
          int nmax, const double x, double * result_array)

     This routine evaluates the ‘m’-th derivative of all probabilist
     Hermite polynomials He_n(x) from orders 0, \dots, \text{nmax} at
     position ‘x’.  The result d^m/dx^m He_n(x) is stored in
     ‘result_array[n]’.  The output ‘result_array’ must have length at
     least ‘nmax + 1’.

 -- Function: int gsl_sf_hermite_prob_deriv_array (const int mmax, const
          int n, const double x, double * result_array)

     This routine evaluates all derivative orders from 0, \dots,
     \text{mmax} of the probabilist Hermite polynomial of order ‘n’,
     He_n, at position ‘x’.  The result d^m/dx^m He_n(x) is stored in
     ‘result_array[m]’.  The output ‘result_array’ must have length at
     least ‘mmax + 1’.


File: gsl-ref.info,  Node: Hermite Functions,  Next: Derivatives of Hermite Functions,  Prev: Derivatives of Hermite Polynomials,  Up: Hermite Polynomials and Functions

7.21.3 Hermite Functions
------------------------

The Hermite functions are defined by

     \psi_n(x) = ( 2^n n! \sqrt{\pi} )^{-1/2} e^{-x^2/2} H_n(x)

and satisfy the Schrödinger equation for a quantum mechanical harmonic
oscillator

     psi''_n(x) + (2n + 1 - x^2) psi_n(x) = 0

They are orthonormal,

     \int_{-\infty}^{\infty} \psi_m(x) \psi_n(x) dx = \delta_{mn}

and form an orthonormal basis of L^2(\mathbb{R}).  The Hermite functions
are also eigenfunctions of the continuous Fourier transform.  GSL offers
two methods for evaluating the Hermite functions.  The first uses the
standard three-term recurrence relation which has O(n) complexity and is
the most accurate.  The second uses a Cauchy integral approach due to
Bunck (2009) which has O(\sqrt{n}) complexity which represents a
significant speed improvement for large n, although it is slightly less
accurate.

 -- Function: double gsl_sf_hermite_func (const int n, const double x)
 -- Function: int gsl_sf_hermite_func_e (const int n, const double x,
          gsl_sf_result * result)

     These routines evaluate the Hermite function \psi_n(x) of order ‘n’
     at position ‘x’ using a three term recurrence relation.  The
     algorithm complexity is O(n).

 -- Function: double gsl_sf_hermite_func_fast (const int n, const
          double x)
 -- Function: int gsl_sf_hermite_func_fast_e (const int n, const
          double x, gsl_sf_result * result)

     These routines evaluate the Hermite function \psi_n(x) of order ‘n’
     at position ‘x’ using a the Cauchy integral algorithm due to Bunck,
     2009.  The algorithm complexity is O(\sqrt{n}).

 -- Function: int gsl_sf_hermite_func_array (const int nmax, const
          double x, double * result_array)

     This routine evaluates all Hermite functions \psi_n(x) for orders n
     = 0, \dots, \textrm{nmax} at position ‘x’, using the recurrence
     relation algorithm.  The results are stored in ‘result_array’ which
     has length at least ‘nmax + 1’.

 -- Function: double gsl_sf_hermite_func_series (const int n, const
          double x, const double * a)
 -- Function: int gsl_sf_hermite_func_series_e (const int n, const
          double x, const double * a, gsl_sf_result * result)

     These routines evaluate the series \sum_{j=0}^n a_j \psi_j(x) with
     \psi_j being the j-th Hermite function using the Clenshaw
     algorithm.


File: gsl-ref.info,  Node: Derivatives of Hermite Functions,  Next: Zeros of Hermite Polynomials and Hermite Functions,  Prev: Hermite Functions,  Up: Hermite Polynomials and Functions

7.21.4 Derivatives of Hermite Functions
---------------------------------------

 -- Function: double gsl_sf_hermite_func_der (const int m, const int n,
          const double x)
 -- Function: int gsl_sf_hermite_func_der_e (const int m, const int n,
          const double x, gsl_sf_result * result)

     These routines evaluate the ‘m’-th derivative of the Hermite
     function \psi_n(x) of order ‘n’ at position ‘x’.


File: gsl-ref.info,  Node: Zeros of Hermite Polynomials and Hermite Functions,  Prev: Derivatives of Hermite Functions,  Up: Hermite Polynomials and Functions

7.21.5 Zeros of Hermite Polynomials and Hermite Functions
---------------------------------------------------------

These routines calculate the s-th zero of the Hermite
polynomial/function of order n.  Since the zeros are symmetrical around
zero, only positive zeros are calculated, ordered from smallest to
largest, starting from index 1.  Only for odd polynomial orders a zeroth
zero exists, its value always being zero.

 -- Function: double gsl_sf_hermite_zero (const int n, const int s)
 -- Function: int gsl_sf_hermite_zero_e (const int n, const int s,
          gsl_sf_result * result)

     These routines evaluate the ‘s’-th zero of the physicist Hermite
     polynomial H_n(x) of order ‘n’.

 -- Function: double gsl_sf_hermite_prob_zero (const int n, const int s)
 -- Function: int gsl_sf_hermite_prob_zero_e (const int n, const int s,
          gsl_sf_result * result)

     These routines evaluate the ‘s’-th zero of the probabilist Hermite
     polynomial He_n(x) of order ‘n’.

 -- Function: double gsl_sf_hermite_func_zero (const int n, const int s)
 -- Function: int gsl_sf_hermite_func_zero_e (const int n, const int s,
          gsl_sf_result * result)

     These routines evaluate the ‘s’-th zero of the Hermite function
     \psi_n(x) of order ‘n’.


File: gsl-ref.info,  Node: Hypergeometric Functions,  Next: Laguerre Functions,  Prev: Hermite Polynomials and Functions,  Up: Special Functions

7.22 Hypergeometric Functions
=============================

Hypergeometric functions are described in Abramowitz & Stegun, Chapters
13 and 15.  These functions are declared in the header file
‘gsl_sf_hyperg.h’.

 -- Function: double gsl_sf_hyperg_0F1 (double c, double x)
 -- Function: int gsl_sf_hyperg_0F1_e (double c, double x, gsl_sf_result
          * result)

     These routines compute the hypergeometric function

          0F1(c,x)

 -- Function: double gsl_sf_hyperg_1F1_int (int m, int n, double x)
 -- Function: int gsl_sf_hyperg_1F1_int_e (int m, int n, double x,
          gsl_sf_result * result)

     These routines compute the confluent hypergeometric function

          1F1(m,n,x) = M(m,n,x)

     for integer parameters ‘m’, ‘n’.

 -- Function: double gsl_sf_hyperg_1F1 (double a, double b, double x)
 -- Function: int gsl_sf_hyperg_1F1_e (double a, double b, double x,
          gsl_sf_result * result)

     These routines compute the confluent hypergeometric function

          1F1(a,b,x) = M(a,b,x)

     for general parameters ‘a’, ‘b’.

 -- Function: double gsl_sf_hyperg_U_int (int m, int n, double x)
 -- Function: int gsl_sf_hyperg_U_int_e (int m, int n, double x,
          gsl_sf_result * result)

     These routines compute the confluent hypergeometric function
     U(m,n,x) for integer parameters ‘m’, ‘n’.

 -- Function: int gsl_sf_hyperg_U_int_e10_e (int m, int n, double x,
          gsl_sf_result_e10 * result)

     This routine computes the confluent hypergeometric function
     U(m,n,x) for integer parameters ‘m’, ‘n’ using the *note
     gsl_sf_result_e10: ce. type to return a result with extended range.

 -- Function: double gsl_sf_hyperg_U (double a, double b, double x)
 -- Function: int gsl_sf_hyperg_U_e (double a, double b, double x,
          gsl_sf_result * result)

     These routines compute the confluent hypergeometric function
     U(a,b,x).

 -- Function: int gsl_sf_hyperg_U_e10_e (double a, double b, double x,
          gsl_sf_result_e10 * result)

     This routine computes the confluent hypergeometric function
     U(a,b,x) using the *note gsl_sf_result_e10: ce. type to return a
     result with extended range.

 -- Function: double gsl_sf_hyperg_2F1 (double a, double b, double c,
          double x)
 -- Function: int gsl_sf_hyperg_2F1_e (double a, double b, double c,
          double x, gsl_sf_result * result)

     These routines compute the Gauss hypergeometric function

          2F1(a,b,c,x) = F(a,b,c,x)

     for |x| < 1.  If the arguments (a,b,c,x) are too close to a
     singularity then the function can return the error code
     ‘GSL_EMAXITER’ when the series approximation converges too slowly.
     This occurs in the region of x = 1, c - a - b = m for integer m.

 -- Function: double gsl_sf_hyperg_2F1_conj (double aR, double aI,
          double c, double x)
 -- Function: int gsl_sf_hyperg_2F1_conj_e (double aR, double aI,
          double c, double x, gsl_sf_result * result)

     These routines compute the Gauss hypergeometric function

          2F1(a_R + i a_I, aR - i aI, c, x)

     with complex parameters for |x| < 1.

 -- Function: double gsl_sf_hyperg_2F1_renorm (double a, double b,
          double c, double x)
 -- Function: int gsl_sf_hyperg_2F1_renorm_e (double a, double b,
          double c, double x, gsl_sf_result * result)

     These routines compute the renormalized Gauss hypergeometric
     function

          2F1(a,b,c,x) / \Gamma(c)

     for |x| < 1.

 -- Function: double gsl_sf_hyperg_2F1_conj_renorm (double aR,
          double aI, double c, double x)
 -- Function: int gsl_sf_hyperg_2F1_conj_renorm_e (double aR, double aI,
          double c, double x, gsl_sf_result * result)

     These routines compute the renormalized Gauss hypergeometric
     function

          2F1(a_R + i a_I, a_R - i a_I, c, x) / \Gamma(c)

     for |x| < 1.

 -- Function: double gsl_sf_hyperg_2F0 (double a, double b, double x)
 -- Function: int gsl_sf_hyperg_2F0_e (double a, double b, double x,
          gsl_sf_result * result)

     These routines compute the hypergeometric function

          2F0(a,b,x)

     The series representation is a divergent hypergeometric series.
     However, for x < 0 we have

          2F0(a,b,x) = (-1/x)^a U(a,1+a-b,-1/x)


File: gsl-ref.info,  Node: Laguerre Functions,  Next: Lambert W Functions,  Prev: Hypergeometric Functions,  Up: Special Functions

7.23 Laguerre Functions
=======================

The generalized Laguerre polynomials, sometimes referred to as
associated Laguerre polynomials, are defined in terms of confluent
hypergeometric functions as

L^a_n(x) = ((a+1)_n / n!)  1F1(-n,a+1,x)

where (a)_n is the *note Pochhammer symbol: 22d. (rising factorial).
They are related to the plain Laguerre polynomials L_n(x) by L^0_n(x) =
L_n(x) and L^k_n(x) = (-1)^k (d^k/dx^k) L_{(n+k)}(x) For more
information see Abramowitz & Stegun, Chapter 22.

The functions described in this section are declared in the header file
‘gsl_sf_laguerre.h’.

 -- Function: double gsl_sf_laguerre_1 (double a, double x)
 -- Function: double gsl_sf_laguerre_2 (double a, double x)
 -- Function: double gsl_sf_laguerre_3 (double a, double x)
 -- Function: int gsl_sf_laguerre_1_e (double a, double x, gsl_sf_result
          * result)
 -- Function: int gsl_sf_laguerre_2_e (double a, double x, gsl_sf_result
          * result)
 -- Function: int gsl_sf_laguerre_3_e (double a, double x, gsl_sf_result
          * result)

     These routines evaluate the generalized Laguerre polynomials
     L^a_1(x), L^a_2(x), L^a_3(x) using explicit representations.

 -- Function: double gsl_sf_laguerre_n (const int n, const double a,
          const double x)
 -- Function: int gsl_sf_laguerre_n_e (int n, double a, double x,
          gsl_sf_result * result)

     These routines evaluate the generalized Laguerre polynomials
     L^a_n(x) for a > -1, n \ge 0.


File: gsl-ref.info,  Node: Lambert W Functions,  Next: Legendre Functions and Spherical Harmonics,  Prev: Laguerre Functions,  Up: Special Functions

7.24 Lambert W Functions
========================

Lambert’s W functions, W(x), are defined to be solutions of the equation
W(x) \exp(W(x)) = x.  This function has multiple branches for x < 0;
however, it has only two real-valued branches.  We define W_0(x) to be
the principal branch, where W > -1 for x < 0, and W_{-1}(x) to be the
other real branch, where W < -1 for x < 0.  The Lambert functions are
declared in the header file ‘gsl_sf_lambert.h’.

 -- Function: double gsl_sf_lambert_W0 (double x)
 -- Function: int gsl_sf_lambert_W0_e (double x, gsl_sf_result * result)

     These compute the principal branch of the Lambert W function,
     W_0(x).

 -- Function: double gsl_sf_lambert_Wm1 (double x)
 -- Function: int gsl_sf_lambert_Wm1_e (double x, gsl_sf_result
          * result)

     These compute the secondary real-valued branch of the Lambert W
     function, W_{-1}(x).


File: gsl-ref.info,  Node: Legendre Functions and Spherical Harmonics,  Next: Logarithm and Related Functions,  Prev: Lambert W Functions,  Up: Special Functions

7.25 Legendre Functions and Spherical Harmonics
===============================================

The Legendre Functions and Legendre Polynomials are described in
Abramowitz & Stegun, Chapter 8.  These functions are declared in the
header file ‘gsl_sf_legendre.h’.

* Menu:

* Legendre Polynomials::
* Associated Legendre Polynomials and Spherical Harmonics::
* Conical Functions::
* Radial Functions for Hyperbolic Space::


File: gsl-ref.info,  Node: Legendre Polynomials,  Next: Associated Legendre Polynomials and Spherical Harmonics,  Up: Legendre Functions and Spherical Harmonics

7.25.1 Legendre Polynomials
---------------------------

 -- Function: double gsl_sf_legendre_P1 (double x)
 -- Function: double gsl_sf_legendre_P2 (double x)
 -- Function: double gsl_sf_legendre_P3 (double x)
 -- Function: int gsl_sf_legendre_P1_e (double x, gsl_sf_result
          * result)
 -- Function: int gsl_sf_legendre_P2_e (double x, gsl_sf_result
          * result)
 -- Function: int gsl_sf_legendre_P3_e (double x, gsl_sf_result
          * result)

     These functions evaluate the Legendre polynomials P_l(x) using
     explicit representations for l = 1, 2, 3.

 -- Function: double gsl_sf_legendre_Pl (int l, double x)
 -- Function: int gsl_sf_legendre_Pl_e (int l, double x, gsl_sf_result
          * result)

     These functions evaluate the Legendre polynomial P_l(x) for a
     specific value of ‘l’, ‘x’ subject to l \ge 0 and |x| \le 1.

 -- Function: int gsl_sf_legendre_Pl_array (int lmax, double x,
          double result_array[])
 -- Function: int gsl_sf_legendre_Pl_deriv_array (int lmax, double x,
          double result_array[], double result_deriv_array[])

     These functions compute arrays of Legendre polynomials P_l(x) and
     derivatives dP_l(x)/dx for l = 0, \dots, lmax and |x| \le 1.

 -- Function: double gsl_sf_legendre_Q0 (double x)
 -- Function: int gsl_sf_legendre_Q0_e (double x, gsl_sf_result
          * result)

     These routines compute the Legendre function Q_0(x) for x > -1 and
     x \ne 1.

 -- Function: double gsl_sf_legendre_Q1 (double x)
 -- Function: int gsl_sf_legendre_Q1_e (double x, gsl_sf_result
          * result)

     These routines compute the Legendre function Q_1(x) for x > -1 and
     x \ne 1.

 -- Function: double gsl_sf_legendre_Ql (int l, double x)
 -- Function: int gsl_sf_legendre_Ql_e (int l, double x, gsl_sf_result
          * result)

     These routines compute the Legendre function Q_l(x) for x > -1, x
     \ne 1 and l \ge 0.


File: gsl-ref.info,  Node: Associated Legendre Polynomials and Spherical Harmonics,  Next: Conical Functions,  Prev: Legendre Polynomials,  Up: Legendre Functions and Spherical Harmonics

7.25.2 Associated Legendre Polynomials and Spherical Harmonics
--------------------------------------------------------------

The following functions compute the associated Legendre polynomials
P_l^m(x) which are solutions of the differential equation

     (1 - x^2) d^2 P_l^m(x) / dx^2 P_l^m(x) - 2x d/dx P_l^m(x) +
     ( l(l+1) - m^2 / (1 - x^2) ) P_l^m(x) = 0

where the degree l and order m satisfy 0 \le l and 0 \le m \le l.  The
functions P_l^m(x) grow combinatorially with l and can overflow for l
larger than about 150.  Alternatively, one may calculate normalized
associated Legendre polynomials.  There are a number of different
normalization conventions, and these functions can be stably computed up
to degree and order 2700.  The following normalizations are provided:

   * Schmidt semi-normalization

     Schmidt semi-normalized associated Legendre polynomials are often
     used in the magnetics community and are defined as

          S_l^0(x) = P_l^0(x)
          S_l^m(x) = (-1)^m \sqrt((2(l-m)! / (l+m)!)) P_l^m(x), m > 0

     The factor of (-1)^m is called the Condon-Shortley phase factor and
     can be excluded if desired by setting the parameter ‘csphase = 1’
     in the functions below.

   * Spherical Harmonic Normalization

     The associated Legendre polynomials suitable for calculating
     spherical harmonics are defined as

          Y_l^m(x) = (-1)^m \sqrt((2l + 1) * (l-m)! / (4 \pi) / (l+m)!) P_l^m(x)

     where again the phase factor (-1)^m can be included or excluded if
     desired.

   * Full Normalization

     The fully normalized associated Legendre polynomials are defined as

          N_l^m(x) = (-1)^m \sqrt((l + 1/2) (l-m)! / (l+m)!) P_l^m(x)

     and have the property

          \int_{-1}^1 N_l^m(x)^2 dx = 1

The normalized associated Legendre routines below use a recurrence
relation which is stable up to a degree and order of about 2700.  Beyond
this, the computed functions could suffer from underflow leading to
incorrect results.  Routines are provided to compute first and second
derivatives dP_l^m(x)/dx and d^2 P_l^m(x)/dx^2 as well as their
alternate versions d P_l^m(\cos{\theta})/d\theta and d^2
P_l^m(\cos{\theta})/d\theta^2.  While there is a simple scaling
relationship between the two forms, the derivatives involving \theta are
heavily used in spherical harmonic expansions and so these routines are
also provided.

In the functions below, a parameter of type *note gsl_sf_legendre_t:
2ad. specifies the type of normalization to use.  The possible values
are

 -- Type: gsl_sf_legendre_t

     Value                                  Description
                                            
     ---------------------------------------------------------------------------------------------------------------------------
                                            
     ‘GSL_SF_LEGENDRE_NONE’                 The unnormalized associated Legendre polynomials P_l^m(x)
                                            
                                            
     ‘GSL_SF_LEGENDRE_SCHMIDT’              The Schmidt semi-normalized associated Legendre polynomials S_l^m(x)
                                            
                                            
     ‘GSL_SF_LEGENDRE_SPHARM’               The spherical harmonic associated Legendre polynomials Y_l^m(x)
                                            
                                            
     ‘GSL_SF_LEGENDRE_FULL’                 The fully normalized associated Legendre polynomials N_l^m(x)
                                            

 -- Function: int gsl_sf_legendre_array (const gsl_sf_legendre_t norm,
          const size_t lmax, const double x, double result_array[])
 -- Function: int gsl_sf_legendre_array_e (const gsl_sf_legendre_t norm,
          const size_t lmax, const double x, const double csphase,
          double result_array[])

     These functions calculate all normalized associated Legendre
     polynomials for 0 \le l \le lmax and 0 \le m \le l for |x| \le 1.
     The ‘norm’ parameter specifies which normalization is used.  The
     normalized P_l^m(x) values are stored in ‘result_array’, whose
     minimum size can be obtained from calling *note
     gsl_sf_legendre_array_n(): 2b0.  The array index of P_l^m(x) is
     obtained from calling ‘gsl_sf_legendre_array_index(l, m)’.  To
     include or exclude the Condon-Shortley phase factor of (-1)^m, set
     the parameter ‘csphase’ to either -1 or 1 respectively in the ‘_e’
     function.  This factor is excluded by default.

 -- Function: int gsl_sf_legendre_deriv_array (const
          gsl_sf_legendre_t norm, const size_t lmax, const double x,
          double result_array[], double result_deriv_array[])
 -- Function: int gsl_sf_legendre_deriv_array_e (const
          gsl_sf_legendre_t norm, const size_t lmax, const double x,
          const double csphase, double result_array[],
          double result_deriv_array[])

     These functions calculate all normalized associated Legendre
     functions and their first derivatives up to degree ‘lmax’ for |x| <
     1.  The parameter ‘norm’ specifies the normalization used.  The
     normalized P_l^m(x) values and their derivatives dP_l^m(x)/dx are
     stored in ‘result_array’ and ‘result_deriv_array’ respectively.  To
     include or exclude the Condon-Shortley phase factor of (-1)^m, set
     the parameter ‘csphase’ to either -1 or 1 respectively in the ‘_e’
     function.  This factor is excluded by default.

 -- Function: int gsl_sf_legendre_deriv_alt_array (const
          gsl_sf_legendre_t norm, const size_t lmax, const double x,
          double result_array[], double result_deriv_array[])
 -- Function: int gsl_sf_legendre_deriv_alt_array_e (const
          gsl_sf_legendre_t norm, const size_t lmax, const double x,
          const double csphase, double result_array[],
          double result_deriv_array[])

     These functions calculate all normalized associated Legendre
     functions and their (alternate) first derivatives up to degree
     ‘lmax’ for |x| < 1.  The normalized P_l^m(x) values and their
     derivatives dP_l^m(\cos{\theta})/d\theta are stored in
     ‘result_array’ and ‘result_deriv_array’ respectively.  To include
     or exclude the Condon-Shortley phase factor of (-1)^m, set the
     parameter ‘csphase’ to either -1 or 1 respectively in the ‘_e’
     function.  This factor is excluded by default.

 -- Function: int gsl_sf_legendre_deriv2_array (const
          gsl_sf_legendre_t norm, const size_t lmax, const double x,
          double result_array[], double result_deriv_array[],
          double result_deriv2_array[])
 -- Function: int gsl_sf_legendre_deriv2_array_e (const
          gsl_sf_legendre_t norm, const size_t lmax, const double x,
          const double csphase, double result_array[],
          double result_deriv_array[], double result_deriv2_array[])

     These functions calculate all normalized associated Legendre
     functions and their first and second derivatives up to degree
     ‘lmax’ for |x| < 1.  The parameter ‘norm’ specifies the
     normalization used.  The normalized P_l^m(x), their first
     derivatives dP_l^m(x)/dx, and their second derivatives d^2
     P_l^m(x)/dx^2 are stored in ‘result_array’, ‘result_deriv_array’,
     and ‘result_deriv2_array’ respectively.  To include or exclude the
     Condon-Shortley phase factor of (-1)^m, set the parameter ‘csphase’
     to either -1 or 1 respectively in the ‘_e’ function.  This factor
     is excluded by default.

 -- Function: int gsl_sf_legendre_deriv2_alt_array (const
          gsl_sf_legendre_t norm, const size_t lmax, const double x,
          double result_array[], double result_deriv_array[],
          double result_deriv2_array[])
 -- Function: int gsl_sf_legendre_deriv2_alt_array_e (const
          gsl_sf_legendre_t norm, const size_t lmax, const double x,
          const double csphase, double result_array[],
          double result_deriv_array[], double result_deriv2_array[])

     These functions calculate all normalized associated Legendre
     functions and their (alternate) first and second derivatives up to
     degree ‘lmax’ for |x| < 1.  The parameter ‘norm’ specifies the
     normalization used.  The normalized P_l^m(x), their first
     derivatives dP_l^m(\cos{\theta})/d\theta, and their second
     derivatives d^2 P_l^m(\cos{\theta})/d\theta^2 are stored in
     ‘result_array’, ‘result_deriv_array’, and ‘result_deriv2_array’
     respectively.  To include or exclude the Condon-Shortley phase
     factor of (-1)^m, set the parameter ‘csphase’ to either -1 or 1
     respectively in the ‘_e’ function.  This factor is excluded by
     default.

 -- Function: size_t gsl_sf_legendre_array_n (const size_t lmax)

     This function returns the minimum array size for maximum degree
     ‘lmax’ needed for the array versions of the associated Legendre
     functions.  Size is calculated as the total number of P_l^m(x)
     functions, plus extra space for precomputing multiplicative factors
     used in the recurrence relations.

 -- Function: size_t gsl_sf_legendre_array_index (const size_t l, const
          size_t m)

     This function returns the index into ‘result_array’,
     ‘result_deriv_array’, or ‘result_deriv2_array’ corresponding to
     P_l^m(x), P_l^{'m}(x), or P_l^{''m}(x).  The index is given by
     l(l+1)/2 + m.

 -- Function: double gsl_sf_legendre_Plm (int l, int m, double x)
 -- Function: int gsl_sf_legendre_Plm_e (int l, int m, double x,
          gsl_sf_result * result)

     These routines compute the associated Legendre polynomial P_l^m(x)
     for m \ge 0, l \ge m, and |x| \le 1.

 -- Function: double gsl_sf_legendre_sphPlm (int l, int m, double x)
 -- Function: int gsl_sf_legendre_sphPlm_e (int l, int m, double x,
          gsl_sf_result * result)

     These routines compute the normalized associated Legendre
     polynomial \sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)
     suitable for use in spherical harmonics.  The parameters must
     satisfy m \ge 0, l \ge m, and |x| \le 1.  These routines avoid the
     overflows that occur for the standard normalization of P_l^m(x).

 -- Function: int gsl_sf_legendre_Plm_array (int lmax, int m, double x,
          double result_array[])
 -- Function: int gsl_sf_legendre_Plm_deriv_array (int lmax, int m,
          double x, double result_array[], double result_deriv_array[])

     These functions are now deprecated and will be removed in a future
     release; see *note gsl_sf_legendre_array(): 2ae. and *note
     gsl_sf_legendre_deriv_array(): 2b1.

 -- Function: int gsl_sf_legendre_sphPlm_array (int lmax, int m,
          double x, double result_array[])
 -- Function: int gsl_sf_legendre_sphPlm_deriv_array (int lmax, int m,
          double x, double result_array[], double result_deriv_array[])

     These functions are now deprecated and will be removed in a future
     release; see *note gsl_sf_legendre_array(): 2ae. and *note
     gsl_sf_legendre_deriv_array(): 2b1.

 -- Function: int gsl_sf_legendre_array_size (const int lmax, const
          int m)

     This function is now deprecated and will be removed in a future
     release.


File: gsl-ref.info,  Node: Conical Functions,  Next: Radial Functions for Hyperbolic Space,  Prev: Associated Legendre Polynomials and Spherical Harmonics,  Up: Legendre Functions and Spherical Harmonics

7.25.3 Conical Functions
------------------------

The Conical Functions P^\mu_{-(1/2)+i\lambda}(x) and
Q^\mu_{-(1/2)+i\lambda} are described in Abramowitz & Stegun, Section
8.12.

 -- Function: double gsl_sf_conicalP_half (double lambda, double x)
 -- Function: int gsl_sf_conicalP_half_e (double lambda, double x,
          gsl_sf_result * result)

     These routines compute the irregular Spherical Conical Function
     P^{1/2}_{-1/2 + i \lambda}(x) for x > -1.

 -- Function: double gsl_sf_conicalP_mhalf (double lambda, double x)
 -- Function: int gsl_sf_conicalP_mhalf_e (double lambda, double x,
          gsl_sf_result * result)

     These routines compute the regular Spherical Conical Function
     P^{-1/2}_{-1/2 + i \lambda}(x) for x > -1.

 -- Function: double gsl_sf_conicalP_0 (double lambda, double x)
 -- Function: int gsl_sf_conicalP_0_e (double lambda, double x,
          gsl_sf_result * result)

     These routines compute the conical function P^0_{-1/2 + i
     \lambda}(x) for x > -1.

 -- Function: double gsl_sf_conicalP_1 (double lambda, double x)
 -- Function: int gsl_sf_conicalP_1_e (double lambda, double x,
          gsl_sf_result * result)

     These routines compute the conical function P^1_{-1/2 + i
     \lambda}(x) for x > -1.

 -- Function: double gsl_sf_conicalP_sph_reg (int l, double lambda,
          double x)
 -- Function: int gsl_sf_conicalP_sph_reg_e (int l, double lambda,
          double x, gsl_sf_result * result)

     These routines compute the Regular Spherical Conical Function
     P^{-1/2-l}_{-1/2 + i \lambda}(x) for x > -1 and l \ge -1.

 -- Function: double gsl_sf_conicalP_cyl_reg (int m, double lambda,
          double x)
 -- Function: int gsl_sf_conicalP_cyl_reg_e (int m, double lambda,
          double x, gsl_sf_result * result)

     These routines compute the Regular Cylindrical Conical Function
     P^{-m}_{-1/2 + i \lambda}(x) for x > -1 and m \ge -1.


File: gsl-ref.info,  Node: Radial Functions for Hyperbolic Space,  Prev: Conical Functions,  Up: Legendre Functions and Spherical Harmonics

7.25.4 Radial Functions for Hyperbolic Space
--------------------------------------------

The following spherical functions are specializations of Legendre
functions which give the regular eigenfunctions of the Laplacian on a
3-dimensional hyperbolic space H^3.  Of particular interest is the flat
limit, \lambda \to \infty, \eta \to 0, \lambda\eta fixed.

 -- Function: double gsl_sf_legendre_H3d_0 (double lambda, double eta)
 -- Function: int gsl_sf_legendre_H3d_0_e (double lambda, double eta,
          gsl_sf_result * result)

     These routines compute the zeroth radial eigenfunction of the
     Laplacian on the 3-dimensional hyperbolic space,

          L^{H3d}_0(\lambda,\eta) := {\sin(\lambda\eta) \over \lambda\sinh(\eta)}

     for \eta \ge 0.  In the flat limit this takes the form
     L^{H3d}_0(\lambda,\eta) = j_0(\lambda\eta).

 -- Function: double gsl_sf_legendre_H3d_1 (double lambda, double eta)
 -- Function: int gsl_sf_legendre_H3d_1_e (double lambda, double eta,
          gsl_sf_result * result)

     These routines compute the first radial eigenfunction of the
     Laplacian on the 3-dimensional hyperbolic space,

          L^{H3d}_1(\lambda,\eta) := {1\over\sqrt{\lambda^2 + 1}} {\left(\sin(\lambda \eta)\over \lambda \sinh(\eta)\right)} \left(\coth(\eta) - \lambda \cot(\lambda\eta)\right)

     for \eta \ge 0 In the flat limit this takes the form
     L^{H3d}_1(\lambda,\eta) = j_1(\lambda\eta).

 -- Function: double gsl_sf_legendre_H3d (int l, double lambda,
          double eta)
 -- Function: int gsl_sf_legendre_H3d_e (int l, double lambda,
          double eta, gsl_sf_result * result)

     These routines compute the ‘l’-th radial eigenfunction of the
     Laplacian on the 3-dimensional hyperbolic space \eta \ge 0 and l
     \ge 0.  In the flat limit this takes the form
     L^{H3d}_l(\lambda,\eta) = j_l(\lambda\eta).

 -- Function: int gsl_sf_legendre_H3d_array (int lmax, double lambda,
          double eta, double result_array[])

     This function computes an array of radial eigenfunctions L^{H3d}_l(
     \lambda, \eta) for 0 \le l \le lmax.


File: gsl-ref.info,  Node: Logarithm and Related Functions,  Next: Mathieu Functions,  Prev: Legendre Functions and Spherical Harmonics,  Up: Special Functions

7.26 Logarithm and Related Functions
====================================

Information on the properties of the Logarithm function can be found in
Abramowitz & Stegun, Chapter 4.  The functions described in this section
are declared in the header file ‘gsl_sf_log.h’.

 -- Function: double gsl_sf_log (double x)
 -- Function: int gsl_sf_log_e (double x, gsl_sf_result * result)

     These routines compute the logarithm of ‘x’, \log(x), for x > 0.

 -- Function: double gsl_sf_log_abs (double x)
 -- Function: int gsl_sf_log_abs_e (double x, gsl_sf_result * result)

     These routines compute the logarithm of the magnitude of ‘x’,
     \log(|x|), for x \ne 0.

 -- Function: int gsl_sf_complex_log_e (double zr, double zi,
          gsl_sf_result * lnr, gsl_sf_result * theta)

     This routine computes the complex logarithm of z = z_r + i z_i.
     The results are returned as ‘lnr’, ‘theta’ such that \exp(lnr + i
     \theta) = z_r + i z_i, where \theta lies in the range [-\pi,\pi].

 -- Function: double gsl_sf_log_1plusx (double x)
 -- Function: int gsl_sf_log_1plusx_e (double x, gsl_sf_result * result)

     These routines compute \log(1 + x) for x > -1 using an algorithm
     that is accurate for small ‘x’.

 -- Function: double gsl_sf_log_1plusx_mx (double x)
 -- Function: int gsl_sf_log_1plusx_mx_e (double x, gsl_sf_result
          * result)

     These routines compute \log(1 + x) - x for x > -1 using an
     algorithm that is accurate for small ‘x’.


File: gsl-ref.info,  Node: Mathieu Functions,  Next: Power Function,  Prev: Logarithm and Related Functions,  Up: Special Functions

7.27 Mathieu Functions
======================

The routines described in this section compute the angular and radial
Mathieu functions, and their characteristic values.  Mathieu functions
are the solutions of the following two differential equations:

     d^2y/dv^2 + (a - 2q\cos 2v)y = 0
     d^2f/du^2 - (a - 2q\cosh 2u)f = 0

The angular Mathieu functions ce_r(x,q), se_r(x,q) are the even and odd
periodic solutions of the first equation, which is known as Mathieu’s
equation.  These exist only for the discrete sequence of characteristic
values a = a_r(q) (even-periodic) and a = b_r(q) (odd-periodic).

The radial Mathieu functions Mc^{(j)}_{r}(z,q) and Ms^{(j)}_{r}(z,q) are
the solutions of the second equation, which is referred to as Mathieu’s
modified equation.  The radial Mathieu functions of the first, second,
third and fourth kind are denoted by the parameter j, which takes the
value 1, 2, 3 or 4.

For more information on the Mathieu functions, see Abramowitz and
Stegun, Chapter 20.  These functions are defined in the header file
‘gsl_sf_mathieu.h’.

* Menu:

* Mathieu Function Workspace::
* Mathieu Function Characteristic Values::
* Angular Mathieu Functions::
* Radial Mathieu Functions::


File: gsl-ref.info,  Node: Mathieu Function Workspace,  Next: Mathieu Function Characteristic Values,  Up: Mathieu Functions

7.27.1 Mathieu Function Workspace
---------------------------------

The Mathieu functions can be computed for a single order or for multiple
orders, using array-based routines.  The array-based routines require a
preallocated workspace.

 -- Type: gsl_sf_mathieu_workspace

     Workspace required for array-based routines

 -- Function: gsl_sf_mathieu_workspace * gsl_sf_mathieu_alloc (size_t n,
          double qmax)

     This function returns a workspace for the array versions of the
     Mathieu routines.  The arguments n and ‘qmax’ specify the maximum
     order and q-value of Mathieu functions which can be computed with
     this workspace.

 -- Function: void gsl_sf_mathieu_free (gsl_sf_mathieu_workspace * work)

     This function frees the workspace ‘work’.


File: gsl-ref.info,  Node: Mathieu Function Characteristic Values,  Next: Angular Mathieu Functions,  Prev: Mathieu Function Workspace,  Up: Mathieu Functions

7.27.2 Mathieu Function Characteristic Values
---------------------------------------------

 -- Function: int gsl_sf_mathieu_a (int n, double q)
 -- Function: int gsl_sf_mathieu_a_e (int n, double q, gsl_sf_result
          * result)
 -- Function: int gsl_sf_mathieu_b (int n, double q)
 -- Function: int gsl_sf_mathieu_b_e (int n, double q, gsl_sf_result
          * result)

     These routines compute the characteristic values a_n(q), b_n(q) of
     the Mathieu functions ce_n(q,x) and se_n(q,x), respectively.

 -- Function: int gsl_sf_mathieu_a_array (int order_min, int order_max,
          double q, gsl_sf_mathieu_workspace * work,
          double result_array[])
 -- Function: int gsl_sf_mathieu_b_array (int order_min, int order_max,
          double q, gsl_sf_mathieu_workspace * work,
          double result_array[])

     These routines compute a series of Mathieu characteristic values
     a_n(q), b_n(q) for n from ‘order_min’ to ‘order_max’ inclusive,
     storing the results in the array ‘result_array’.


File: gsl-ref.info,  Node: Angular Mathieu Functions,  Next: Radial Mathieu Functions,  Prev: Mathieu Function Characteristic Values,  Up: Mathieu Functions

7.27.3 Angular Mathieu Functions
--------------------------------

 -- Function: int gsl_sf_mathieu_ce (int n, double q, double x)
 -- Function: int gsl_sf_mathieu_ce_e (int n, double q, double x,
          gsl_sf_result * result)
 -- Function: int gsl_sf_mathieu_se (int n, double q, double x)
 -- Function: int gsl_sf_mathieu_se_e (int n, double q, double x,
          gsl_sf_result * result)

     These routines compute the angular Mathieu functions ce_n(q,x) and
     se_n(q,x), respectively.

 -- Function: int gsl_sf_mathieu_ce_array (int nmin, int nmax, double q,
          double x, gsl_sf_mathieu_workspace * work,
          double result_array[])
 -- Function: int gsl_sf_mathieu_se_array (int nmin, int nmax, double q,
          double x, gsl_sf_mathieu_workspace * work,
          double result_array[])

     These routines compute a series of the angular Mathieu functions
     ce_n(q,x) and se_n(q,x) of order n from ‘nmin’ to ‘nmax’ inclusive,
     storing the results in the array ‘result_array’.


File: gsl-ref.info,  Node: Radial Mathieu Functions,  Prev: Angular Mathieu Functions,  Up: Mathieu Functions

7.27.4 Radial Mathieu Functions
-------------------------------

 -- Function: int gsl_sf_mathieu_Mc (int j, int n, double q, double x)
 -- Function: int gsl_sf_mathieu_Mc_e (int j, int n, double q, double x,
          gsl_sf_result * result)
 -- Function: int gsl_sf_mathieu_Ms (int j, int n, double q, double x)
 -- Function: int gsl_sf_mathieu_Ms_e (int j, int n, double q, double x,
          gsl_sf_result * result)

     These routines compute the radial ‘j’-th kind Mathieu functions
     Mc_n^{(j)}(q,x) and Ms_n^{(j)}(q,x) of order ‘n’.

     The allowed values of ‘j’ are 1 and 2.  The functions for j = 3,4
     can be computed as M_n^{(3)} = M_n^{(1)} + iM_n^{(2)} and M_n^{(4)}
     = M_n^{(1)} - iM_n^{(2)}, where M_n^{(j)} = Mc_n^{(j)} or
     Ms_n^{(j)}.

 -- Function: int gsl_sf_mathieu_Mc_array (int j, int nmin, int nmax,
          double q, double x, gsl_sf_mathieu_workspace * work,
          double result_array[])
 -- Function: int gsl_sf_mathieu_Ms_array (int j, int nmin, int nmax,
          double q, double x, gsl_sf_mathieu_workspace * work,
          double result_array[])

     These routines compute a series of the radial Mathieu functions of
     kind ‘j’, with order from ‘nmin’ to ‘nmax’ inclusive, storing the
     results in the array ‘result_array’.


File: gsl-ref.info,  Node: Power Function,  Next: Psi Digamma Function,  Prev: Mathieu Functions,  Up: Special Functions

7.28 Power Function
===================

The following functions are equivalent to the function *note
gsl_pow_int(): 4a. with an error estimate.  These functions are declared
in the header file ‘gsl_sf_pow_int.h’.

 -- Function: double gsl_sf_pow_int (double x, int n)
 -- Function: int gsl_sf_pow_int_e (double x, int n, gsl_sf_result
          * result)

     These routines compute the power x^n for integer ‘n’.  The power is
     computed using the minimum number of multiplications.  For example,
     x^8 is computed as ((x^2)^2)^2, requiring only 3 multiplications.
     For reasons of efficiency, these functions do not check for
     overflow or underflow conditions.  The following is a simple
     example:

          #include <gsl/gsl_sf_pow_int.h>
          /* compute 3.0**12 */
          double y = gsl_sf_pow_int(3.0, 12);


File: gsl-ref.info,  Node: Psi Digamma Function,  Next: Synchrotron Functions,  Prev: Power Function,  Up: Special Functions

7.29 Psi (Digamma) Function
===========================

The polygamma functions of order n are defined by

     \psi^{(n)}(x) = (d/dx)^n \psi(x) = (d/dx)^{n+1} \log(\Gamma(x))

where \psi(x) = \Gamma'(x)/\Gamma(x) is known as the digamma function.
These functions are declared in the header file ‘gsl_sf_psi.h’.

* Menu:

* Digamma Function::
* Trigamma Function::
* Polygamma Function::


File: gsl-ref.info,  Node: Digamma Function,  Next: Trigamma Function,  Up: Psi Digamma Function

7.29.1 Digamma Function
-----------------------

 -- Function: double gsl_sf_psi_int (int n)
 -- Function: int gsl_sf_psi_int_e (int n, gsl_sf_result * result)

     These routines compute the digamma function \psi(n) for positive
     integer ‘n’.  The digamma function is also called the Psi function.

 -- Function: double gsl_sf_psi (double x)
 -- Function: int gsl_sf_psi_e (double x, gsl_sf_result * result)

     These routines compute the digamma function \psi(x) for general
     ‘x’, x \ne 0.

 -- Function: double gsl_sf_psi_1piy (double y)
 -- Function: int gsl_sf_psi_1piy_e (double y, gsl_sf_result * result)

     These routines compute the real part of the digamma function on the
     line 1 + i y, \Re[\psi(1 + i y)].


File: gsl-ref.info,  Node: Trigamma Function,  Next: Polygamma Function,  Prev: Digamma Function,  Up: Psi Digamma Function

7.29.2 Trigamma Function
------------------------

 -- Function: double gsl_sf_psi_1_int (int n)
 -- Function: int gsl_sf_psi_1_int_e (int n, gsl_sf_result * result)

     These routines compute the Trigamma function \psi'(n) for positive
     integer n.

 -- Function: double gsl_sf_psi_1 (double x)
 -- Function: int gsl_sf_psi_1_e (double x, gsl_sf_result * result)

     These routines compute the Trigamma function \psi'(x) for general
     ‘x’.


File: gsl-ref.info,  Node: Polygamma Function,  Prev: Trigamma Function,  Up: Psi Digamma Function

7.29.3 Polygamma Function
-------------------------

 -- Function: double gsl_sf_psi_n (int n, double x)
 -- Function: int gsl_sf_psi_n_e (int n, double x, gsl_sf_result
          * result)

     These routines compute the polygamma function \psi^{(n)}(x) for n
     \ge 0, x > 0.


File: gsl-ref.info,  Node: Synchrotron Functions,  Next: Transport Functions,  Prev: Psi Digamma Function,  Up: Special Functions

7.30 Synchrotron Functions
==========================

The functions described in this section are declared in the header file
‘gsl_sf_synchrotron.h’.

 -- Function: double gsl_sf_synchrotron_1 (double x)
 -- Function: int gsl_sf_synchrotron_1_e (double x, gsl_sf_result
          * result)

     These routines compute the first synchrotron function x
     \int_x^\infty dt K_{5/3}(t) for x \ge 0.

 -- Function: double gsl_sf_synchrotron_2 (double x)
 -- Function: int gsl_sf_synchrotron_2_e (double x, gsl_sf_result
          * result)

     These routines compute the second synchrotron function x K_{2/3}(x)
     for x \ge 0.


File: gsl-ref.info,  Node: Transport Functions,  Next: Trigonometric Functions,  Prev: Synchrotron Functions,  Up: Special Functions

7.31 Transport Functions
========================

The transport functions J(n,x) are defined by the integral
representations

     J(n,x) = \int_0^x t^n e^t /(e^t - 1)^2 dt

They are declared in the header file ‘gsl_sf_transport.h’.

 -- Function: double gsl_sf_transport_2 (double x)
 -- Function: int gsl_sf_transport_2_e (double x, gsl_sf_result
          * result)

     These routines compute the transport function J(2,x).

 -- Function: double gsl_sf_transport_3 (double x)
 -- Function: int gsl_sf_transport_3_e (double x, gsl_sf_result
          * result)

     These routines compute the transport function J(3,x).

 -- Function: double gsl_sf_transport_4 (double x)
 -- Function: int gsl_sf_transport_4_e (double x, gsl_sf_result
          * result)

     These routines compute the transport function J(4,x).

 -- Function: double gsl_sf_transport_5 (double x)
 -- Function: int gsl_sf_transport_5_e (double x, gsl_sf_result
          * result)

     These routines compute the transport function J(5,x).


File: gsl-ref.info,  Node: Trigonometric Functions,  Next: Zeta Functions,  Prev: Transport Functions,  Up: Special Functions

7.32 Trigonometric Functions
============================

The library includes its own trigonometric functions in order to provide
consistency across platforms and reliable error estimates.  These
functions are declared in the header file ‘gsl_sf_trig.h’.

* Menu:

* Circular Trigonometric Functions::
* Trigonometric Functions for Complex Arguments::
* Hyperbolic Trigonometric Functions::
* Conversion Functions::
* Restriction Functions::
* Trigonometric Functions With Error Estimates::


File: gsl-ref.info,  Node: Circular Trigonometric Functions,  Next: Trigonometric Functions for Complex Arguments,  Up: Trigonometric Functions

7.32.1 Circular Trigonometric Functions
---------------------------------------

 -- Function: double gsl_sf_sin (double x)
 -- Function: int gsl_sf_sin_e (double x, gsl_sf_result * result)

     These routines compute the sine function \sin(x).

 -- Function: double gsl_sf_cos (double x)
 -- Function: int gsl_sf_cos_e (double x, gsl_sf_result * result)

     These routines compute the cosine function \cos(x).

 -- Function: double gsl_sf_hypot (double x, double y)
 -- Function: int gsl_sf_hypot_e (double x, double y, gsl_sf_result
          * result)

     These routines compute the hypotenuse function \sqrt{x^2 + y^2}
     avoiding overflow and underflow.

 -- Function: double gsl_sf_sinc (double x)
 -- Function: int gsl_sf_sinc_e (double x, gsl_sf_result * result)

     These routines compute \sinc(x) = \sin(\pi x) / (\pi x) for any
     value of ‘x’.


File: gsl-ref.info,  Node: Trigonometric Functions for Complex Arguments,  Next: Hyperbolic Trigonometric Functions,  Prev: Circular Trigonometric Functions,  Up: Trigonometric Functions

7.32.2 Trigonometric Functions for Complex Arguments
----------------------------------------------------

 -- Function: int gsl_sf_complex_sin_e (double zr, double zi,
          gsl_sf_result * szr, gsl_sf_result * szi)

     This function computes the complex sine, \sin(z_r + i z_i) storing
     the real and imaginary parts in ‘szr’, ‘szi’.

 -- Function: int gsl_sf_complex_cos_e (double zr, double zi,
          gsl_sf_result * czr, gsl_sf_result * czi)

     This function computes the complex cosine, \cos(z_r + i z_i)
     storing the real and imaginary parts in ‘czr’, ‘czi’.

 -- Function: int gsl_sf_complex_logsin_e (double zr, double zi,
          gsl_sf_result * lszr, gsl_sf_result * lszi)

     This function computes the logarithm of the complex sine,
     \log(\sin(z_r + i z_i)) storing the real and imaginary parts in
     ‘lszr’, ‘lszi’.


File: gsl-ref.info,  Node: Hyperbolic Trigonometric Functions,  Next: Conversion Functions,  Prev: Trigonometric Functions for Complex Arguments,  Up: Trigonometric Functions

7.32.3 Hyperbolic Trigonometric Functions
-----------------------------------------

 -- Function: double gsl_sf_lnsinh (double x)
 -- Function: int gsl_sf_lnsinh_e (double x, gsl_sf_result * result)

     These routines compute \log(\sinh(x)) for x > 0.

 -- Function: double gsl_sf_lncosh (double x)
 -- Function: int gsl_sf_lncosh_e (double x, gsl_sf_result * result)

     These routines compute \log(\cosh(x)) for any ‘x’.


File: gsl-ref.info,  Node: Conversion Functions,  Next: Restriction Functions,  Prev: Hyperbolic Trigonometric Functions,  Up: Trigonometric Functions

7.32.4 Conversion Functions
---------------------------

 -- Function: int gsl_sf_polar_to_rect (double r, double theta,
          gsl_sf_result * x, gsl_sf_result * y)

     This function converts the polar coordinates (‘r’, ‘theta’) to
     rectilinear coordinates (‘x’, ‘y’), x = r\cos(\theta), y =
     r\sin(\theta).

 -- Function: int gsl_sf_rect_to_polar (double x, double y,
          gsl_sf_result * r, gsl_sf_result * theta)

     This function converts the rectilinear coordinates (‘x’, ‘y’) to
     polar coordinates (‘r’, ‘theta’), such that x = r\cos(\theta), y =
     r\sin(\theta).  The argument ‘theta’ lies in the range [-\pi, \pi].


File: gsl-ref.info,  Node: Restriction Functions,  Next: Trigonometric Functions With Error Estimates,  Prev: Conversion Functions,  Up: Trigonometric Functions

7.32.5 Restriction Functions
----------------------------

 -- Function: double gsl_sf_angle_restrict_symm (double theta)
 -- Function: int gsl_sf_angle_restrict_symm_e (double * theta)

     These routines force the angle ‘theta’ to lie in the range
     (-\pi,\pi].

     Note that the mathematical value of \pi is slightly greater than
     ‘M_PI’, so the machine numbers ‘M_PI’ and ‘-M_PI’ are included in
     the range.

 -- Function: double gsl_sf_angle_restrict_pos (double theta)
 -- Function: int gsl_sf_angle_restrict_pos_e (double * theta)

     These routines force the angle ‘theta’ to lie in the range [0,
     2\pi).

     Note that the mathematical value of 2\pi is slightly greater than
     ‘2*M_PI’, so the machine number ‘2*M_PI’ is included in the range.


File: gsl-ref.info,  Node: Trigonometric Functions With Error Estimates,  Prev: Restriction Functions,  Up: Trigonometric Functions

7.32.6 Trigonometric Functions With Error Estimates
---------------------------------------------------

 -- Function: int gsl_sf_sin_err_e (double x, double dx, gsl_sf_result
          * result)

     This routine computes the sine of an angle ‘x’ with an associated
     absolute error ‘dx’, \sin(x \pm dx).  Note that this function is
     provided in the error-handling form only since its purpose is to
     compute the propagated error.

 -- Function: int gsl_sf_cos_err_e (double x, double dx, gsl_sf_result
          * result)

     This routine computes the cosine of an angle ‘x’ with an associated
     absolute error ‘dx’, \cos(x \pm dx).  Note that this function is
     provided in the error-handling form only since its purpose is to
     compute the propagated error.


File: gsl-ref.info,  Node: Zeta Functions,  Next: Examples<3>,  Prev: Trigonometric Functions,  Up: Special Functions

7.33 Zeta Functions
===================

The Riemann zeta function is defined in Abramowitz & Stegun, Section
23.2.  The functions described in this section are declared in the
header file ‘gsl_sf_zeta.h’.

* Menu:

* Riemann Zeta Function::
* Riemann Zeta Function Minus One::
* Hurwitz Zeta Function::
* Eta Function::


File: gsl-ref.info,  Node: Riemann Zeta Function,  Next: Riemann Zeta Function Minus One,  Up: Zeta Functions

7.33.1 Riemann Zeta Function
----------------------------

The Riemann zeta function is defined by the infinite sum

     \zeta(s) = \sum_{k=1}^\infty k^{-s}

 -- Function: double gsl_sf_zeta_int (int n)
 -- Function: int gsl_sf_zeta_int_e (int n, gsl_sf_result * result)

     These routines compute the Riemann zeta function \zeta(n) for
     integer ‘n’, n \ne 1.

 -- Function: double gsl_sf_zeta (double s)
 -- Function: int gsl_sf_zeta_e (double s, gsl_sf_result * result)

     These routines compute the Riemann zeta function \zeta(s) for
     arbitrary ‘s’, s \ne 1.


File: gsl-ref.info,  Node: Riemann Zeta Function Minus One,  Next: Hurwitz Zeta Function,  Prev: Riemann Zeta Function,  Up: Zeta Functions

7.33.2 Riemann Zeta Function Minus One
--------------------------------------

For large positive argument, the Riemann zeta function approaches one.
In this region the fractional part is interesting, and therefore we need
a function to evaluate it explicitly.

 -- Function: double gsl_sf_zetam1_int (int n)
 -- Function: int gsl_sf_zetam1_int_e (int n, gsl_sf_result * result)

     These routines compute \zeta(n) - 1 for integer ‘n’, n \ne 1.

 -- Function: double gsl_sf_zetam1 (double s)
 -- Function: int gsl_sf_zetam1_e (double s, gsl_sf_result * result)

     These routines compute \zeta(s) - 1 for arbitrary ‘s’, s \ne 1.


File: gsl-ref.info,  Node: Hurwitz Zeta Function,  Next: Eta Function,  Prev: Riemann Zeta Function Minus One,  Up: Zeta Functions

7.33.3 Hurwitz Zeta Function
----------------------------

The Hurwitz zeta function is defined by

     \zeta(s,q) = \sum_0^\infty (k+q)^{-s}

 -- Function: double gsl_sf_hzeta (double s, double q)
 -- Function: int gsl_sf_hzeta_e (double s, double q, gsl_sf_result
          * result)

     These routines compute the Hurwitz zeta function \zeta(s,q) for s >
     1, q > 0.


File: gsl-ref.info,  Node: Eta Function,  Prev: Hurwitz Zeta Function,  Up: Zeta Functions

7.33.4 Eta Function
-------------------

The eta function is defined by

     \eta(s) = (1-2^{1-s}) \zeta(s)

 -- Function: double gsl_sf_eta_int (int n)
 -- Function: int gsl_sf_eta_int_e (int n, gsl_sf_result * result)

     These routines compute the eta function \eta(n) for integer ‘n’.

 -- Function: double gsl_sf_eta (double s)
 -- Function: int gsl_sf_eta_e (double s, gsl_sf_result * result)

     These routines compute the eta function \eta(s) for arbitrary ‘s’.


File: gsl-ref.info,  Node: Examples<3>,  Next: References and Further Reading<3>,  Prev: Zeta Functions,  Up: Special Functions

7.34 Examples
=============

The following example demonstrates the use of the error handling form of
the special functions, in this case to compute the Bessel function
J_0(5.0),

     #include <stdio.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_sf_bessel.h>

     int
     main (void)
     {
       double x = 5.0;
       gsl_sf_result result;

       double expected = -0.17759677131433830434739701;

       int status = gsl_sf_bessel_J0_e (x, &result);

       printf ("status  = %s\n", gsl_strerror(status));
       printf ("J0(5.0) = %.18f\n"
               "      +/- % .18f\n",
               result.val, result.err);
       printf ("exact   = %.18f\n", expected);
       return status;
     }

Here are the results of running the program,

     status  = success
     J0(5.0) = -0.177596771314338264
           +/-  0.000000000000000193
     exact   = -0.177596771314338292

The next program computes the same quantity using the natural form of
the function.  In this case the error term ‘result.err’ and return
status are not accessible.

     #include <stdio.h>
     #include <gsl/gsl_sf_bessel.h>

     int
     main (void)
     {
       double x = 5.0;
       double expected = -0.17759677131433830434739701;

       double y = gsl_sf_bessel_J0 (x);

       printf ("J0(5.0) = %.18f\n", y);
       printf ("exact   = %.18f\n", expected);
       return 0;
     }

The results of the function are the same,

     J0(5.0) = -0.177596771314338264
     exact   = -0.177596771314338292


File: gsl-ref.info,  Node: References and Further Reading<3>,  Prev: Examples<3>,  Up: Special Functions

7.35 References and Further Reading
===================================

The library follows the conventions of the following book where
possible,

   * Handbook of Mathematical Functions, edited by Abramowitz & Stegun,
     Dover, ISBN 0486612724.

The following papers contain information on the algorithms used to
compute the special functions,

   * Allan J. MacLeod, MISCFUN: A software package to compute uncommon
     special functions.  ACM Trans.  Math.  Soft., vol.: 22, 1996,
     288–301

   * Bunck, B. F., A fast algorithm for evaluation of normalized Hermite
     functions, BIT Numer.  Math, 49: 281-295, 2009.

   * G.N. Watson, A Treatise on the Theory of Bessel Functions, 2nd
     Edition (Cambridge University Press, 1944).

   * G. Nemeth, Mathematical Approximations of Special Functions, Nova
     Science Publishers, ISBN 1-56072-052-2

   * B.C. Carlson, Special Functions of Applied Mathematics (1977)

   * N. M. Temme, Special Functions: An Introduction to the Classical
     Functions of Mathematical Physics (1996), ISBN 978-0471113133.

   * W.J. Thompson, Atlas for Computing Mathematical Functions, John
     Wiley & Sons, New York (1997).

   * Y.Y. Luke, Algorithms for the Computation of Mathematical
     Functions, Academic Press, New York (1977).

   * S. A. Holmes and W. E. Featherstone, A unified approach to the
     Clenshaw summation and the recursive computation of very high
     degree and order normalised associated Legendre functions, Journal
     of Geodesy, 76, pg.  279-299, 2002.


File: gsl-ref.info,  Node: Vectors and Matrices,  Next: Permutations,  Prev: Special Functions,  Up: Top

8 Vectors and Matrices
**********************

The functions described in this chapter provide a simple vector and
matrix interface to ordinary C arrays.  The memory management of these
arrays is implemented using a single underlying type, known as a block.
By writing your functions in terms of vectors and matrices you can pass
a single structure containing both data and dimensions as an argument
without needing additional function parameters.  The structures are
compatible with the vector and matrix formats used by BLAS routines.

* Menu:

* Data types::
* Blocks::
* Vectors::
* Matrices::


File: gsl-ref.info,  Node: Data types,  Next: Blocks,  Up: Vectors and Matrices

8.1 Data types
==============

All the functions are available for each of the standard data-types.
The versions for ‘double’ have the prefix ‘gsl_block’, ‘gsl_vector’ and
‘gsl_matrix’.  Similarly the versions for single-precision ‘float’
arrays have the prefix ‘gsl_block_float’, ‘gsl_vector_float’ and
‘gsl_matrix_float’.  The full list of available types is given below,

Prefix                              Type
                                    
-------------------------------------------------------------
                                    
gsl_block                           double
                                    
                                    
gsl_block_float                     float
                                    
                                    
gsl_block_long_double               long double
                                    
                                    
gsl_block_int                       int
                                    
                                    
gsl_block_uint                      unsigned int
                                    
                                    
gsl_block_long                      long
                                    
                                    
gsl_block_ulong                     unsigned long
                                    
                                    
gsl_block_short                     short
                                    
                                    
gsl_block_ushort                    unsigned short
                                    
                                    
gsl_block_char                      char
                                    
                                    
gsl_block_uchar                     unsigned char
                                    
                                    
gsl_block_complex                   complex double
                                    
                                    
gsl_block_complex_float             complex float
                                    
                                    
gsl_block_complex_long_double       complex long double
                                    

Corresponding types exist for the ‘gsl_vector’ and ‘gsl_matrix’
functions.


File: gsl-ref.info,  Node: Blocks,  Next: Vectors,  Prev: Data types,  Up: Vectors and Matrices

8.2 Blocks
==========

For consistency all memory is allocated through a ‘gsl_block’ structure.
The structure contains two components, the size of an area of memory and
a pointer to the memory.  The ‘gsl_block’ structure looks like this,

 -- Type: gsl_block

          typedef struct
          {
            size_t size;
            double * data;
          } gsl_block;

Vectors and matrices are made by `slicing' an underlying block.  A slice
is a set of elements formed from an initial offset and a combination of
indices and step-sizes.  In the case of a matrix the step-size for the
column index represents the row-length.  The step-size for a vector is
known as the `stride'.

The functions for allocating and deallocating blocks are defined in
‘gsl_block.h’.

* Menu:

* Block allocation::
* Reading and writing blocks::
* Example programs for blocks::


File: gsl-ref.info,  Node: Block allocation,  Next: Reading and writing blocks,  Up: Blocks

8.2.1 Block allocation
----------------------

The functions for allocating memory to a block follow the style of
‘malloc’ and ‘free’.  In addition they also perform their own error
checking.  If there is insufficient memory available to allocate a block
then the functions call the GSL error handler (with an error number of
*note GSL_ENOMEM: 2a.) in addition to returning a null pointer.  Thus if
you use the library error handler to abort your program then it isn’t
necessary to check every ‘alloc’.

 -- Function: gsl_block * gsl_block_alloc (size_t n)

     This function allocates memory for a block of ‘n’ double-precision
     elements, returning a pointer to the block struct.  The block is
     not initialized and so the values of its elements are undefined.
     Use the function *note gsl_block_calloc(): 356. if you want to
     ensure that all the elements are initialized to zero.

     Zero-sized requests are valid and return a non-null result.  A null
     pointer is returned if insufficient memory is available to create
     the block.

 -- Function: gsl_block * gsl_block_calloc (size_t n)

     This function allocates memory for a block and initializes all the
     elements of the block to zero.

 -- Function: void gsl_block_free (gsl_block * b)

     This function frees the memory used by a block ‘b’ previously
     allocated with *note gsl_block_alloc(): 355. or *note
     gsl_block_calloc(): 356.


File: gsl-ref.info,  Node: Reading and writing blocks,  Next: Example programs for blocks,  Prev: Block allocation,  Up: Blocks

8.2.2 Reading and writing blocks
--------------------------------

The library provides functions for reading and writing blocks to a file
as binary data or formatted text.

 -- Function: int gsl_block_fwrite (FILE * stream, const gsl_block * b)

     This function writes the elements of the block ‘b’ to the stream
     ‘stream’ in binary format.  The return value is 0 for success and
     ‘GSL_EFAILED’ if there was a problem writing to the file.  Since
     the data is written in the native binary format it may not be
     portable between different architectures.

 -- Function: int gsl_block_fread (FILE * stream, gsl_block * b)

     This function reads into the block ‘b’ from the open stream
     ‘stream’ in binary format.  The block ‘b’ must be preallocated with
     the correct length since the function uses the size of ‘b’ to
     determine how many bytes to read.  The return value is 0 for
     success and ‘GSL_EFAILED’ if there was a problem reading from the
     file.  The data is assumed to have been written in the native
     binary format on the same architecture.

 -- Function: int gsl_block_fprintf (FILE * stream, const gsl_block * b,
          const char * format)

     This function writes the elements of the block ‘b’ line-by-line to
     the stream ‘stream’ using the format specifier ‘format’, which
     should be one of the ‘%g’, ‘%e’ or ‘%f’ formats for floating point
     numbers and ‘%d’ for integers.  The function returns 0 for success
     and ‘GSL_EFAILED’ if there was a problem writing to the file.

 -- Function: int gsl_block_fscanf (FILE * stream, gsl_block * b)

     This function reads formatted data from the stream ‘stream’ into
     the block ‘b’.  The block ‘b’ must be preallocated with the correct
     length since the function uses the size of ‘b’ to determine how
     many numbers to read.  The function returns 0 for success and
     ‘GSL_EFAILED’ if there was a problem reading from the file.


File: gsl-ref.info,  Node: Example programs for blocks,  Prev: Reading and writing blocks,  Up: Blocks

8.2.3 Example programs for blocks
---------------------------------

The following program shows how to allocate a block,

     #include <stdio.h>
     #include <gsl/gsl_block.h>

     int
     main (void)
     {
       gsl_block * b = gsl_block_alloc (100);

       printf ("length of block = %zu\n", b->size);
       printf ("block data address = %p\n", b->data);

       gsl_block_free (b);
       return 0;
     }

Here is the output from the program,

     length of block = 100
     block data address = 0x804b0d8


File: gsl-ref.info,  Node: Vectors,  Next: Matrices,  Prev: Blocks,  Up: Vectors and Matrices

8.3 Vectors
===========

Vectors are defined by a *note gsl_vector: 35f. structure which
describes a slice of a block.  Different vectors can be created which
point to the same block.  A vector slice is a set of equally-spaced
elements of an area of memory.

The *note gsl_vector: 35f. structure contains five components, the
`size', the `stride', a pointer to the memory where the elements are
stored, ‘data’, a pointer to the block owned by the vector, ‘block’, if
any, and an ownership flag, ‘owner’.  The structure is very simple and
looks like this,

 -- Type: gsl_vector

          typedef struct
          {
            size_t size;
            size_t stride;
            double * data;
            gsl_block * block;
            int owner;
          } gsl_vector;

The ‘size’ is simply the number of vector elements.  The range of valid
indices runs from 0 to ‘size-1’.  The ‘stride’ is the step-size from one
element to the next in physical memory, measured in units of the
appropriate datatype.  The pointer ‘data’ gives the location of the
first element of the vector in memory.  The pointer ‘block’ stores the
location of the memory block in which the vector elements are located
(if any).  If the vector owns this block then the ‘owner’ field is set
to one and the block will be deallocated when the vector is freed.  If
the vector points to a block owned by another object then the ‘owner’
field is zero and any underlying block will not be deallocated with the
vector.

The functions for allocating and accessing vectors are defined in
‘gsl_vector.h’.

* Menu:

* Vector allocation::
* Accessing vector elements::
* Initializing vector elements::
* Reading and writing vectors::
* Vector views::
* Copying vectors::
* Exchanging elements::
* Vector operations::
* Finding maximum and minimum elements of vectors::
* Vector properties::
* Example programs for vectors::


File: gsl-ref.info,  Node: Vector allocation,  Next: Accessing vector elements,  Up: Vectors

8.3.1 Vector allocation
-----------------------

The functions for allocating memory to a vector follow the style of
‘malloc’ and ‘free’.  In addition they also perform their own error
checking.  If there is insufficient memory available to allocate a
vector then the functions call the GSL error handler (with an error
number of *note GSL_ENOMEM: 2a.) in addition to returning a null
pointer.  Thus if you use the library error handler to abort your
program then it isn’t necessary to check every ‘alloc’.

 -- Function: gsl_vector * gsl_vector_alloc (size_t n)

     This function creates a vector of length n, returning a pointer to
     a newly initialized vector struct.  A new block is allocated for
     the elements of the vector, and stored in the ‘block’ component of
     the vector struct.  The block is “owned” by the vector, and will be
     deallocated when the vector is deallocated.  Zero-sized requests
     are valid and return a non-null result.

 -- Function: gsl_vector * gsl_vector_calloc (size_t n)

     This function allocates memory for a vector of length ‘n’ and
     initializes all the elements of the vector to zero.

 -- Function: void gsl_vector_free (gsl_vector * v)

     This function frees a previously allocated vector ‘v’.  If the
     vector was created using *note gsl_vector_alloc(): 361. then the
     block underlying the vector will also be deallocated.  If the
     vector has been created from another object then the memory is
     still owned by that object and will not be deallocated.


File: gsl-ref.info,  Node: Accessing vector elements,  Next: Initializing vector elements,  Prev: Vector allocation,  Up: Vectors

8.3.2 Accessing vector elements
-------------------------------

Unlike Fortran compilers, C compilers do not usually provide support for
range checking of vectors and matrices.  (1) The functions *note
gsl_vector_get(): 365. and *note gsl_vector_set(): 366. can perform
portable range checking for you and report an error if you attempt to
access elements outside the allowed range.

The functions for accessing the elements of a vector or matrix are
defined in ‘gsl_vector.h’ and declared ‘extern inline’ to eliminate
function-call overhead.  You must compile your program with the
preprocessor macro ‘HAVE_INLINE’ defined to use these functions.

 -- Macro: GSL_RANGE_CHECK_OFF

     If necessary you can turn off range checking completely without
     modifying any source files by recompiling your program with the
     preprocessor definition *note GSL_RANGE_CHECK_OFF: 367.  Provided
     your compiler supports inline functions the effect of turning off
     range checking is to replace calls to ‘gsl_vector_get(v,i)’ by
     ‘v->data[i*v->stride]’ and calls to ‘gsl_vector_set(v,i,x)’ by
     ‘v->data[i*v->stride]=x’.  Thus there should be no performance
     penalty for using the range checking functions when range checking
     is turned off.

 -- Macro: GSL_C99_INLINE

     If you use a C99 compiler which requires inline functions in header
     files to be declared ‘inline’ instead of ‘extern inline’, define
     the macro *note GSL_C99_INLINE: 368. (see *note Inline functions:
     15.).  With GCC this is selected automatically when compiling in
     C99 mode (‘-std=c99’).

 -- Variable: gsl_check_range

     If inline functions are not used, calls to the functions *note
     gsl_vector_get(): 365. and *note gsl_vector_set(): 366. will link
     to the compiled versions of these functions in the library itself.
     The range checking in these functions is controlled by the global
     integer variable ‘gsl_check_range’.  It is enabled by default—to
     disable it, set ‘gsl_check_range’ to zero.  Due to function-call
     overhead, there is less benefit in disabling range checking here
     than for inline functions.

 -- Function: double gsl_vector_get (const gsl_vector * v, const
          size_t i)

     This function returns the ‘i’-th element of a vector ‘v’.  If ‘i’
     lies outside the allowed range of 0 to ‘size - 1’ then the error
     handler is invoked and 0 is returned.  An inline version of this
     function is used when ‘HAVE_INLINE’ is defined.

 -- Function: void gsl_vector_set (gsl_vector * v, const size_t i,
          double x)

     This function sets the value of the ‘i’-th element of a vector ‘v’
     to ‘x’.  If ‘i’ lies outside the allowed range of 0 to ‘size - 1’
     then the error handler is invoked.  An inline version of this
     function is used when ‘HAVE_INLINE’ is defined.

 -- Function: double * gsl_vector_ptr (gsl_vector * v, size_t i)
 -- Function: const double * gsl_vector_const_ptr (const gsl_vector * v,
          size_t i)

     These functions return a pointer to the ‘i’-th element of a vector
     ‘v’.  If ‘i’ lies outside the allowed range of 0 to ‘size - 1’ then
     the error handler is invoked and a null pointer is returned.
     Inline versions of these functions are used when ‘HAVE_INLINE’ is
     defined.

   ---------- Footnotes ----------

   (1) Range checking is available in the GNU C Compiler bounds-checking
extension, but it is not part of the default installation of GCC. Memory
accesses can also be checked with Valgrind or the ‘gcc -fmudflap’ memory
protection option.


File: gsl-ref.info,  Node: Initializing vector elements,  Next: Reading and writing vectors,  Prev: Accessing vector elements,  Up: Vectors

8.3.3 Initializing vector elements
----------------------------------

 -- Function: void gsl_vector_set_all (gsl_vector * v, double x)

     This function sets all the elements of the vector ‘v’ to the value
     ‘x’.

 -- Function: void gsl_vector_set_zero (gsl_vector * v)

     This function sets all the elements of the vector ‘v’ to zero.

 -- Function: int gsl_vector_set_basis (gsl_vector * v, size_t i)

     This function makes a basis vector by setting all the elements of
     the vector ‘v’ to zero except for the ‘i’-th element which is set
     to one.


File: gsl-ref.info,  Node: Reading and writing vectors,  Next: Vector views,  Prev: Initializing vector elements,  Up: Vectors

8.3.4 Reading and writing vectors
---------------------------------

The library provides functions for reading and writing vectors to a file
as binary data or formatted text.

 -- Function: int gsl_vector_fwrite (FILE * stream, const gsl_vector
          * v)

     This function writes the elements of the vector ‘v’ to the stream
     ‘stream’ in binary format.  The return value is 0 for success and
     ‘GSL_EFAILED’ if there was a problem writing to the file.  Since
     the data is written in the native binary format it may not be
     portable between different architectures.

 -- Function: int gsl_vector_fread (FILE * stream, gsl_vector * v)

     This function reads into the vector ‘v’ from the open stream
     ‘stream’ in binary format.  The vector ‘v’ must be preallocated
     with the correct length since the function uses the size of ‘v’ to
     determine how many bytes to read.  The return value is 0 for
     success and ‘GSL_EFAILED’ if there was a problem reading from the
     file.  The data is assumed to have been written in the native
     binary format on the same architecture.

 -- Function: int gsl_vector_fprintf (FILE * stream, const gsl_vector
          * v, const char * format)

     This function writes the elements of the vector ‘v’ line-by-line to
     the stream ‘stream’ using the format specifier ‘format’, which
     should be one of the ‘%g’, ‘%e’ or ‘%f’ formats for floating point
     numbers and ‘%d’ for integers.  The function returns 0 for success
     and ‘GSL_EFAILED’ if there was a problem writing to the file.

 -- Function: int gsl_vector_fscanf (FILE * stream, gsl_vector * v)

     This function reads formatted data from the stream ‘stream’ into
     the vector ‘v’.  The vector ‘v’ must be preallocated with the
     correct length since the function uses the size of ‘v’ to determine
     how many numbers to read.  The function returns 0 for success and
     ‘GSL_EFAILED’ if there was a problem reading from the file.


File: gsl-ref.info,  Node: Vector views,  Next: Copying vectors,  Prev: Reading and writing vectors,  Up: Vectors

8.3.5 Vector views
------------------

In addition to creating vectors from slices of blocks it is also
possible to slice vectors and create vector views.  For example, a
subvector of another vector can be described with a view, or two views
can be made which provide access to the even and odd elements of a
vector.

 -- Type: gsl_vector_view
 -- Type: gsl_vector_const_view

     A vector view is a temporary object, stored on the stack, which can
     be used to operate on a subset of vector elements.  Vector views
     can be defined for both constant and non-constant vectors, using
     separate types that preserve constness.  A vector view has the type
     *note gsl_vector_view: 376. and a constant vector view has the type
     *note gsl_vector_const_view: 377.  In both cases the elements of
     the view can be accessed as a *note gsl_vector: 35f. using the
     ‘vector’ component of the view object.  A pointer to a vector of
     type ‘gsl_vector *’ or ‘const gsl_vector *’ can be obtained by
     taking the address of this component with the ‘&’ operator.

     When using this pointer it is important to ensure that the view
     itself remains in scope—the simplest way to do so is by always
     writing the pointer as ‘&view.vector’, and never storing this value
     in another variable.

 -- Function: gsl_vector_view gsl_vector_subvector (gsl_vector * v,
          size_t offset, size_t n)
 -- Function: gsl_vector_const_view gsl_vector_const_subvector (const
          gsl_vector * v, size_t offset, size_t n)

     These functions return a vector view of a subvector of another
     vector ‘v’.  The start of the new vector is offset by ‘offset’
     elements from the start of the original vector.  The new vector has
     ‘n’ elements.  Mathematically, the ‘i’-th element of the new vector
     ‘v'’ is given by:

          v'(i) = v->data[(offset + i)*v->stride]

     where the index ‘i’ runs from 0 to ‘n - 1’.

     The ‘data’ pointer of the returned vector struct is set to null if
     the combined parameters (‘offset’, ‘n’) overrun the end of the
     original vector.

     The new vector is only a view of the block underlying the original
     vector, ‘v’.  The block containing the elements of ‘v’ is not owned
     by the new vector.  When the view goes out of scope the original
     vector ‘v’ and its block will continue to exist.  The original
     memory can only be deallocated by freeing the original vector.  Of
     course, the original vector should not be deallocated while the
     view is still in use.

     The function *note gsl_vector_const_subvector(): 379. is equivalent
     to *note gsl_vector_subvector(): 378. but can be used for vectors
     which are declared ‘const’.

 -- Function: gsl_vector_view gsl_vector_subvector_with_stride
          (gsl_vector * v, size_t offset, size_t stride, size_t n)
 -- Function: gsl_vector_const_view
          gsl_vector_const_subvector_with_stride (const gsl_vector * v,
          size_t offset, size_t stride, size_t n)

     These functions return a vector view of a subvector of another
     vector ‘v’ with an additional stride argument.  The subvector is
     formed in the same way as for *note gsl_vector_subvector(): 378.
     but the new vector has ‘n’ elements with a step-size of ‘stride’
     from one element to the next in the original vector.
     Mathematically, the ‘i’-th element of the new vector ‘v'’ is given
     by:

          v'(i) = v->data[(offset + i*stride)*v->stride]

     where the index ‘i’ runs from 0 to ‘n - 1’.

     Note that subvector views give direct access to the underlying
     elements of the original vector.  For example, the following code
     will zero the even elements of the vector ‘v’ of length ‘n’, while
     leaving the odd elements untouched:

          gsl_vector_view v_even = gsl_vector_subvector_with_stride (v, 0, 2, n/2);
          gsl_vector_set_zero (&v_even.vector);

     A vector view can be passed to any subroutine which takes a vector
     argument just as a directly allocated vector would be, using
     ‘&view.vector’.  For example, the following code computes the norm
     of the odd elements of ‘v’ using the BLAS routine ‘dnrm2’:

          gsl_vector_view v_odd = gsl_vector_subvector_with_stride (v, 1, 2, n/2);
          double r = gsl_blas_dnrm2 (&v_odd.vector);

     The function *note gsl_vector_const_subvector_with_stride(): 37b.
     is equivalent to *note gsl_vector_subvector_with_stride(): 37a. but
     can be used for vectors which are declared ‘const’.

 -- Function: gsl_vector_view gsl_vector_complex_real
          (gsl_vector_complex * v)
 -- Function: gsl_vector_const_view gsl_vector_complex_const_real (const
          gsl_vector_complex * v)

     These functions return a vector view of the real parts of the
     complex vector ‘v’.

     The function *note gsl_vector_complex_const_real(): 37d. is
     equivalent to *note gsl_vector_complex_real(): 37c. but can be used
     for vectors which are declared ‘const’.

 -- Function: gsl_vector_view gsl_vector_complex_imag
          (gsl_vector_complex * v)
 -- Function: gsl_vector_const_view gsl_vector_complex_const_imag (const
          gsl_vector_complex * v)

     These functions return a vector view of the imaginary parts of the
     complex vector ‘v’.

     The function *note gsl_vector_complex_const_imag(): 37f. is
     equivalent to *note gsl_vector_complex_imag(): 37e. but can be used
     for vectors which are declared ‘const’.

 -- Function: gsl_vector_view gsl_vector_view_array (double * base,
          size_t n)
 -- Function: gsl_vector_const_view gsl_vector_const_view_array (const
          double * base, size_t n)

     These functions return a vector view of an array.  The start of the
     new vector is given by ‘base’ and has ‘n’ elements.
     Mathematically, the ‘i’-th element of the new vector ‘v'’ is given
     by:

          v'(i) = base[i]

     where the index ‘i’ runs from 0 to ‘n - 1’.

     The array containing the elements of ‘v’ is not owned by the new
     vector view.  When the view goes out of scope the original array
     will continue to exist.  The original memory can only be
     deallocated by freeing the original pointer ‘base’.  Of course, the
     original array should not be deallocated while the view is still in
     use.

     The function *note gsl_vector_const_view_array(): 381. is
     equivalent to *note gsl_vector_view_array(): 380. but can be used
     for arrays which are declared ‘const’.

 -- Function: gsl_vector_view gsl_vector_view_array_with_stride (double
          * base, size_t stride, size_t n)
 -- Function: gsl_vector_const_view
          gsl_vector_const_view_array_with_stride (const double * base,
          size_t stride, size_t n)

     These functions return a vector view of an array ‘base’ with an
     additional stride argument.  The subvector is formed in the same
     way as for *note gsl_vector_view_array(): 380. but the new vector
     has ‘n’ elements with a step-size of ‘stride’ from one element to
     the next in the original array.  Mathematically, the ‘i’-th element
     of the new vector ‘v'’ is given by:

          v'(i) = base[i*stride]

     where the index ‘i’ runs from 0 to ‘n - 1’.

     Note that the view gives direct access to the underlying elements
     of the original array.  A vector view can be passed to any
     subroutine which takes a vector argument just as a directly
     allocated vector would be, using ‘&view.vector’.

     The function *note gsl_vector_const_view_array_with_stride(): 383.
     is equivalent to *note gsl_vector_view_array_with_stride(): 382.
     but can be used for arrays which are declared ‘const’.


File: gsl-ref.info,  Node: Copying vectors,  Next: Exchanging elements,  Prev: Vector views,  Up: Vectors

8.3.6 Copying vectors
---------------------

Common operations on vectors such as addition and multiplication are
available in the BLAS part of the library (see *note BLAS Support: 11.).
However, it is useful to have a small number of utility functions which
do not require the full BLAS code.  The following functions fall into
this category.

 -- Function: int gsl_vector_memcpy (gsl_vector * dest, const gsl_vector
          * src)

     This function copies the elements of the vector ‘src’ into the
     vector ‘dest’.  The two vectors must have the same length.

 -- Function: int gsl_vector_swap (gsl_vector * v, gsl_vector * w)

     This function exchanges the elements of the vectors ‘v’ and ‘w’ by
     copying.  The two vectors must have the same length.


File: gsl-ref.info,  Node: Exchanging elements,  Next: Vector operations,  Prev: Copying vectors,  Up: Vectors

8.3.7 Exchanging elements
-------------------------

The following functions can be used to exchange, or permute, the
elements of a vector.

 -- Function: int gsl_vector_swap_elements (gsl_vector * v, size_t i,
          size_t j)

     This function exchanges the ‘i’-th and ‘j’-th elements of the
     vector ‘v’ in-place.

 -- Function: int gsl_vector_reverse (gsl_vector * v)

     This function reverses the order of the elements of the vector ‘v’.


File: gsl-ref.info,  Node: Vector operations,  Next: Finding maximum and minimum elements of vectors,  Prev: Exchanging elements,  Up: Vectors

8.3.8 Vector operations
-----------------------

 -- Function: int gsl_vector_add (gsl_vector * a, const gsl_vector * b)

     This function adds the elements of vector ‘b’ to the elements of
     vector ‘a’.  The result a_i \leftarrow a_i + b_i is stored in ‘a’
     and ‘b’ remains unchanged.  The two vectors must have the same
     length.

 -- Function: int gsl_vector_sub (gsl_vector * a, const gsl_vector * b)

     This function subtracts the elements of vector ‘b’ from the
     elements of vector ‘a’.  The result a_i \leftarrow a_i - b_i is
     stored in ‘a’ and ‘b’ remains unchanged.  The two vectors must have
     the same length.

 -- Function: int gsl_vector_mul (gsl_vector * a, const gsl_vector * b)

     This function multiplies the elements of vector ‘a’ by the elements
     of vector ‘b’.  The result a_i \leftarrow a_i * b_i is stored in
     ‘a’ and ‘b’ remains unchanged.  The two vectors must have the same
     length.

 -- Function: int gsl_vector_div (gsl_vector * a, const gsl_vector * b)

     This function divides the elements of vector ‘a’ by the elements of
     vector ‘b’.  The result a_i \leftarrow a_i / b_i is stored in ‘a’
     and ‘b’ remains unchanged.  The two vectors must have the same
     length.

 -- Function: int gsl_vector_scale (gsl_vector * a, const double x)

     This function multiplies the elements of vector ‘a’ by the constant
     factor ‘x’.  The result a_i \leftarrow x a_i is stored in ‘a’.

 -- Function: int gsl_vector_add_constant (gsl_vector * a, const
          double x)

     This function adds the constant value ‘x’ to the elements of the
     vector ‘a’.  The result a_i \leftarrow a_i + x is stored in ‘a’.

 -- Function: int gsl_vector_axpby (const double alpha, const gsl_vector
          * x, const double beta, gsl_vector * y)

     This function performs the operation y \leftarrow \alpha x + \beta
     y.  The vectors ‘x’ and ‘y’ must have the same length.


File: gsl-ref.info,  Node: Finding maximum and minimum elements of vectors,  Next: Vector properties,  Prev: Vector operations,  Up: Vectors

8.3.9 Finding maximum and minimum elements of vectors
-----------------------------------------------------

The following operations are only defined for real vectors.

 -- Function: double gsl_vector_max (const gsl_vector * v)

     This function returns the maximum value in the vector ‘v’.

 -- Function: double gsl_vector_min (const gsl_vector * v)

     This function returns the minimum value in the vector ‘v’.

 -- Function: void gsl_vector_minmax (const gsl_vector * v, double
          * min_out, double * max_out)

     This function returns the minimum and maximum values in the vector
     ‘v’, storing them in ‘min_out’ and ‘max_out’.

 -- Function: size_t gsl_vector_max_index (const gsl_vector * v)

     This function returns the index of the maximum value in the vector
     ‘v’.  When there are several equal maximum elements then the lowest
     index is returned.

 -- Function: size_t gsl_vector_min_index (const gsl_vector * v)

     This function returns the index of the minimum value in the vector
     ‘v’.  When there are several equal minimum elements then the lowest
     index is returned.

 -- Function: void gsl_vector_minmax_index (const gsl_vector * v, size_t
          * imin, size_t * imax)

     This function returns the indices of the minimum and maximum values
     in the vector ‘v’, storing them in ‘imin’ and ‘imax’.  When there
     are several equal minimum or maximum elements then the lowest
     indices are returned.


File: gsl-ref.info,  Node: Vector properties,  Next: Example programs for vectors,  Prev: Finding maximum and minimum elements of vectors,  Up: Vectors

8.3.10 Vector properties
------------------------

The following functions are defined for real and complex vectors.  For
complex vectors both the real and imaginary parts must satisfy the
conditions.

 -- Function: int gsl_vector_isnull (const gsl_vector * v)
 -- Function: int gsl_vector_ispos (const gsl_vector * v)
 -- Function: int gsl_vector_isneg (const gsl_vector * v)
 -- Function: int gsl_vector_isnonneg (const gsl_vector * v)

     These functions return 1 if all the elements of the vector ‘v’ are
     zero, strictly positive, strictly negative, or non-negative
     respectively, and 0 otherwise.

 -- Function: int gsl_vector_equal (const gsl_vector * u, const
          gsl_vector * v)

     This function returns 1 if the vectors ‘u’ and ‘v’ are equal (by
     comparison of element values) and 0 otherwise.


File: gsl-ref.info,  Node: Example programs for vectors,  Prev: Vector properties,  Up: Vectors

8.3.11 Example programs for vectors
-----------------------------------

This program shows how to allocate, initialize and read from a vector
using the functions *note gsl_vector_alloc(): 361, *note
gsl_vector_set(): 366. and *note gsl_vector_get(): 365.

     #include <stdio.h>
     #include <gsl/gsl_vector.h>

     int
     main (void)
     {
       int i;
       gsl_vector * v = gsl_vector_alloc (3);

       for (i = 0; i < 3; i++)
         {
           gsl_vector_set (v, i, 1.23 + i);
         }

       for (i = 0; i < 100; i++) /* OUT OF RANGE ERROR */
         {
           printf ("v_%d = %g\n", i, gsl_vector_get (v, i));
         }

       gsl_vector_free (v);
       return 0;
     }

Here is the output from the program.  The final loop attempts to read
outside the range of the vector ‘v’, and the error is trapped by the
range-checking code in *note gsl_vector_get(): 365.

     $ ./a.out
     v_0 = 1.23
     v_1 = 2.23
     v_2 = 3.23
     gsl: vector_source.c:12: ERROR: index out of range
     Default GSL error handler invoked.
     Aborted (core dumped)

The next program shows how to write a vector to a file.

     #include <stdio.h>
     #include <gsl/gsl_vector.h>

     int
     main (void)
     {
       int i;
       gsl_vector * v = gsl_vector_alloc (100);

       for (i = 0; i < 100; i++)
         {
           gsl_vector_set (v, i, 1.23 + i);
         }

       {
          FILE * f = fopen ("test.dat", "w");
          gsl_vector_fprintf (f, v, "%.5g");
          fclose (f);
       }

       gsl_vector_free (v);
       return 0;
     }

After running this program the file ‘test.dat’ should contain the
elements of ‘v’, written using the format specifier ‘%.5g’.  The vector
could then be read back in using the function ‘gsl_vector_fscanf (f, v)’
as follows:

     #include <stdio.h>
     #include <gsl/gsl_vector.h>

     int
     main (void)
     {
       int i;
       gsl_vector * v = gsl_vector_alloc (10);

       {
          FILE * f = fopen ("test.dat", "r");
          gsl_vector_fscanf (f, v);
          fclose (f);
       }

       for (i = 0; i < 10; i++)
         {
           printf ("%g\n", gsl_vector_get(v, i));
         }

       gsl_vector_free (v);
       return 0;
     }


File: gsl-ref.info,  Node: Matrices,  Prev: Vectors,  Up: Vectors and Matrices

8.4 Matrices
============

Matrices are defined by a *note gsl_matrix: 3a1. structure which
describes a generalized slice of a block.  Like a vector it represents a
set of elements in an area of memory, but uses two indices instead of
one.

 -- Type: gsl_matrix

     The *note gsl_matrix: 3a1. structure contains six components, the
     two dimensions of the matrix, a physical dimension, a pointer to
     the memory where the elements of the matrix are stored, ‘data’, a
     pointer to the block owned by the matrix ‘block’, if any, and an
     ownership flag, ‘owner’.  The physical dimension determines the
     memory layout and can differ from the matrix dimension to allow the
     use of submatrices.  The *note gsl_matrix: 3a1. structure is very
     simple and looks like this:

          typedef struct
          {
            size_t size1;
            size_t size2;
            size_t tda;
            double * data;
            gsl_block * block;
            int owner;
          } gsl_matrix;

Matrices are stored in row-major order, meaning that each row of
elements forms a contiguous block in memory.  This is the standard
“C-language ordering” of two-dimensional arrays.  Note that Fortran
stores arrays in column-major order.  The number of rows is ‘size1’.
The range of valid row indices runs from 0 to ‘size1 - 1’.  Similarly
‘size2’ is the number of columns.  The range of valid column indices
runs from 0 to ‘size2 - 1’.  The physical row dimension ‘tda’, or
`trailing dimension', specifies the size of a row of the matrix as laid
out in memory.

For example, in the following matrix ‘size1’ is 3, ‘size2’ is 4, and
‘tda’ is 8.  The physical memory layout of the matrix begins in the top
left hand-corner and proceeds from left to right along each row in turn.

     00 01 02 03 XX XX XX XX
     10 11 12 13 XX XX XX XX
     20 21 22 23 XX XX XX XX

Each unused memory location is represented by “‘XX’”.  The pointer
‘data’ gives the location of the first element of the matrix in memory.
The pointer ‘block’ stores the location of the memory block in which the
elements of the matrix are located (if any).  If the matrix owns this
block then the ‘owner’ field is set to one and the block will be
deallocated when the matrix is freed.  If the matrix is only a slice of
a block owned by another object then the ‘owner’ field is zero and any
underlying block will not be freed.

The functions for allocating and accessing matrices are defined in
‘gsl_matrix.h’.

* Menu:

* Matrix allocation::
* Accessing matrix elements::
* Initializing matrix elements::
* Reading and writing matrices::
* Matrix views::
* Creating row and column views::
* Copying matrices::
* Copying rows and columns::
* Exchanging rows and columns::
* Matrix operations::
* Finding maximum and minimum elements of matrices::
* Matrix properties::
* Example programs for matrices::
* References and Further Reading: References and Further Reading<4>.


File: gsl-ref.info,  Node: Matrix allocation,  Next: Accessing matrix elements,  Up: Matrices

8.4.1 Matrix allocation
-----------------------

The functions for allocating memory to a matrix follow the style of
‘malloc’ and ‘free’.  They also perform their own error checking.  If
there is insufficient memory available to allocate a matrix then the
functions call the GSL error handler (with an error number of *note
GSL_ENOMEM: 2a.) in addition to returning a null pointer.  Thus if you
use the library error handler to abort your program then it isn’t
necessary to check every ‘alloc’.

 -- Function: gsl_matrix * gsl_matrix_alloc (size_t n1, size_t n2)

     This function creates a matrix of size ‘n1’ rows by ‘n2’ columns,
     returning a pointer to a newly initialized matrix struct.  A new
     block is allocated for the elements of the matrix, and stored in
     the ‘block’ component of the matrix struct.  The block is “owned”
     by the matrix, and will be deallocated when the matrix is
     deallocated.  Requesting zero for ‘n1’ or ‘n2’ is valid and returns
     a non-null result.

 -- Function: gsl_matrix * gsl_matrix_calloc (size_t n1, size_t n2)

     This function allocates memory for a matrix of size ‘n1’ rows by
     ‘n2’ columns and initializes all the elements of the matrix to
     zero.

 -- Function: void gsl_matrix_free (gsl_matrix * m)

     This function frees a previously allocated matrix ‘m’.  If the
     matrix was created using *note gsl_matrix_alloc(): 3a3. then the
     block underlying the matrix will also be deallocated.  If the
     matrix has been created from another object then the memory is
     still owned by that object and will not be deallocated.


File: gsl-ref.info,  Node: Accessing matrix elements,  Next: Initializing matrix elements,  Prev: Matrix allocation,  Up: Matrices

8.4.2 Accessing matrix elements
-------------------------------

The functions for accessing the elements of a matrix use the same range
checking system as vectors.  You can turn off range checking by
recompiling your program with the preprocessor definition *note
GSL_RANGE_CHECK_OFF: 367.

The elements of the matrix are stored in “C-order”, where the second
index moves continuously through memory.  More precisely, the element
accessed by the function ‘gsl_matrix_get(m,i,j)’ and
‘gsl_matrix_set(m,i,j,x)’ is:

     m->data[i * m->tda + j]

where ‘tda’ is the physical row-length of the matrix.

 -- Function: double gsl_matrix_get (const gsl_matrix * m, const
          size_t i, const size_t j)

     This function returns the (i,j)-th element of a matrix ‘m’.  If ‘i’
     or ‘j’ lie outside the allowed range of 0 to ‘n1 - 1’ and 0 to ‘n2
     - 1’ then the error handler is invoked and 0 is returned.  An
     inline version of this function is used when ‘HAVE_INLINE’ is
     defined.

 -- Function: void gsl_matrix_set (gsl_matrix * m, const size_t i, const
          size_t j, double x)

     This function sets the value of the (i,j)-th element of a matrix
     ‘m’ to ‘x’.  If ‘i’ or ‘j’ lies outside the allowed range of 0 to
     ‘n1 - 1’ and 0 to ‘n2 - 1’ then the error handler is invoked.  An
     inline version of this function is used when ‘HAVE_INLINE’ is
     defined.

 -- Function: double * gsl_matrix_ptr (gsl_matrix * m, size_t i,
          size_t j)
 -- Function: const double * gsl_matrix_const_ptr (const gsl_matrix * m,
          size_t i, size_t j)

     These functions return a pointer to the (i,j)-th element of a
     matrix ‘m’.  If ‘i’ or ‘j’ lie outside the allowed range of 0 to
     ‘n1 - 1’ and 0 to ‘n2 - 1’ then the error handler is invoked and a
     null pointer is returned.  Inline versions of these functions are
     used when ‘HAVE_INLINE’ is defined.


File: gsl-ref.info,  Node: Initializing matrix elements,  Next: Reading and writing matrices,  Prev: Accessing matrix elements,  Up: Matrices

8.4.3 Initializing matrix elements
----------------------------------

 -- Function: void gsl_matrix_set_all (gsl_matrix * m, double x)

     This function sets all the elements of the matrix ‘m’ to the value
     ‘x’.

 -- Function: void gsl_matrix_set_zero (gsl_matrix * m)

     This function sets all the elements of the matrix ‘m’ to zero.

 -- Function: void gsl_matrix_set_identity (gsl_matrix * m)

     This function sets the elements of the matrix ‘m’ to the
     corresponding elements of the identity matrix, m(i,j) =
     \delta(i,j), i.e.  a unit diagonal with all off-diagonal elements
     zero.  This applies to both square and rectangular matrices.


File: gsl-ref.info,  Node: Reading and writing matrices,  Next: Matrix views,  Prev: Initializing matrix elements,  Up: Matrices

8.4.4 Reading and writing matrices
----------------------------------

The library provides functions for reading and writing matrices to a
file as binary data or formatted text.

 -- Function: int gsl_matrix_fwrite (FILE * stream, const gsl_matrix
          * m)

     This function writes the elements of the matrix ‘m’ to the stream
     ‘stream’ in binary format.  The return value is 0 for success and
     ‘GSL_EFAILED’ if there was a problem writing to the file.  Since
     the data is written in the native binary format it may not be
     portable between different architectures.

 -- Function: int gsl_matrix_fread (FILE * stream, gsl_matrix * m)

     This function reads into the matrix ‘m’ from the open stream
     ‘stream’ in binary format.  The matrix ‘m’ must be preallocated
     with the correct dimensions since the function uses the size of ‘m’
     to determine how many bytes to read.  The return value is 0 for
     success and ‘GSL_EFAILED’ if there was a problem reading from the
     file.  The data is assumed to have been written in the native
     binary format on the same architecture.

 -- Function: int gsl_matrix_fprintf (FILE * stream, const gsl_matrix
          * m, const char * format)

     This function writes the elements of the matrix ‘m’ line-by-line to
     the stream ‘stream’ using the format specifier ‘format’, which
     should be one of the ‘%g’, ‘%e’ or ‘%f’ formats for floating point
     numbers and ‘%d’ for integers.  The function returns 0 for success
     and ‘GSL_EFAILED’ if there was a problem writing to the file.

 -- Function: int gsl_matrix_fscanf (FILE * stream, gsl_matrix * m)

     This function reads formatted data from the stream ‘stream’ into
     the matrix ‘m’.  The matrix ‘m’ must be preallocated with the
     correct dimensions since the function uses the size of ‘m’ to
     determine how many numbers to read.  The function returns 0 for
     success and ‘GSL_EFAILED’ if there was a problem reading from the
     file.


File: gsl-ref.info,  Node: Matrix views,  Next: Creating row and column views,  Prev: Reading and writing matrices,  Up: Matrices

8.4.5 Matrix views
------------------

 -- Type: gsl_matrix_view
 -- Type: gsl_matrix_const_view

     A matrix view is a temporary object, stored on the stack, which can
     be used to operate on a subset of matrix elements.  Matrix views
     can be defined for both constant and non-constant matrices using
     separate types that preserve constness.  A matrix view has the type
     *note gsl_matrix_view: 3b5. and a constant matrix view has the type
     *note gsl_matrix_const_view: 3b6.  In both cases the elements of
     the view can by accessed using the ‘matrix’ component of the view
     object.  A pointer ‘gsl_matrix *’ or ‘const gsl_matrix *’ can be
     obtained by taking the address of the ‘matrix’ component with the
     ‘&’ operator.  In addition to matrix views it is also possible to
     create vector views of a matrix, such as row or column views.

 -- Function: gsl_matrix_view gsl_matrix_submatrix (gsl_matrix * m,
          size_t k1, size_t k2, size_t n1, size_t n2)
 -- Function: gsl_matrix_const_view gsl_matrix_const_submatrix (const
          gsl_matrix * m, size_t k1, size_t k2, size_t n1, size_t n2)

     These functions return a matrix view of a submatrix of the matrix
     ‘m’.  The upper-left element of the submatrix is the element (‘k1’,
     ‘k2’) of the original matrix.  The submatrix has ‘n1’ rows and ‘n2’
     columns.  The physical number of columns in memory given by ‘tda’
     is unchanged.  Mathematically, the (i,j)-th element of the new
     matrix is given by:

          m'(i,j) = m->data[(k1*m->tda + k2) + i*m->tda + j]

     where the index ‘i’ runs from 0 to ‘n1 - 1’ and the index ‘j’ runs
     from 0 to ‘n2 - 1’.

     The ‘data’ pointer of the returned matrix struct is set to null if
     the combined parameters (‘i’, ‘j’, ‘n1’, ‘n2’, ‘tda’) overrun the
     ends of the original matrix.

     The new matrix view is only a view of the block underlying the
     existing matrix, ‘m’.  The block containing the elements of ‘m’ is
     not owned by the new matrix view.  When the view goes out of scope
     the original matrix ‘m’ and its block will continue to exist.  The
     original memory can only be deallocated by freeing the original
     matrix.  Of course, the original matrix should not be deallocated
     while the view is still in use.

     The function *note gsl_matrix_const_submatrix(): 3b8. is equivalent
     to *note gsl_matrix_submatrix(): 3b7. but can be used for matrices
     which are declared ‘const’.

 -- Function: gsl_matrix_view gsl_matrix_view_array (double * base,
          size_t n1, size_t n2)
 -- Function: gsl_matrix_const_view gsl_matrix_const_view_array (const
          double * base, size_t n1, size_t n2)

     These functions return a matrix view of the array ‘base’.  The
     matrix has ‘n1’ rows and ‘n2’ columns.  The physical number of
     columns in memory is also given by ‘n2’.  Mathematically, the
     (i,j)-th element of the new matrix is given by:

          m'(i,j) = base[i*n2 + j]

     where the index ‘i’ runs from 0 to ‘n1 - 1’ and the index ‘j’ runs
     from 0 to ‘n2 - 1’.

     The new matrix is only a view of the array ‘base’.  When the view
     goes out of scope the original array ‘base’ will continue to exist.
     The original memory can only be deallocated by freeing the original
     array.  Of course, the original array should not be deallocated
     while the view is still in use.

     The function *note gsl_matrix_const_view_array(): 3ba. is
     equivalent to *note gsl_matrix_view_array(): 3b9. but can be used
     for matrices which are declared ‘const’.

 -- Function: gsl_matrix_view gsl_matrix_view_array_with_tda (double
          * base, size_t n1, size_t n2, size_t tda)
 -- Function: gsl_matrix_const_view gsl_matrix_const_view_array_with_tda
          (const double * base, size_t n1, size_t n2, size_t tda)

     These functions return a matrix view of the array ‘base’ with a
     physical number of columns ‘tda’ which may differ from the
     corresponding dimension of the matrix.  The matrix has ‘n1’ rows
     and ‘n2’ columns, and the physical number of columns in memory is
     given by ‘tda’.  Mathematically, the (i,j)-th element of the new
     matrix is given by:

          m'(i,j) = base[i*tda + j]

     where the index ‘i’ runs from 0 to ‘n1 - 1’ and the index ‘j’ runs
     from 0 to ‘n2 - 1’.

     The new matrix is only a view of the array ‘base’.  When the view
     goes out of scope the original array ‘base’ will continue to exist.
     The original memory can only be deallocated by freeing the original
     array.  Of course, the original array should not be deallocated
     while the view is still in use.

     The function *note gsl_matrix_const_view_array_with_tda(): 3bc. is
     equivalent to *note gsl_matrix_view_array_with_tda(): 3bb. but can
     be used for matrices which are declared ‘const’.

 -- Function: gsl_matrix_view gsl_matrix_view_vector (gsl_vector * v,
          size_t n1, size_t n2)
 -- Function: gsl_matrix_const_view gsl_matrix_const_view_vector (const
          gsl_vector * v, size_t n1, size_t n2)

     These functions return a matrix view of the vector ‘v’.  The matrix
     has ‘n1’ rows and ‘n2’ columns.  The vector must have unit stride.
     The physical number of columns in memory is also given by ‘n2’.
     Mathematically, the (i,j)-th element of the new matrix is given by:

          m'(i,j) = v->data[i*n2 + j]

     where the index ‘i’ runs from 0 to ‘n1 - 1’ and the index ‘j’ runs
     from 0 to ‘n2 - 1’.

     The new matrix is only a view of the vector ‘v’.  When the view
     goes out of scope the original vector ‘v’ will continue to exist.
     The original memory can only be deallocated by freeing the original
     vector.  Of course, the original vector should not be deallocated
     while the view is still in use.

     The function *note gsl_matrix_const_view_vector(): 3be. is
     equivalent to *note gsl_matrix_view_vector(): 3bd. but can be used
     for matrices which are declared ‘const’.

 -- Function: gsl_matrix_view gsl_matrix_view_vector_with_tda
          (gsl_vector * v, size_t n1, size_t n2, size_t tda)
 -- Function: gsl_matrix_const_view
          gsl_matrix_const_view_vector_with_tda (const gsl_vector * v,
          size_t n1, size_t n2, size_t tda)

     These functions return a matrix view of the vector ‘v’ with a
     physical number of columns ‘tda’ which may differ from the
     corresponding matrix dimension.  The vector must have unit stride.
     The matrix has ‘n1’ rows and ‘n2’ columns, and the physical number
     of columns in memory is given by ‘tda’.  Mathematically, the
     (i,j)-th element of the new matrix is given by:

          m'(i,j) = v->data[i*tda + j]

     where the index ‘i’ runs from 0 to ‘n1 - 1’ and the index ‘j’ runs
     from 0 to ‘n2 - 1’.

     The new matrix is only a view of the vector ‘v’.  When the view
     goes out of scope the original vector ‘v’ will continue to exist.
     The original memory can only be deallocated by freeing the original
     vector.  Of course, the original vector should not be deallocated
     while the view is still in use.

     The function *note gsl_matrix_const_view_vector_with_tda(): 3c0. is
     equivalent to *note gsl_matrix_view_vector_with_tda(): 3bf. but can
     be used for matrices which are declared ‘const’.


File: gsl-ref.info,  Node: Creating row and column views,  Next: Copying matrices,  Prev: Matrix views,  Up: Matrices

8.4.6 Creating row and column views
-----------------------------------

In general there are two ways to access an object, by reference or by
copying.  The functions described in this section create vector views
which allow access to a row or column of a matrix by reference.
Modifying elements of the view is equivalent to modifying the matrix,
since both the vector view and the matrix point to the same memory
block.

 -- Function: gsl_vector_view gsl_matrix_row (gsl_matrix * m, size_t i)
 -- Function: gsl_vector_const_view gsl_matrix_const_row (const
          gsl_matrix * m, size_t i)

     These functions return a vector view of the ‘i’-th row of the
     matrix ‘m’.  The ‘data’ pointer of the new vector is set to null if
     ‘i’ is out of range.

     The function *note gsl_matrix_const_row(): 3c3. is equivalent to
     *note gsl_matrix_row(): 3c2. but can be used for matrices which are
     declared ‘const’.

 -- Function: gsl_vector_view gsl_matrix_column (gsl_matrix * m,
          size_t j)
 -- Function: gsl_vector_const_view gsl_matrix_const_column (const
          gsl_matrix * m, size_t j)

     These functions return a vector view of the ‘j’-th column of the
     matrix ‘m’.  The ‘data’ pointer of the new vector is set to null if
     ‘j’ is out of range.

     The function *note gsl_matrix_const_column(): 3c5. is equivalent to
     *note gsl_matrix_column(): 3c4. but can be used for matrices which
     are declared ‘const’.

 -- Function: gsl_vector_view gsl_matrix_subrow (gsl_matrix * m,
          size_t i, size_t offset, size_t n)
 -- Function: gsl_vector_const_view gsl_matrix_const_subrow (const
          gsl_matrix * m, size_t i, size_t offset, size_t n)

     These functions return a vector view of the ‘i’-th row of the
     matrix ‘m’ beginning at ‘offset’ elements past the first column and
     containing ‘n’ elements.  The ‘data’ pointer of the new vector is
     set to null if ‘i’, ‘offset’, or ‘n’ are out of range.

     The function *note gsl_matrix_const_subrow(): 3c7. is equivalent to
     *note gsl_matrix_subrow(): 3c6. but can be used for matrices which
     are declared ‘const’.

 -- Function: gsl_vector_view gsl_matrix_subcolumn (gsl_matrix * m,
          size_t j, size_t offset, size_t n)
 -- Function: gsl_vector_const_view gsl_matrix_const_subcolumn (const
          gsl_matrix * m, size_t j, size_t offset, size_t n)

     These functions return a vector view of the ‘j’-th column of the
     matrix ‘m’ beginning at ‘offset’ elements past the first row and
     containing ‘n’ elements.  The ‘data’ pointer of the new vector is
     set to null if ‘j’, ‘offset’, or ‘n’ are out of range.

     The function *note gsl_matrix_const_subcolumn(): 3c9. is equivalent
     to *note gsl_matrix_subcolumn(): 3c8. but can be used for matrices
     which are declared ‘const’.

 -- Function: gsl_vector_view gsl_matrix_diagonal (gsl_matrix * m)
 -- Function: gsl_vector_const_view gsl_matrix_const_diagonal (const
          gsl_matrix * m)

     These functions return a vector view of the diagonal of the matrix
     ‘m’.  The matrix ‘m’ is not required to be square.  For a
     rectangular matrix the length of the diagonal is the same as the
     smaller dimension of the matrix.

     The function *note gsl_matrix_const_diagonal(): 3cb. is equivalent
     to *note gsl_matrix_diagonal(): 3ca. but can be used for matrices
     which are declared ‘const’.

 -- Function: gsl_vector_view gsl_matrix_subdiagonal (gsl_matrix * m,
          size_t k)
 -- Function: gsl_vector_const_view gsl_matrix_const_subdiagonal (const
          gsl_matrix * m, size_t k)

     These functions return a vector view of the ‘k’-th subdiagonal of
     the matrix ‘m’.  The matrix ‘m’ is not required to be square.  The
     diagonal of the matrix corresponds to k = 0.

     The function *note gsl_matrix_const_subdiagonal(): 3cd. is
     equivalent to *note gsl_matrix_subdiagonal(): 3cc. but can be used
     for matrices which are declared ‘const’.

 -- Function: gsl_vector_view gsl_matrix_superdiagonal (gsl_matrix * m,
          size_t k)
 -- Function: gsl_vector_const_view gsl_matrix_const_superdiagonal
          (const gsl_matrix * m, size_t k)

     These functions return a vector view of the ‘k’-th superdiagonal of
     the matrix ‘m’.  The matrix ‘m’ is not required to be square.  The
     diagonal of the matrix corresponds to k = 0.

     The function *note gsl_matrix_const_superdiagonal(): 3cf. is
     equivalent to *note gsl_matrix_superdiagonal(): 3ce. but can be
     used for matrices which are declared ‘const’.


File: gsl-ref.info,  Node: Copying matrices,  Next: Copying rows and columns,  Prev: Creating row and column views,  Up: Matrices

8.4.7 Copying matrices
----------------------

 -- Function: int gsl_matrix_memcpy (gsl_matrix * dest, const gsl_matrix
          * src)

     This function copies the elements of the matrix ‘src’ into the
     matrix ‘dest’.  The two matrices must have the same size.

 -- Function: int gsl_matrix_swap (gsl_matrix * m1, gsl_matrix * m2)

     This function exchanges the elements of the matrices ‘m1’ and ‘m2’
     by copying.  The two matrices must have the same size.


File: gsl-ref.info,  Node: Copying rows and columns,  Next: Exchanging rows and columns,  Prev: Copying matrices,  Up: Matrices

8.4.8 Copying rows and columns
------------------------------

The functions described in this section copy a row or column of a matrix
into a vector.  This allows the elements of the vector and the matrix to
be modified independently.  Note that if the matrix and the vector point
to overlapping regions of memory then the result will be undefined.  The
same effect can be achieved with more generality using *note
gsl_vector_memcpy(): 385. with vector views of rows and columns.

 -- Function: int gsl_matrix_get_row (gsl_vector * v, const gsl_matrix
          * m, size_t i)

     This function copies the elements of the ‘i’-th row of the matrix
     ‘m’ into the vector ‘v’.  The length of the vector must be the same
     as the length of the row.

 -- Function: int gsl_matrix_get_col (gsl_vector * v, const gsl_matrix
          * m, size_t j)

     This function copies the elements of the ‘j’-th column of the
     matrix ‘m’ into the vector ‘v’.  The length of the vector must be
     the same as the length of the column.

 -- Function: int gsl_matrix_set_row (gsl_matrix * m, size_t i, const
          gsl_vector * v)

     This function copies the elements of the vector ‘v’ into the ‘i’-th
     row of the matrix ‘m’.  The length of the vector must be the same
     as the length of the row.

 -- Function: int gsl_matrix_set_col (gsl_matrix * m, size_t j, const
          gsl_vector * v)

     This function copies the elements of the vector ‘v’ into the ‘j’-th
     column of the matrix ‘m’.  The length of the vector must be the
     same as the length of the column.


File: gsl-ref.info,  Node: Exchanging rows and columns,  Next: Matrix operations,  Prev: Copying rows and columns,  Up: Matrices

8.4.9 Exchanging rows and columns
---------------------------------

The following functions can be used to exchange the rows and columns of
a matrix.

 -- Function: int gsl_matrix_swap_rows (gsl_matrix * m, size_t i,
          size_t j)

     This function exchanges the ‘i’-th and ‘j’-th rows of the matrix
     ‘m’ in-place.

 -- Function: int gsl_matrix_swap_columns (gsl_matrix * m, size_t i,
          size_t j)

     This function exchanges the ‘i’-th and ‘j’-th columns of the matrix
     ‘m’ in-place.

 -- Function: int gsl_matrix_swap_rowcol (gsl_matrix * m, size_t i,
          size_t j)

     This function exchanges the ‘i’-th row and ‘j’-th column of the
     matrix ‘m’ in-place.  The matrix must be square for this operation
     to be possible.

 -- Function: int gsl_matrix_transpose_memcpy (gsl_matrix * dest, const
          gsl_matrix * src)

     This function makes the matrix ‘dest’ the transpose of the matrix
     ‘src’ by copying the elements of ‘src’ into ‘dest’.  This function
     works for all matrices provided that the dimensions of the matrix
     ‘dest’ match the transposed dimensions of the matrix ‘src’.

 -- Function: int gsl_matrix_transpose (gsl_matrix * m)

     This function replaces the matrix ‘m’ by its transpose by copying
     the elements of the matrix in-place.  The matrix must be square for
     this operation to be possible.


File: gsl-ref.info,  Node: Matrix operations,  Next: Finding maximum and minimum elements of matrices,  Prev: Exchanging rows and columns,  Up: Matrices

8.4.10 Matrix operations
------------------------

The following operations are defined for real and complex matrices.

 -- Function: int gsl_matrix_add (gsl_matrix * a, const gsl_matrix * b)

     This function adds the elements of matrix ‘b’ to the elements of
     matrix ‘a’.  The result a(i,j) \leftarrow a(i,j) + b(i,j) is stored
     in ‘a’ and ‘b’ remains unchanged.  The two matrices must have the
     same dimensions.

 -- Function: int gsl_matrix_sub (gsl_matrix * a, const gsl_matrix * b)

     This function subtracts the elements of matrix ‘b’ from the
     elements of matrix ‘a’.  The result a(i,j) \leftarrow a(i,j) -
     b(i,j) is stored in ‘a’ and ‘b’ remains unchanged.  The two
     matrices must have the same dimensions.

 -- Function: int gsl_matrix_mul_elements (gsl_matrix * a, const
          gsl_matrix * b)

     This function multiplies the elements of matrix ‘a’ by the elements
     of matrix ‘b’.  The result a(i,j) \leftarrow a(i,j) * b(i,j) is
     stored in ‘a’ and ‘b’ remains unchanged.  The two matrices must
     have the same dimensions.

 -- Function: int gsl_matrix_div_elements (gsl_matrix * a, const
          gsl_matrix * b)

     This function divides the elements of matrix ‘a’ by the elements of
     matrix ‘b’.  The result a(i,j) \leftarrow a(i,j) / b(i,j) is stored
     in ‘a’ and ‘b’ remains unchanged.  The two matrices must have the
     same dimensions.

 -- Function: int gsl_matrix_scale (gsl_matrix * a, const double x)

     This function multiplies the elements of matrix ‘a’ by the constant
     factor ‘x’.  The result a(i,j) \leftarrow x a(i,j) is stored in
     ‘a’.

 -- Function: int gsl_matrix_add_constant (gsl_matrix * a, const
          double x)

     This function adds the constant value ‘x’ to the elements of the
     matrix ‘a’.  The result a(i,j) \leftarrow a(i,j) + x is stored in
     ‘a’.


File: gsl-ref.info,  Node: Finding maximum and minimum elements of matrices,  Next: Matrix properties,  Prev: Matrix operations,  Up: Matrices

8.4.11 Finding maximum and minimum elements of matrices
-------------------------------------------------------

The following operations are only defined for real matrices.

 -- Function: double gsl_matrix_max (const gsl_matrix * m)

     This function returns the maximum value in the matrix ‘m’.

 -- Function: double gsl_matrix_min (const gsl_matrix * m)

     This function returns the minimum value in the matrix ‘m’.

 -- Function: void gsl_matrix_minmax (const gsl_matrix * m, double
          * min_out, double * max_out)

     This function returns the minimum and maximum values in the matrix
     ‘m’, storing them in ‘min_out’ and ‘max_out’.

 -- Function: void gsl_matrix_max_index (const gsl_matrix * m, size_t
          * imax, size_t * jmax)

     This function returns the indices of the maximum value in the
     matrix ‘m’, storing them in ‘imax’ and ‘jmax’.  When there are
     several equal maximum elements then the first element found is
     returned, searching in row-major order.

 -- Function: void gsl_matrix_min_index (const gsl_matrix * m, size_t
          * imin, size_t * jmin)

     This function returns the indices of the minimum value in the
     matrix ‘m’, storing them in ‘imin’ and ‘jmin’.  When there are
     several equal minimum elements then the first element found is
     returned, searching in row-major order.

 -- Function: void gsl_matrix_minmax_index (const gsl_matrix * m, size_t
          * imin, size_t * jmin, size_t * imax, size_t * jmax)

     This function returns the indices of the minimum and maximum values
     in the matrix ‘m’, storing them in (‘imin’, ‘jmin’) and (‘imax’,
     ‘jmax’).  When there are several equal minimum or maximum elements
     then the first elements found are returned, searching in row-major
     order.


File: gsl-ref.info,  Node: Matrix properties,  Next: Example programs for matrices,  Prev: Finding maximum and minimum elements of matrices,  Up: Matrices

8.4.12 Matrix properties
------------------------

The following functions are defined for real and complex matrices.  For
complex matrices both the real and imaginary parts must satisfy the
conditions.

 -- Function: int gsl_matrix_isnull (const gsl_matrix * m)
 -- Function: int gsl_matrix_ispos (const gsl_matrix * m)
 -- Function: int gsl_matrix_isneg (const gsl_matrix * m)
 -- Function: int gsl_matrix_isnonneg (const gsl_matrix * m)

     These functions return 1 if all the elements of the matrix ‘m’ are
     zero, strictly positive, strictly negative, or non-negative
     respectively, and 0 otherwise.  To test whether a matrix is
     positive-definite, use the *note Cholesky decomposition: 3f1.

 -- Function: int gsl_matrix_equal (const gsl_matrix * a, const
          gsl_matrix * b)

     This function returns 1 if the matrices ‘a’ and ‘b’ are equal (by
     comparison of element values) and 0 otherwise.


File: gsl-ref.info,  Node: Example programs for matrices,  Next: References and Further Reading<4>,  Prev: Matrix properties,  Up: Matrices

8.4.13 Example programs for matrices
------------------------------------

The program below shows how to allocate, initialize and read from a
matrix using the functions *note gsl_matrix_alloc(): 3a3, *note
gsl_matrix_set(): 3a8. and *note gsl_matrix_get(): 3a7.

     #include <stdio.h>
     #include <gsl/gsl_matrix.h>

     int
     main (void)
     {
       int i, j;
       gsl_matrix * m = gsl_matrix_alloc (10, 3);

       for (i = 0; i < 10; i++)
         for (j = 0; j < 3; j++)
           gsl_matrix_set (m, i, j, 0.23 + 100*i + j);

       for (i = 0; i < 100; i++)  /* OUT OF RANGE ERROR */
         for (j = 0; j < 3; j++)
           printf ("m(%d,%d) = %g\n", i, j,
                   gsl_matrix_get (m, i, j));

       gsl_matrix_free (m);

       return 0;
     }

Here is the output from the program.  The final loop attempts to read
outside the range of the matrix ‘m’, and the error is trapped by the
range-checking code in *note gsl_matrix_get(): 3a7.

     $ ./a.out
     m(0,0) = 0.23
     m(0,1) = 1.23
     m(0,2) = 2.23
     m(1,0) = 100.23
     m(1,1) = 101.23
     m(1,2) = 102.23
     ...
     m(9,2) = 902.23
     gsl: matrix_source.c:13: ERROR: first index out of range
     Default GSL error handler invoked.
     Aborted (core dumped)

The next program shows how to write a matrix to a file.

     #include <stdio.h>
     #include <gsl/gsl_matrix.h>

     int
     main (void)
     {
       int i, j, k = 0;
       gsl_matrix * m = gsl_matrix_alloc (100, 100);
       gsl_matrix * a = gsl_matrix_alloc (100, 100);

       for (i = 0; i < 100; i++)
         for (j = 0; j < 100; j++)
           gsl_matrix_set (m, i, j, 0.23 + i + j);

       {
          FILE * f = fopen ("test.dat", "wb");
          gsl_matrix_fwrite (f, m);
          fclose (f);
       }

       {
          FILE * f = fopen ("test.dat", "rb");
          gsl_matrix_fread (f, a);
          fclose (f);
       }

       for (i = 0; i < 100; i++)
         for (j = 0; j < 100; j++)
           {
             double mij = gsl_matrix_get (m, i, j);
             double aij = gsl_matrix_get (a, i, j);
             if (mij != aij) k++;
           }

       gsl_matrix_free (m);
       gsl_matrix_free (a);

       printf ("differences = %d (should be zero)\n", k);
       return (k > 0);
     }

After running this program the file ‘test.dat’ should contain the
elements of ‘m’, written in binary format.  The matrix which is read
back in using the function *note gsl_matrix_fread(): 3b1. should be
exactly equal to the original matrix.

The following program demonstrates the use of vector views.  The program
computes the column norms of a matrix.

     #include <math.h>
     #include <stdio.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_blas.h>

     int
     main (void)
     {
       size_t i,j;

       gsl_matrix *m = gsl_matrix_alloc (10, 10);

       for (i = 0; i < 10; i++)
         for (j = 0; j < 10; j++)
           gsl_matrix_set (m, i, j, sin (i) + cos (j));

       for (j = 0; j < 10; j++)
         {
           gsl_vector_view column = gsl_matrix_column (m, j);
           double d;

           d = gsl_blas_dnrm2 (&column.vector);

           printf ("matrix column %zu, norm = %g\n", j, d);
         }

       gsl_matrix_free (m);

       return 0;
     }

Here is the output of the program,

     matrix column 0, norm = 4.31461
     matrix column 1, norm = 3.1205
     matrix column 2, norm = 2.19316
     matrix column 3, norm = 3.26114
     matrix column 4, norm = 2.53416
     matrix column 5, norm = 2.57281
     matrix column 6, norm = 4.20469
     matrix column 7, norm = 3.65202
     matrix column 8, norm = 2.08524
     matrix column 9, norm = 3.07313

The results can be confirmed using GNU octave:

     $ octave
     GNU Octave, version 2.0.16.92
     octave> m = sin(0:9)' * ones(1,10)
                    + ones(10,1) * cos(0:9);
     octave> sqrt(sum(m.^2))
     ans =
       4.3146  3.1205  2.1932  3.2611  2.5342  2.5728
       4.2047  3.6520  2.0852  3.0731


File: gsl-ref.info,  Node: References and Further Reading<4>,  Prev: Example programs for matrices,  Up: Matrices

8.4.14 References and Further Reading
-------------------------------------

The block, vector and matrix objects in GSL follow the ‘valarray’ model
of C++.  A description of this model can be found in the following
reference,

   * B. Stroustrup, The C++ Programming Language (3rd Ed), Section 22.4
     Vector Arithmetic.  Addison-Wesley 1997, ISBN 0-201-88954-4.


File: gsl-ref.info,  Node: Permutations,  Next: Combinations,  Prev: Vectors and Matrices,  Up: Top

9 Permutations
**************

This chapter describes functions for creating and manipulating
permutations.  A permutation p is represented by an array of n integers
in the range 0 to n-1, where each value p_i occurs once and only once.
The application of a permutation p to a vector v yields a new vector v'
where v'_i = v_{p_i}.  For example, the array (0,1,3,2) represents a
permutation which exchanges the last two elements of a four element
vector.  The corresponding identity permutation is (0,1,2,3).

Note that the permutations produced by the linear algebra routines
correspond to the exchange of matrix columns, and so should be
considered as applying to row-vectors in the form v' = v P rather than
column-vectors, when permuting the elements of a vector.

The functions described in this chapter are defined in the header file
‘gsl_permutation.h’.

* Menu:

* The Permutation struct::
* Permutation allocation::
* Accessing permutation elements::
* Permutation properties::
* Permutation functions::
* Applying Permutations::
* Reading and writing permutations::
* Permutations in cyclic form::
* Examples: Examples<4>.
* References and Further Reading: References and Further Reading<5>.


File: gsl-ref.info,  Node: The Permutation struct,  Next: Permutation allocation,  Up: Permutations

9.1 The Permutation struct
==========================

 -- Type: gsl_permutation

     A permutation is defined by a structure containing two components,
     the size of the permutation and a pointer to the permutation array.
     The elements of the permutation array are all of type ‘size_t’.
     The *note gsl_permutation: 3f8. structure looks like this:

          typedef struct
          {
            size_t size;
            size_t * data;
          } gsl_permutation;


File: gsl-ref.info,  Node: Permutation allocation,  Next: Accessing permutation elements,  Prev: The Permutation struct,  Up: Permutations

9.2 Permutation allocation
==========================

 -- Function: gsl_permutation * gsl_permutation_alloc (size_t n)

     This function allocates memory for a new permutation of size ‘n’.
     The permutation is not initialized and its elements are undefined.
     Use the function *note gsl_permutation_calloc(): 3fb. if you want
     to create a permutation which is initialized to the identity.  A
     null pointer is returned if insufficient memory is available to
     create the permutation.

 -- Function: gsl_permutation * gsl_permutation_calloc (size_t n)

     This function allocates memory for a new permutation of size ‘n’
     and initializes it to the identity.  A null pointer is returned if
     insufficient memory is available to create the permutation.

 -- Function: void gsl_permutation_init (gsl_permutation * p)

     This function initializes the permutation ‘p’ to the identity, i.e.
     (0, 1, 2, \dots, n - 1).

 -- Function: void gsl_permutation_free (gsl_permutation * p)

     This function frees all the memory used by the permutation ‘p’.

 -- Function: int gsl_permutation_memcpy (gsl_permutation * dest, const
          gsl_permutation * src)

     This function copies the elements of the permutation ‘src’ into the
     permutation ‘dest’.  The two permutations must have the same size.


File: gsl-ref.info,  Node: Accessing permutation elements,  Next: Permutation properties,  Prev: Permutation allocation,  Up: Permutations

9.3 Accessing permutation elements
==================================

The following functions can be used to access and manipulate
permutations.

 -- Function: size_t gsl_permutation_get (const gsl_permutation * p,
          const size_t i)

     This function returns the value of the ‘i’-th element of the
     permutation ‘p’.  If ‘i’ lies outside the allowed range of 0 to n -
     1 then the error handler is invoked and 0 is returned.  An inline
     version of this function is used when ‘HAVE_INLINE’ is defined.

 -- Function: int gsl_permutation_swap (gsl_permutation * p, const
          size_t i, const size_t j)

     This function exchanges the ‘i’-th and ‘j’-th elements of the
     permutation ‘p’.


File: gsl-ref.info,  Node: Permutation properties,  Next: Permutation functions,  Prev: Accessing permutation elements,  Up: Permutations

9.4 Permutation properties
==========================

 -- Function: size_t gsl_permutation_size (const gsl_permutation * p)

     This function returns the size of the permutation ‘p’.

 -- Function: size_t * gsl_permutation_data (const gsl_permutation * p)

     This function returns a pointer to the array of elements in the
     permutation ‘p’.

 -- Function: int gsl_permutation_valid (const gsl_permutation * p)

     This function checks that the permutation ‘p’ is valid.  The ‘n’
     elements should contain each of the numbers 0 to ‘n - 1’ once and
     only once.


File: gsl-ref.info,  Node: Permutation functions,  Next: Applying Permutations,  Prev: Permutation properties,  Up: Permutations

9.5 Permutation functions
=========================

 -- Function: void gsl_permutation_reverse (gsl_permutation * p)

     This function reverses the elements of the permutation ‘p’.

 -- Function: int gsl_permutation_inverse (gsl_permutation * inv, const
          gsl_permutation * p)

     This function computes the inverse of the permutation ‘p’, storing
     the result in ‘inv’.

 -- Function: int gsl_permutation_next (gsl_permutation * p)

     This function advances the permutation ‘p’ to the next permutation
     in lexicographic order and returns ‘GSL_SUCCESS’.  If no further
     permutations are available it returns ‘GSL_FAILURE’ and leaves ‘p’
     unmodified.  Starting with the identity permutation and repeatedly
     applying this function will iterate through all possible
     permutations of a given order.

 -- Function: int gsl_permutation_prev (gsl_permutation * p)

     This function steps backwards from the permutation ‘p’ to the
     previous permutation in lexicographic order, returning
     ‘GSL_SUCCESS’.  If no previous permutation is available it returns
     ‘GSL_FAILURE’ and leaves ‘p’ unmodified.


File: gsl-ref.info,  Node: Applying Permutations,  Next: Reading and writing permutations,  Prev: Permutation functions,  Up: Permutations

9.6 Applying Permutations
=========================

The following functions are defined in the header files ‘gsl_permute.h’
and ‘gsl_permute_vector.h’.

 -- Function: int gsl_permute (const size_t * p, double * data,
          size_t stride, size_t n)

     This function applies the permutation ‘p’ to the array ‘data’ of
     size ‘n’ with stride ‘stride’.

 -- Function: int gsl_permute_inverse (const size_t * p, double * data,
          size_t stride, size_t n)

     This function applies the inverse of the permutation ‘p’ to the
     array ‘data’ of size ‘n’ with stride ‘stride’.

 -- Function: int gsl_permute_vector (const gsl_permutation * p,
          gsl_vector * v)

     This function applies the permutation ‘p’ to the elements of the
     vector ‘v’, considered as a row-vector acted on by a permutation
     matrix from the right, v' = v P. The j-th column of the permutation
     matrix P is given by the p_j-th column of the identity matrix.  The
     permutation ‘p’ and the vector ‘v’ must have the same length.

 -- Function: int gsl_permute_vector_inverse (const gsl_permutation * p,
          gsl_vector * v)

     This function applies the inverse of the permutation ‘p’ to the
     elements of the vector ‘v’, considered as a row-vector acted on by
     an inverse permutation matrix from the right, v' = v P^T. Note that
     for permutation matrices the inverse is the same as the transpose.
     The j-th column of the permutation matrix P is given by the p_j-th
     column of the identity matrix.  The permutation ‘p’ and the vector
     ‘v’ must have the same length.

 -- Function: int gsl_permute_matrix (const gsl_permutation * p,
          gsl_matrix * A)

     This function applies the permutation ‘p’ to the matrix ‘A’ from
     the right, A' = A P. The j-th column of the permutation matrix P is
     given by the p_j-th column of the identity matrix.  This
     effectively permutes the columns of ‘A’ according to the
     permutation ‘p’, and so the number of columns of ‘A’ must equal the
     size of the permutation ‘p’.

 -- Function: int gsl_permutation_mul (gsl_permutation * p, const
          gsl_permutation * pa, const gsl_permutation * pb)

     This function combines the two permutations ‘pa’ and ‘pb’ into a
     single permutation ‘p’, where p = pa * pb The permutation ‘p’ is
     equivalent to applying ‘pb’ first and then ‘pa’.


File: gsl-ref.info,  Node: Reading and writing permutations,  Next: Permutations in cyclic form,  Prev: Applying Permutations,  Up: Permutations

9.7 Reading and writing permutations
====================================

The library provides functions for reading and writing permutations to a
file as binary data or formatted text.

 -- Function: int gsl_permutation_fwrite (FILE * stream, const
          gsl_permutation * p)

     This function writes the elements of the permutation ‘p’ to the
     stream ‘stream’ in binary format.  The function returns
     ‘GSL_EFAILED’ if there was a problem writing to the file.  Since
     the data is written in the native binary format it may not be
     portable between different architectures.

 -- Function: int gsl_permutation_fread (FILE * stream, gsl_permutation
          * p)

     This function reads into the permutation ‘p’ from the open stream
     ‘stream’ in binary format.  The permutation ‘p’ must be
     preallocated with the correct length since the function uses the
     size of ‘p’ to determine how many bytes to read.  The function
     returns ‘GSL_EFAILED’ if there was a problem reading from the file.
     The data is assumed to have been written in the native binary
     format on the same architecture.

 -- Function: int gsl_permutation_fprintf (FILE * stream, const
          gsl_permutation * p, const char * format)

     This function writes the elements of the permutation ‘p’
     line-by-line to the stream ‘stream’ using the format specifier
     ‘format’, which should be suitable for a type of ‘size_t’.  In ISO
     C99 the type modifier ‘z’ represents ‘size_t’, so ‘"%zu\n"’ is a
     suitable format (1).  The function returns ‘GSL_EFAILED’ if there
     was a problem writing to the file.

 -- Function: int gsl_permutation_fscanf (FILE * stream, gsl_permutation
          * p)

     This function reads formatted data from the stream ‘stream’ into
     the permutation ‘p’.  The permutation ‘p’ must be preallocated with
     the correct length since the function uses the size of ‘p’ to
     determine how many numbers to read.  The function returns
     ‘GSL_EFAILED’ if there was a problem reading from the file.

   ---------- Footnotes ----------

   (1) In versions of the GNU C library prior to the ISO C99 standard,
the type modifier ‘Z’ was used instead.


File: gsl-ref.info,  Node: Permutations in cyclic form,  Next: Examples<4>,  Prev: Reading and writing permutations,  Up: Permutations

9.8 Permutations in cyclic form
===============================

A permutation can be represented in both `linear' and `cyclic'
notations.  The functions described in this section convert between the
two forms.  The linear notation is an index mapping, and has already
been described above.  The cyclic notation expresses a permutation as a
series of circular rearrangements of groups of elements, or `cycles'.

For example, under the cycle (1 2 3), 1 is replaced by 2, 2 is replaced
by 3 and 3 is replaced by 1 in a circular fashion.  Cycles of different
sets of elements can be combined independently, for example (1 2 3) (4
5) combines the cycle (1 2 3) with the cycle (4 5), which is an exchange
of elements 4 and 5.  A cycle of length one represents an element which
is unchanged by the permutation and is referred to as a `singleton'.

It can be shown that every permutation can be decomposed into
combinations of cycles.  The decomposition is not unique, but can always
be rearranged into a standard `canonical form' by a reordering of
elements.  The library uses the canonical form defined in Knuth’s `Art
of Computer Programming' (Vol 1, 3rd Ed, 1997) Section 1.3.3, p.178.

The procedure for obtaining the canonical form given by Knuth is,

  1. Write all singleton cycles explicitly

  2. Within each cycle, put the smallest number first

  3. Order the cycles in decreasing order of the first number in the
     cycle.

For example, the linear representation (2 4 3 0 1) is represented as (1
4) (0 2 3) in canonical form.  The permutation corresponds to an
exchange of elements 1 and 4, and rotation of elements 0, 2 and 3.

The important property of the canonical form is that it can be
reconstructed from the contents of each cycle without the brackets.  In
addition, by removing the brackets it can be considered as a linear
representation of a different permutation.  In the example given above
the permutation (2 4 3 0 1) would become (1 4 0 2 3).  This mapping has
many applications in the theory of permutations.

 -- Function: int gsl_permutation_linear_to_canonical (gsl_permutation
          * q, const gsl_permutation * p)

     This function computes the canonical form of the permutation ‘p’
     and stores it in the output argument ‘q’.

 -- Function: int gsl_permutation_canonical_to_linear (gsl_permutation
          * p, const gsl_permutation * q)

     This function converts a permutation ‘q’ in canonical form back
     into linear form storing it in the output argument ‘p’.

 -- Function: size_t gsl_permutation_inversions (const gsl_permutation
          * p)

     This function counts the number of inversions in the permutation
     ‘p’.  An inversion is any pair of elements that are not in order.
     For example, the permutation 2031 has three inversions,
     corresponding to the pairs (2,0) (2,1) and (3,1).  The identity
     permutation has no inversions.

 -- Function: size_t gsl_permutation_linear_cycles (const
          gsl_permutation * p)

     This function counts the number of cycles in the permutation ‘p’,
     given in linear form.

 -- Function: size_t gsl_permutation_canonical_cycles (const
          gsl_permutation * q)

     This function counts the number of cycles in the permutation ‘q’,
     given in canonical form.


File: gsl-ref.info,  Node: Examples<4>,  Next: References and Further Reading<5>,  Prev: Permutations in cyclic form,  Up: Permutations

9.9 Examples
============

The example program below creates a random permutation (by shuffling the
elements of the identity) and finds its inverse.

     #include <stdio.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_permutation.h>

     int
     main (void)
     {
       const size_t N = 10;
       const gsl_rng_type * T;
       gsl_rng * r;

       gsl_permutation * p = gsl_permutation_alloc (N);
       gsl_permutation * q = gsl_permutation_alloc (N);

       gsl_rng_env_setup();
       T = gsl_rng_default;
       r = gsl_rng_alloc (T);

       printf ("initial permutation:");
       gsl_permutation_init (p);
       gsl_permutation_fprintf (stdout, p, " %u");
       printf ("\n");

       printf (" random permutation:");
       gsl_ran_shuffle (r, p->data, N, sizeof(size_t));
       gsl_permutation_fprintf (stdout, p, " %u");
       printf ("\n");

       printf ("inverse permutation:");
       gsl_permutation_inverse (q, p);
       gsl_permutation_fprintf (stdout, q, " %u");
       printf ("\n");

       gsl_permutation_free (p);
       gsl_permutation_free (q);
       gsl_rng_free (r);

       return 0;
     }

Here is the output from the program:

     $ ./a.out
     initial permutation: 0 1 2 3 4 5 6 7 8 9
      random permutation: 1 3 5 2 7 6 0 4 9 8
     inverse permutation: 6 0 3 1 7 2 5 4 9 8

The random permutation ‘p[i]’ and its inverse ‘q[i]’ are related through
the identity ‘p[q[i]] = i’, which can be verified from the output.

The next example program steps forwards through all possible third order
permutations, starting from the identity,

     #include <stdio.h>
     #include <gsl/gsl_permutation.h>

     int
     main (void)
     {
       gsl_permutation * p = gsl_permutation_alloc (3);

       gsl_permutation_init (p);

       do
        {
           gsl_permutation_fprintf (stdout, p, " %u");
           printf ("\n");
        }
       while (gsl_permutation_next(p) == GSL_SUCCESS);

       gsl_permutation_free (p);

       return 0;
     }

Here is the output from the program:

     $ ./a.out
      0 1 2
      0 2 1
      1 0 2
      1 2 0
      2 0 1
      2 1 0

The permutations are generated in lexicographic order.  To reverse the
sequence, begin with the final permutation (which is the reverse of the
identity) and replace *note gsl_permutation_next(): 409. with *note
gsl_permutation_prev(): 40a.


File: gsl-ref.info,  Node: References and Further Reading<5>,  Prev: Examples<4>,  Up: Permutations

9.10 References and Further Reading
===================================

The subject of permutations is covered extensively in the following,

   * Donald E. Knuth, The Art of Computer Programming: Sorting and
     Searching (Vol 3, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896850.

For the definition of the `canonical form' see,

   * Donald E. Knuth, The Art of Computer Programming: Fundamental
     Algorithms (Vol 1, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896850.
     Section 1.3.3, An Unusual Correspondence, p.178–179.


File: gsl-ref.info,  Node: Combinations,  Next: Multisets,  Prev: Permutations,  Up: Top

10 Combinations
***************

This chapter describes functions for creating and manipulating
combinations.  A combination c is represented by an array of k integers
in the range 0 to n - 1, where each value c_i occurs at most once.  The
combination c corresponds to indices of k elements chosen from an n
element vector.  Combinations are useful for iterating over all
k-element subsets of a set.

The functions described in this chapter are defined in the header file
‘gsl_combination.h’.

* Menu:

* The Combination struct::
* Combination allocation::
* Accessing combination elements::
* Combination properties::
* Combination functions::
* Reading and writing combinations::
* Examples: Examples<5>.
* References and Further Reading: References and Further Reading<6>.


File: gsl-ref.info,  Node: The Combination struct,  Next: Combination allocation,  Up: Combinations

10.1 The Combination struct
===========================

 -- Type: gsl_combination

     A combination is defined by a structure containing three
     components, the values of n and k, and a pointer to the combination
     array.  The elements of the combination array are all of type
     ‘size_t’, and are stored in increasing order.  The *note
     gsl_combination: 422. structure looks like this:

          typedef struct
          {
            size_t n;
            size_t k;
            size_t *data;
          } gsl_combination;


File: gsl-ref.info,  Node: Combination allocation,  Next: Accessing combination elements,  Prev: The Combination struct,  Up: Combinations

10.2 Combination allocation
===========================

 -- Function: gsl_combination * gsl_combination_alloc (size_t n,
          size_t k)

     This function allocates memory for a new combination with
     parameters ‘n’, ‘k’.  The combination is not initialized and its
     elements are undefined.  Use the function *note
     gsl_combination_calloc(): 425. if you want to create a combination
     which is initialized to the lexicographically first combination.  A
     null pointer is returned if insufficient memory is available to
     create the combination.

 -- Function: gsl_combination * gsl_combination_calloc (size_t n,
          size_t k)

     This function allocates memory for a new combination with
     parameters ‘n’, ‘k’ and initializes it to the lexicographically
     first combination.  A null pointer is returned if insufficient
     memory is available to create the combination.

 -- Function: void gsl_combination_init_first (gsl_combination * c)

     This function initializes the combination ‘c’ to the
     lexicographically first combination, i.e.  (0, 1, 2, \dots, k - 1).

 -- Function: void gsl_combination_init_last (gsl_combination * c)

     This function initializes the combination ‘c’ to the
     lexicographically last combination, i.e.  (n - k, n - k + 1, \dots,
     n - 1).

 -- Function: void gsl_combination_free (gsl_combination * c)

     This function frees all the memory used by the combination ‘c’.

 -- Function: int gsl_combination_memcpy (gsl_combination * dest, const
          gsl_combination * src)

     This function copies the elements of the combination ‘src’ into the
     combination ‘dest’.  The two combinations must have the same size.


File: gsl-ref.info,  Node: Accessing combination elements,  Next: Combination properties,  Prev: Combination allocation,  Up: Combinations

10.3 Accessing combination elements
===================================

The following function can be used to access the elements of a
combination.

 -- Function: size_t gsl_combination_get (const gsl_combination * c,
          const size_t i)

     This function returns the value of the ‘i’-th element of the
     combination ‘c’.  If ‘i’ lies outside the allowed range of 0 to k -
     1 then the error handler is invoked and 0 is returned.  An inline
     version of this function is used when ‘HAVE_INLINE’ is defined.


File: gsl-ref.info,  Node: Combination properties,  Next: Combination functions,  Prev: Accessing combination elements,  Up: Combinations

10.4 Combination properties
===========================

 -- Function: size_t gsl_combination_n (const gsl_combination * c)

     This function returns the range (n) of the combination c.

 -- Function: size_t gsl_combination_k (const gsl_combination * c)

     This function returns the number of elements (k) in the combination
     ‘c’.

 -- Function: size_t * gsl_combination_data (const gsl_combination * c)

     This function returns a pointer to the array of elements in the
     combination ‘c’.

 -- Function: int gsl_combination_valid (gsl_combination * c)

     This function checks that the combination ‘c’ is valid.  The ‘k’
     elements should lie in the range 0 to n - 1, with each value
     occurring once at most and in increasing order.


File: gsl-ref.info,  Node: Combination functions,  Next: Reading and writing combinations,  Prev: Combination properties,  Up: Combinations

10.5 Combination functions
==========================

 -- Function: int gsl_combination_next (gsl_combination * c)

     This function advances the combination ‘c’ to the next combination
     in lexicographic order and returns ‘GSL_SUCCESS’.  If no further
     combinations are available it returns ‘GSL_FAILURE’ and leaves ‘c’
     unmodified.  Starting with the first combination and repeatedly
     applying this function will iterate through all possible
     combinations of a given order.

 -- Function: int gsl_combination_prev (gsl_combination * c)

     This function steps backwards from the combination ‘c’ to the
     previous combination in lexicographic order, returning
     ‘GSL_SUCCESS’.  If no previous combination is available it returns
     ‘GSL_FAILURE’ and leaves ‘c’ unmodified.


File: gsl-ref.info,  Node: Reading and writing combinations,  Next: Examples<5>,  Prev: Combination functions,  Up: Combinations

10.6 Reading and writing combinations
=====================================

The library provides functions for reading and writing combinations to a
file as binary data or formatted text.

 -- Function: int gsl_combination_fwrite (FILE * stream, const
          gsl_combination * c)

     This function writes the elements of the combination ‘c’ to the
     stream ‘stream’ in binary format.  The function returns
     ‘GSL_EFAILED’ if there was a problem writing to the file.  Since
     the data is written in the native binary format it may not be
     portable between different architectures.

 -- Function: int gsl_combination_fread (FILE * stream, gsl_combination
          * c)

     This function reads elements from the open stream ‘stream’ into the
     combination ‘c’ in binary format.  The combination ‘c’ must be
     preallocated with correct values of n and k since the function uses
     the size of ‘c’ to determine how many bytes to read.  The function
     returns ‘GSL_EFAILED’ if there was a problem reading from the file.
     The data is assumed to have been written in the native binary
     format on the same architecture.

 -- Function: int gsl_combination_fprintf (FILE * stream, const
          gsl_combination * c, const char * format)

     This function writes the elements of the combination ‘c’
     line-by-line to the stream ‘stream’ using the format specifier
     ‘format’, which should be suitable for a type of ‘size_t’.  In ISO
     C99 the type modifier ‘z’ represents ‘size_t’, so ‘"%zu\n"’ is a
     suitable format (1).  The function returns ‘GSL_EFAILED’ if there
     was a problem writing to the file.

 -- Function: int gsl_combination_fscanf (FILE * stream, gsl_combination
          * c)

     This function reads formatted data from the stream ‘stream’ into
     the combination ‘c’.  The combination ‘c’ must be preallocated with
     correct values of n and k since the function uses the size of ‘c’
     to determine how many numbers to read.  The function returns
     ‘GSL_EFAILED’ if there was a problem reading from the file.

   ---------- Footnotes ----------

   (1) In versions of the GNU C library prior to the ISO C99 standard,
the type modifier ‘Z’ was used instead.


File: gsl-ref.info,  Node: Examples<5>,  Next: References and Further Reading<6>,  Prev: Reading and writing combinations,  Up: Combinations

10.7 Examples
=============

The example program below prints all subsets of the set {0,1,2,3}
ordered by size.  Subsets of the same size are ordered
lexicographically.

     #include <stdio.h>
     #include <gsl/gsl_combination.h>

     int
     main (void)
     {
       gsl_combination * c;
       size_t i;

       printf ("All subsets of {0,1,2,3} by size:\n") ;
       for (i = 0; i <= 4; i++)
         {
           c = gsl_combination_calloc (4, i);
           do
             {
               printf ("{");
               gsl_combination_fprintf (stdout, c, " %u");
               printf (" }\n");
             }
           while (gsl_combination_next (c) == GSL_SUCCESS);
           gsl_combination_free (c);
         }

       return 0;
     }

Here is the output from the program,

     All subsets of {0,1,2,3} by size:
     { }
     { 0 }
     { 1 }
     { 2 }
     { 3 }
     { 0 1 }
     { 0 2 }
     { 0 3 }
     { 1 2 }
     { 1 3 }
     { 2 3 }
     { 0 1 2 }
     { 0 1 3 }
     { 0 2 3 }
     { 1 2 3 }
     { 0 1 2 3 }

All 16 subsets are generated, and the subsets of each size are sorted
lexicographically.


File: gsl-ref.info,  Node: References and Further Reading<6>,  Prev: Examples<5>,  Up: Combinations

10.8 References and Further Reading
===================================

Further information on combinations can be found in,

   * Donald L. Kreher, Douglas R. Stinson, Combinatorial Algorithms:
     Generation, Enumeration and Search, 1998, CRC Press LLC, ISBN
     084933988X


File: gsl-ref.info,  Node: Multisets,  Next: Sorting,  Prev: Combinations,  Up: Top

11 Multisets
************

This chapter describes functions for creating and manipulating
multisets.  A multiset c is represented by an array of k integers in the
range 0 to n - 1, where each value c_i may occur more than once.  The
multiset c corresponds to indices of k elements chosen from an n element
vector with replacement.  In mathematical terms, n is the cardinality of
the multiset while k is the maximum multiplicity of any value.
Multisets are useful, for example, when iterating over the indices of a
k-th order symmetric tensor in n-space.

The functions described in this chapter are defined in the header file
‘gsl_multiset.h’.

* Menu:

* The Multiset struct::
* Multiset allocation::
* Accessing multiset elements::
* Multiset properties::
* Multiset functions::
* Reading and writing multisets::
* Examples: Examples<6>.


File: gsl-ref.info,  Node: The Multiset struct,  Next: Multiset allocation,  Up: Multisets

11.1 The Multiset struct
========================

 -- Type: gsl_multiset

     A multiset is defined by a structure containing three components,
     the values of n and k, and a pointer to the multiset array.  The
     elements of the multiset array are all of type ‘size_t’, and are
     stored in increasing order.  The *note gsl_multiset: 43e. structure
     looks like this:

          typedef struct
          {
            size_t n;
            size_t k;
            size_t *data;
          } gsl_multiset;


File: gsl-ref.info,  Node: Multiset allocation,  Next: Accessing multiset elements,  Prev: The Multiset struct,  Up: Multisets

11.2 Multiset allocation
========================

 -- Function: gsl_multiset * gsl_multiset_alloc (size_t n, size_t k)

     This function allocates memory for a new multiset with parameters
     ‘n’, ‘k’.  The multiset is not initialized and its elements are
     undefined.  Use the function *note gsl_multiset_calloc(): 441. if
     you want to create a multiset which is initialized to the
     lexicographically first multiset element.  A null pointer is
     returned if insufficient memory is available to create the
     multiset.

 -- Function: gsl_multiset * gsl_multiset_calloc (size_t n, size_t k)

     This function allocates memory for a new multiset with parameters
     ‘n’, ‘k’ and initializes it to the lexicographically first multiset
     element.  A null pointer is returned if insufficient memory is
     available to create the multiset.

 -- Function: void gsl_multiset_init_first (gsl_multiset * c)

     This function initializes the multiset ‘c’ to the lexicographically
     first multiset element, i.e.  0 repeated k times.

 -- Function: void gsl_multiset_init_last (gsl_multiset * c)

     This function initializes the multiset ‘c’ to the lexicographically
     last multiset element, i.e.  n-1 repeated k times.

 -- Function: void gsl_multiset_free (gsl_multiset * c)

     This function frees all the memory used by the multiset ‘c’.

 -- Function: int gsl_multiset_memcpy (gsl_multiset * dest, const
          gsl_multiset * src)

     This function copies the elements of the multiset ‘src’ into the
     multiset ‘dest’.  The two multisets must have the same size.


File: gsl-ref.info,  Node: Accessing multiset elements,  Next: Multiset properties,  Prev: Multiset allocation,  Up: Multisets

11.3 Accessing multiset elements
================================

The following function can be used to access the elements of a multiset.

 -- Function: size_t gsl_multiset_get (const gsl_multiset * c, const
          size_t i)

     This function returns the value of the ‘i’-th element of the
     multiset ‘c’.  If ‘i’ lies outside the allowed range of 0 to k - 1
     then the error handler is invoked and 0 is returned.  An inline
     version of this function is used when ‘HAVE_INLINE’ is defined.


File: gsl-ref.info,  Node: Multiset properties,  Next: Multiset functions,  Prev: Accessing multiset elements,  Up: Multisets

11.4 Multiset properties
========================

 -- Function: size_t gsl_multiset_n (const gsl_multiset * c)

     This function returns the range (n) of the multiset ‘c’.

 -- Function: size_t gsl_multiset_k (const gsl_multiset * c)

     This function returns the number of elements (k) in the multiset
     ‘c’.

 -- Function: size_t * gsl_multiset_data (const gsl_multiset * c)

     This function returns a pointer to the array of elements in the
     multiset ‘c’.

 -- Function: int gsl_multiset_valid (gsl_multiset * c)

     This function checks that the multiset ‘c’ is valid.  The ‘k’
     elements should lie in the range 0 to n - 1, with each value
     occurring in nondecreasing order.


File: gsl-ref.info,  Node: Multiset functions,  Next: Reading and writing multisets,  Prev: Multiset properties,  Up: Multisets

11.5 Multiset functions
=======================

 -- Function: int gsl_multiset_next (gsl_multiset * c)

     This function advances the multiset ‘c’ to the next multiset
     element in lexicographic order and returns ‘GSL_SUCCESS’.  If no
     further multisets elements are available it returns ‘GSL_FAILURE’
     and leaves ‘c’ unmodified.  Starting with the first multiset and
     repeatedly applying this function will iterate through all possible
     multisets of a given order.

 -- Function: int gsl_multiset_prev (gsl_multiset * c)

     This function steps backwards from the multiset ‘c’ to the previous
     multiset element in lexicographic order, returning ‘GSL_SUCCESS’.
     If no previous multiset is available it returns ‘GSL_FAILURE’ and
     leaves ‘c’ unmodified.


File: gsl-ref.info,  Node: Reading and writing multisets,  Next: Examples<6>,  Prev: Multiset functions,  Up: Multisets

11.6 Reading and writing multisets
==================================

The library provides functions for reading and writing multisets to a
file as binary data or formatted text.

 -- Function: int gsl_multiset_fwrite (FILE * stream, const gsl_multiset
          * c)

     This function writes the elements of the multiset ‘c’ to the stream
     ‘stream’ in binary format.  The function returns ‘GSL_EFAILED’ if
     there was a problem writing to the file.  Since the data is written
     in the native binary format it may not be portable between
     different architectures.

 -- Function: int gsl_multiset_fread (FILE * stream, gsl_multiset * c)

     This function reads elements from the open stream ‘stream’ into the
     multiset ‘c’ in binary format.  The multiset ‘c’ must be
     preallocated with correct values of n and k since the function uses
     the size of ‘c’ to determine how many bytes to read.  The function
     returns ‘GSL_EFAILED’ if there was a problem reading from the file.
     The data is assumed to have been written in the native binary
     format on the same architecture.

 -- Function: int gsl_multiset_fprintf (FILE * stream, const
          gsl_multiset * c, const char * format)

     This function writes the elements of the multiset ‘c’ line-by-line
     to the stream ‘stream’ using the format specifier ‘format’, which
     should be suitable for a type of ‘size_t’.  In ISO C99 the type
     modifier ‘z’ represents ‘size_t’, so ‘"%zu\n"’ is a suitable format
     (1).  The function returns ‘GSL_EFAILED’ if there was a problem
     writing to the file.

 -- Function: int gsl_multiset_fscanf (FILE * stream, gsl_multiset * c)

     This function reads formatted data from the stream ‘stream’ into
     the multiset ‘c’.  The multiset ‘c’ must be preallocated with
     correct values of n and k since the function uses the size of ‘c’
     to determine how many numbers to read.  The function returns
     ‘GSL_EFAILED’ if there was a problem reading from the file.

   ---------- Footnotes ----------

   (1) In versions of the GNU C library prior to the ISO C99 standard,
the type modifier ‘Z’ was used instead.


File: gsl-ref.info,  Node: Examples<6>,  Prev: Reading and writing multisets,  Up: Multisets

11.7 Examples
=============

The example program below prints all multisets elements containing the
values {0,1,2,3} ordered by size.  Multiset elements of the same size
are ordered lexicographically.

     #include <stdio.h>
     #include <gsl/gsl_multiset.h>

     int
     main (void)
     {
       gsl_multiset * c;
       size_t i;

       printf ("All multisets of {0,1,2,3} by size:\n") ;
       for (i = 0; i <= 4; i++)
         {
           c = gsl_multiset_calloc (4, i);
           do
             {
               printf ("{");
               gsl_multiset_fprintf (stdout, c, " %u");
               printf (" }\n");
             }
           while (gsl_multiset_next (c) == GSL_SUCCESS);
           gsl_multiset_free (c);
         }

       return 0;
     }

Here is the output from the program,

     All multisets of {0,1,2,3} by size:
     { }
     { 0 }
     { 1 }
     { 2 }
     { 3 }
     { 0 0 }
     { 0 1 }
     { 0 2 }
     { 0 3 }
     { 1 1 }
     { 1 2 }
     { 1 3 }
     { 2 2 }
     { 2 3 }
     { 3 3 }
     { 0 0 0 }
     { 0 0 1 }
     { 0 0 2 }
     { 0 0 3 }
     { 0 1 1 }
     { 0 1 2 }
     { 0 1 3 }
     { 0 2 2 }
     { 0 2 3 }
     { 0 3 3 }
     { 1 1 1 }
     { 1 1 2 }
     { 1 1 3 }
     { 1 2 2 }
     { 1 2 3 }
     { 1 3 3 }
     { 2 2 2 }
     { 2 2 3 }
     { 2 3 3 }
     { 3 3 3 }
     { 0 0 0 0 }
     { 0 0 0 1 }
     { 0 0 0 2 }
     { 0 0 0 3 }
     { 0 0 1 1 }
     { 0 0 1 2 }
     { 0 0 1 3 }
     { 0 0 2 2 }
     { 0 0 2 3 }
     { 0 0 3 3 }
     { 0 1 1 1 }
     { 0 1 1 2 }
     { 0 1 1 3 }
     { 0 1 2 2 }
     { 0 1 2 3 }
     { 0 1 3 3 }
     { 0 2 2 2 }
     { 0 2 2 3 }
     { 0 2 3 3 }
     { 0 3 3 3 }
     { 1 1 1 1 }
     { 1 1 1 2 }
     { 1 1 1 3 }
     { 1 1 2 2 }
     { 1 1 2 3 }
     { 1 1 3 3 }
     { 1 2 2 2 }
     { 1 2 2 3 }
     { 1 2 3 3 }
     { 1 3 3 3 }
     { 2 2 2 2 }
     { 2 2 2 3 }
     { 2 2 3 3 }
     { 2 3 3 3 }
     { 3 3 3 3 }

All 70 multisets are generated and sorted lexicographically.


File: gsl-ref.info,  Node: Sorting,  Next: BLAS Support,  Prev: Multisets,  Up: Top

12 Sorting
**********

This chapter describes functions for sorting data, both directly and
indirectly (using an index).  All the functions use the `heapsort'
algorithm.  Heapsort is an O(N \log N) algorithm which operates in-place
and does not require any additional storage.  It also provides
consistent performance, the running time for its worst-case (ordered
data) being not significantly longer than the average and best cases.
Note that the heapsort algorithm does not preserve the relative ordering
of equal elements—it is an `unstable' sort.  However the resulting order
of equal elements will be consistent across different platforms when
using these functions.

* Menu:

* Sorting objects::
* Sorting vectors::
* Selecting the k smallest or largest elements::
* Computing the rank::
* Examples: Examples<7>.
* References and Further Reading: References and Further Reading<7>.


File: gsl-ref.info,  Node: Sorting objects,  Next: Sorting vectors,  Up: Sorting

12.1 Sorting objects
====================

The following function provides a simple alternative to the standard
library function ‘qsort()’.  It is intended for systems lacking
‘qsort()’, not as a replacement for it.  The function ‘qsort()’ should
be used whenever possible, as it will be faster and can provide stable
ordering of equal elements.  Documentation for ‘qsort()’ is available in
the GNU C Library Reference Manual.

The functions described in this section are defined in the header file
‘gsl_heapsort.h’.

 -- Function: void gsl_heapsort (void * array, size_t count,
          size_t size, gsl_comparison_fn_t compare)

     This function sorts the ‘count’ elements of the array ‘array’, each
     of size ‘size’, into ascending order using the comparison function
     ‘compare’.  The type of the comparison function is defined by

      -- Type: gsl_comparison_fn_t

               int (*gsl_comparison_fn_t) (const void * a, const void * b)

     A comparison function should return a negative integer if the first
     argument is less than the second argument, ‘0’ if the two arguments
     are equal and a positive integer if the first argument is greater
     than the second argument.

     For example, the following function can be used to sort doubles
     into ascending numerical order.

          int
          compare_doubles (const double * a, const double * b)
          {
            if (*a > *b)
              return 1;
            else if (*a < *b)
              return -1;
            else
              return 0;
          }

     The appropriate function call to perform the sort is:

          gsl_heapsort (array, count, sizeof(double), compare_doubles);

     Note that unlike ‘qsort()’ the heapsort algorithm cannot be made
     into a stable sort by pointer arithmetic.  The trick of comparing
     pointers for equal elements in the comparison function does not
     work for the heapsort algorithm.  The heapsort algorithm performs
     an internal rearrangement of the data which destroys its initial
     ordering.

 -- Function: int gsl_heapsort_index (size_t * p, const void * array,
          size_t count, size_t size, gsl_comparison_fn_t compare)

     This function indirectly sorts the ‘count’ elements of the array
     ‘array’, each of size ‘size’, into ascending order using the
     comparison function ‘compare’.  The resulting permutation is stored
     in ‘p’, an array of length ‘n’.  The elements of ‘p’ give the index
     of the array element which would have been stored in that position
     if the array had been sorted in place.  The first element of ‘p’
     gives the index of the least element in ‘array’, and the last
     element of ‘p’ gives the index of the greatest element in ‘array’.
     The array itself is not changed.


File: gsl-ref.info,  Node: Sorting vectors,  Next: Selecting the k smallest or largest elements,  Prev: Sorting objects,  Up: Sorting

12.2 Sorting vectors
====================

The following functions will sort the elements of an array or vector,
either directly or indirectly.  They are defined for all real and
integer types using the normal suffix rules.  For example, the ‘float’
versions of the array functions are ‘gsl_sort_float()’ and
‘gsl_sort_float_index()’.  The corresponding vector functions are
‘gsl_sort_vector_float()’ and ‘gsl_sort_vector_float_index()’.  The
prototypes are available in the header files ‘gsl_sort_float.h’
‘gsl_sort_vector_float.h’.  The complete set of prototypes can be
included using the header files ‘gsl_sort.h’ and ‘gsl_sort_vector.h’.

There are no functions for sorting complex arrays or vectors, since the
ordering of complex numbers is not uniquely defined.  To sort a complex
vector by magnitude compute a real vector containing the magnitudes of
the complex elements, and sort this vector indirectly.  The resulting
index gives the appropriate ordering of the original complex vector.

 -- Function: void gsl_sort (double * data, const size_t stride,
          size_t n)

     This function sorts the ‘n’ elements of the array ‘data’ with
     stride ‘stride’ into ascending numerical order.

 -- Function: void gsl_sort2 (double * data1, const size_t stride1,
          double * data2, const size_t stride2, size_t n)

     This function sorts the ‘n’ elements of the array ‘data1’ with
     stride ‘stride1’ into ascending numerical order, while making the
     same rearrangement of the array ‘data2’ with stride ‘stride2’, also
     of size ‘n’.

 -- Function: void gsl_sort_vector (gsl_vector * v)

     This function sorts the elements of the vector ‘v’ into ascending
     numerical order.

 -- Function: void gsl_sort_vector2 (gsl_vector * v1, gsl_vector * v2)

     This function sorts the elements of the vector ‘v1’ into ascending
     numerical order, while making the same rearrangement of the vector
     ‘v2’.

 -- Function: void gsl_sort_index (size_t * p, const double * data,
          size_t stride, size_t n)

     This function indirectly sorts the ‘n’ elements of the array ‘data’
     with stride ‘stride’ into ascending order, storing the resulting
     permutation in ‘p’.  The array ‘p’ must be allocated with a
     sufficient length to store the ‘n’ elements of the permutation.
     The elements of ‘p’ give the index of the array element which would
     have been stored in that position if the array had been sorted in
     place.  The array ‘data’ is not changed.

 -- Function: int gsl_sort_vector_index (gsl_permutation * p, const
          gsl_vector * v)

     This function indirectly sorts the elements of the vector ‘v’ into
     ascending order, storing the resulting permutation in ‘p’.  The
     elements of ‘p’ give the index of the vector element which would
     have been stored in that position if the vector had been sorted in
     place.  The first element of ‘p’ gives the index of the least
     element in ‘v’, and the last element of ‘p’ gives the index of the
     greatest element in ‘v’.  The vector ‘v’ is not changed.


File: gsl-ref.info,  Node: Selecting the k smallest or largest elements,  Next: Computing the rank,  Prev: Sorting vectors,  Up: Sorting

12.3 Selecting the k smallest or largest elements
=================================================

The functions described in this section select the k smallest or largest
elements of a data set of size N. The routines use an O(kN) direct
insertion algorithm which is suited to subsets that are small compared
with the total size of the dataset.  For example, the routines are
useful for selecting the 10 largest values from one million data points,
but not for selecting the largest 100,000 values.  If the subset is a
significant part of the total dataset it may be faster to sort all the
elements of the dataset directly with an O(N \log N) algorithm and
obtain the smallest or largest values that way.

 -- Function: int gsl_sort_smallest (double * dest, size_t k, const
          double * src, size_t stride, size_t n)

     This function copies the ‘k’ smallest elements of the array ‘src’,
     of size ‘n’ and stride ‘stride’, in ascending numerical order into
     the array ‘dest’.  The size ‘k’ of the subset must be less than or
     equal to ‘n’.  The data ‘src’ is not modified by this operation.

 -- Function: int gsl_sort_largest (double * dest, size_t k, const
          double * src, size_t stride, size_t n)

     This function copies the ‘k’ largest elements of the array ‘src’,
     of size ‘n’ and stride ‘stride’, in descending numerical order into
     the array ‘dest’.  ‘k’ must be less than or equal to ‘n’.  The data
     ‘src’ is not modified by this operation.

 -- Function: int gsl_sort_vector_smallest (double * dest, size_t k,
          const gsl_vector * v)
 -- Function: int gsl_sort_vector_largest (double * dest, size_t k,
          const gsl_vector * v)

     These functions copy the ‘k’ smallest or largest elements of the
     vector ‘v’ into the array ‘dest’.  ‘k’ must be less than or equal
     to the length of the vector ‘v’.

The following functions find the indices of the k smallest or largest
elements of a dataset.

 -- Function: int gsl_sort_smallest_index (size_t * p, size_t k, const
          double * src, size_t stride, size_t n)

     This function stores the indices of the ‘k’ smallest elements of
     the array ‘src’, of size ‘n’ and stride ‘stride’, in the array ‘p’.
     The indices are chosen so that the corresponding data is in
     ascending numerical order.  ‘k’ must be less than or equal to ‘n’.
     The data ‘src’ is not modified by this operation.

 -- Function: int gsl_sort_largest_index (size_t * p, size_t k, const
          double * src, size_t stride, size_t n)

     This function stores the indices of the ‘k’ largest elements of the
     array ‘src’, of size ‘n’ and stride ‘stride’, in the array ‘p’.
     The indices are chosen so that the corresponding data is in
     descending numerical order.  ‘k’ must be less than or equal to ‘n’.
     The data ‘src’ is not modified by this operation.

 -- Function: int gsl_sort_vector_smallest_index (size_t * p, size_t k,
          const gsl_vector * v)
 -- Function: int gsl_sort_vector_largest_index (size_t * p, size_t k,
          const gsl_vector * v)

     These functions store the indices of the ‘k’ smallest or largest
     elements of the vector ‘v’ in the array ‘p’.  ‘k’ must be less than
     or equal to the length of the vector ‘v’.


File: gsl-ref.info,  Node: Computing the rank,  Next: Examples<7>,  Prev: Selecting the k smallest or largest elements,  Up: Sorting

12.4 Computing the rank
=======================

The `rank' of an element is its order in the sorted data.  The rank is
the inverse of the index permutation, p.  It can be computed using the
following algorithm:

     for (i = 0; i < p->size; i++)
       {
         size_t pi = p->data[i];
         rank->data[pi] = i;
       }

This can be computed directly from the function
‘gsl_permutation_inverse(rank,p)’.

The following function will print the rank of each element of the vector
v:

     void
     print_rank (gsl_vector * v)
     {
       size_t i;
       size_t n = v->size;
       gsl_permutation * perm = gsl_permutation_alloc(n);
       gsl_permutation * rank = gsl_permutation_alloc(n);

       gsl_sort_vector_index (perm, v);
       gsl_permutation_inverse (rank, perm);

       for (i = 0; i < n; i++)
         {
           double vi = gsl_vector_get(v, i);
           printf ("element = %d, value = %g, rank = %d\n",
                    i, vi, rank->data[i]);
         }

       gsl_permutation_free (perm);
       gsl_permutation_free (rank);
     }


File: gsl-ref.info,  Node: Examples<7>,  Next: References and Further Reading<7>,  Prev: Computing the rank,  Up: Sorting

12.5 Examples
=============

The following example shows how to use the permutation p to print the
elements of the vector v in ascending order:

     gsl_sort_vector_index (p, v);

     for (i = 0; i < v->size; i++)
       {
         double vpi = gsl_vector_get (v, p->data[i]);
         printf ("order = %d, value = %g\n", i, vpi);
       }

The next example uses the function *note gsl_sort_smallest(): 464. to
select the 5 smallest numbers from 100000 uniform random variates stored
in an array,

     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_sort_double.h>

     int
     main (void)
     {
       const gsl_rng_type * T;
       gsl_rng * r;

       size_t i, k = 5, N = 100000;

       double * x = malloc (N * sizeof(double));
       double * small = malloc (k * sizeof(double));

       gsl_rng_env_setup();

       T = gsl_rng_default;
       r = gsl_rng_alloc (T);

       for (i = 0; i < N; i++)
         {
           x[i] = gsl_rng_uniform(r);
         }

       gsl_sort_smallest (small, k, x, 1, N);

       printf ("%zu smallest values from %zu\n", k, N);

       for (i = 0; i < k; i++)
         {
           printf ("%zu: %.18f\n", i, small[i]);
         }

       free (x);
       free (small);
       gsl_rng_free (r);
       return 0;
     }

The output lists the 5 smallest values, in ascending order,

     5 smallest values from 100000
     0: 0.000003489200025797
     1: 0.000008199829608202
     2: 0.000008953968062997
     3: 0.000010712770745158
     4: 0.000033531803637743


File: gsl-ref.info,  Node: References and Further Reading<7>,  Prev: Examples<7>,  Up: Sorting

12.6 References and Further Reading
===================================

The subject of sorting is covered extensively in the following,

   * Donald E. Knuth, The Art of Computer Programming: Sorting and
     Searching (Vol 3, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896850.

The Heapsort algorithm is described in the following book,

   * Robert Sedgewick, Algorithms in C, Addison-Wesley, ISBN 0201514257.


File: gsl-ref.info,  Node: BLAS Support,  Next: Linear Algebra,  Prev: Sorting,  Up: Top

13 BLAS Support
***************

The Basic Linear Algebra Subprograms (BLAS) define a set of fundamental
operations on vectors and matrices which can be used to create optimized
higher-level linear algebra functionality.

The library provides a low-level layer which corresponds directly to the
C-language BLAS standard, referred to here as “CBLAS”, and a
higher-level interface for operations on GSL vectors and matrices.
Users who are interested in simple operations on GSL vector and matrix
objects should use the high-level layer described in this chapter.  The
functions are declared in the file ‘gsl_blas.h’ and should satisfy the
needs of most users.

Note that GSL matrices are implemented using dense-storage so the
interface only includes the corresponding dense-storage BLAS functions.
The full BLAS functionality for band-format and packed-format matrices
is available through the low-level CBLAS interface.  Similarly, GSL
vectors are restricted to positive strides, whereas the low-level CBLAS
interface supports negative strides as specified in the BLAS standard
(1).

The interface for the ‘gsl_cblas’ layer is specified in the file
‘gsl_cblas.h’.  This interface corresponds to the BLAS Technical Forum’s
standard for the C interface to legacy BLAS implementations.  Users who
have access to other conforming CBLAS implementations can use these in
place of the version provided by the library.  Note that users who have
only a Fortran BLAS library can use a CBLAS conformant wrapper to
convert it into a CBLAS library.  A reference CBLAS wrapper for legacy
Fortran implementations exists as part of the CBLAS standard and can be
obtained from Netlib.  The complete set of CBLAS functions is listed in
an *note appendix: 471.

There are three levels of BLAS operations,

`Level 1'       Vector operations, e.g.  y = \alpha x + y
                
                
`Level 2'       Matrix-vector operations, e.g.  y = \alpha A x + \beta y
                
                
`Level 3'       Matrix-matrix operations, e.g.  C = \alpha A B + C
                

Each routine has a name which specifies the operation, the type of
matrices involved and their precisions.  Some of the most common
operations and their names are given below,

`DOT'        scalar product, x^T y
             
             
`AXPY'       vector sum, \alpha x + y
             
             
`MV'         matrix-vector product, A x
             
             
`SV'         matrix-vector solve, inv(A) x
             
             
`MM'         matrix-matrix product, A B
             
             
`SM'         matrix-matrix solve, inv(A) B
             

The types of matrices are,

`GE'       general
           
           
`GB'       general band
           
           
`SY'       symmetric
           
           
`SB'       symmetric band
           
           
`SP'       symmetric packed
           
           
`HE'       hermitian
           
           
`HB'       hermitian band
           
           
`HP'       hermitian packed
           
           
`TR'       triangular
           
           
`TB'       triangular band
           
           
`TP'       triangular packed
           

Each operation is defined for four precisions,

`S'       single real
          
          
`D'       double real
          
          
`C'       single complex
          
          
`Z'       double complex
          

Thus, for example, the name SGEMM stands for “single-precision general
matrix-matrix multiply” and ZGEMM stands for “double-precision complex
matrix-matrix multiply”.

Note that the vector and matrix arguments to BLAS functions must not be
aliased, as the results are undefined when the underlying arrays overlap
(*note Aliasing of arrays: 1f.).

* Menu:

* GSL BLAS Interface::
* Examples: Examples<8>.
* References and Further Reading: References and Further Reading<8>.

   ---------- Footnotes ----------

   (1) In the low-level CBLAS interface, a negative stride accesses the
vector elements in reverse order, i.e.  the i-th element is given by
(N-i)*|incx| for incx < 0.


File: gsl-ref.info,  Node: GSL BLAS Interface,  Next: Examples<8>,  Up: BLAS Support

13.1 GSL BLAS Interface
=======================

GSL provides dense vector and matrix objects, based on the relevant
built-in types.  The library provides an interface to the BLAS
operations which apply to these objects.  The interface to this
functionality is given in the file ‘gsl_blas.h’.

* Menu:

* Level 1::
* Level 2::
* Level 3::


File: gsl-ref.info,  Node: Level 1,  Next: Level 2,  Up: GSL BLAS Interface

13.1.1 Level 1
--------------

 -- Function: int gsl_blas_sdsdot (float alpha, const gsl_vector_float
          * x, const gsl_vector_float * y, float * result)

     This function computes the sum \alpha + x^T y for the vectors ‘x’
     and ‘y’, returning the result in ‘result’.

 -- Function: int gsl_blas_sdot (const gsl_vector_float * x, const
          gsl_vector_float * y, float * result)
 -- Function: int gsl_blas_dsdot (const gsl_vector_float * x, const
          gsl_vector_float * y, double * result)
 -- Function: int gsl_blas_ddot (const gsl_vector * x, const gsl_vector
          * y, double * result)

     These functions compute the scalar product x^T y for the vectors
     ‘x’ and ‘y’, returning the result in ‘result’.

 -- Function: int gsl_blas_cdotu (const gsl_vector_complex_float * x,
          const gsl_vector_complex_float * y, gsl_complex_float * dotu)
 -- Function: int gsl_blas_zdotu (const gsl_vector_complex * x, const
          gsl_vector_complex * y, gsl_complex * dotu)

     These functions compute the complex scalar product x^T y for the
     vectors ‘x’ and ‘y’, returning the result in ‘dotu’

 -- Function: int gsl_blas_cdotc (const gsl_vector_complex_float * x,
          const gsl_vector_complex_float * y, gsl_complex_float * dotc)
 -- Function: int gsl_blas_zdotc (const gsl_vector_complex * x, const
          gsl_vector_complex * y, gsl_complex * dotc)

     These functions compute the complex conjugate scalar product x^H y
     for the vectors ‘x’ and ‘y’, returning the result in ‘dotc’

 -- Function: float gsl_blas_snrm2 (const gsl_vector_float * x)
 -- Function: double gsl_blas_dnrm2 (const gsl_vector * x)

     These functions compute the Euclidean norm ||x||_2 = \sqrt{\sum
     x_i^2} of the vector ‘x’.

 -- Function: float gsl_blas_scnrm2 (const gsl_vector_complex_float * x)
 -- Function: double gsl_blas_dznrm2 (const gsl_vector_complex * x)

     These functions compute the Euclidean norm of the complex vector
     ‘x’,

          ||x||_2 = \sqrt{\sum (\Re(x_i)^2 + \Im(x_i)^2)}.

 -- Function: float gsl_blas_sasum (const gsl_vector_float * x)
 -- Function: double gsl_blas_dasum (const gsl_vector * x)

     These functions compute the absolute sum \sum |x_i| of the elements
     of the vector ‘x’.

 -- Function: float gsl_blas_scasum (const gsl_vector_complex_float * x)
 -- Function: double gsl_blas_dzasum (const gsl_vector_complex * x)

     These functions compute the sum of the magnitudes of the real and
     imaginary parts of the complex vector ‘x’, \sum \left( |\Re(x_i)| +
     |\Im(x_i)| \right).

 -- Function: CBLAS_INDEX_t gsl_blas_isamax (const gsl_vector_float * x)
 -- Function: CBLAS_INDEX_t gsl_blas_idamax (const gsl_vector * x)
 -- Function: CBLAS_INDEX_t gsl_blas_icamax (const
          gsl_vector_complex_float * x)
 -- Function: CBLAS_INDEX_t gsl_blas_izamax (const gsl_vector_complex
          * x)

     These functions return the index of the largest element of the
     vector ‘x’.  The largest element is determined by its absolute
     magnitude for real vectors and by the sum of the magnitudes of the
     real and imaginary parts |\Re(x_i)| + |\Im(x_i)| for complex
     vectors.  If the largest value occurs several times then the index
     of the first occurrence is returned.

 -- Function: int gsl_blas_sswap (gsl_vector_float * x, gsl_vector_float
          * y)
 -- Function: int gsl_blas_dswap (gsl_vector * x, gsl_vector * y)
 -- Function: int gsl_blas_cswap (gsl_vector_complex_float * x,
          gsl_vector_complex_float * y)
 -- Function: int gsl_blas_zswap (gsl_vector_complex * x,
          gsl_vector_complex * y)

     These functions exchange the elements of the vectors ‘x’ and ‘y’.

 -- Function: int gsl_blas_scopy (const gsl_vector_float * x,
          gsl_vector_float * y)
 -- Function: int gsl_blas_dcopy (const gsl_vector * x, gsl_vector * y)
 -- Function: int gsl_blas_ccopy (const gsl_vector_complex_float * x,
          gsl_vector_complex_float * y)
 -- Function: int gsl_blas_zcopy (const gsl_vector_complex * x,
          gsl_vector_complex * y)

     These functions copy the elements of the vector ‘x’ into the vector
     ‘y’.

 -- Function: int gsl_blas_saxpy (float alpha, const gsl_vector_float
          * x, gsl_vector_float * y)
 -- Function: int gsl_blas_daxpy (double alpha, const gsl_vector * x,
          gsl_vector * y)
 -- Function: int gsl_blas_caxpy (const gsl_complex_float alpha, const
          gsl_vector_complex_float * x, gsl_vector_complex_float * y)
 -- Function: int gsl_blas_zaxpy (const gsl_complex alpha, const
          gsl_vector_complex * x, gsl_vector_complex * y)

     These functions compute the sum y = \alpha x + y for the vectors
     ‘x’ and ‘y’.

 -- Function: void gsl_blas_sscal (float alpha, gsl_vector_float * x)
 -- Function: void gsl_blas_dscal (double alpha, gsl_vector * x)
 -- Function: void gsl_blas_cscal (const gsl_complex_float alpha,
          gsl_vector_complex_float * x)
 -- Function: void gsl_blas_zscal (const gsl_complex alpha,
          gsl_vector_complex * x)
 -- Function: void gsl_blas_csscal (float alpha,
          gsl_vector_complex_float * x)
 -- Function: void gsl_blas_zdscal (double alpha, gsl_vector_complex
          * x)

     These functions rescale the vector ‘x’ by the multiplicative factor
     *note alpha: 49a.

 -- Function: int gsl_blas_srotg (float a[], float b[], float c[],
          float s[])
 -- Function: int gsl_blas_drotg (double a[], double b[], double c[],
          double s[])

     These functions compute a Givens rotation (c,s) which zeroes the
     vector (a,b),

          [  c  s ] [ a ] = [ r ]
          [ -s  c ] [ b ]   [ 0 ]

     The variables ‘a’ and ‘b’ are overwritten by the routine.

 -- Function: int gsl_blas_srot (gsl_vector_float * x, gsl_vector_float
          * y, float c, float s)
 -- Function: int gsl_blas_drot (gsl_vector * x, gsl_vector * y, const
          double c, const double s)

     These functions apply a Givens rotation (x', y') = (c x + s y, -s x
     + c y) to the vectors ‘x’, ‘y’.

 -- Function: int gsl_blas_srotmg (float d1[], float d2[], float b1[],
          float b2, float P[])
 -- Function: int gsl_blas_drotmg (double d1[], double d2[],
          double b1[], double b2, double P[])

     These functions compute a modified Givens transformation.  The
     modified Givens transformation is defined in the original Level-1
     BLAS specification, given in the references.

 -- Function: int gsl_blas_srotm (gsl_vector_float * x, gsl_vector_float
          * y, const float P[])
 -- Function: int gsl_blas_drotm (gsl_vector * x, gsl_vector * y, const
          double P[])

     These functions apply a modified Givens transformation.


File: gsl-ref.info,  Node: Level 2,  Next: Level 3,  Prev: Level 1,  Up: GSL BLAS Interface

13.1.2 Level 2
--------------

 -- Function: int gsl_blas_sgemv (CBLAS_TRANSPOSE_t TransA, float alpha,
          const gsl_matrix_float * A, const gsl_vector_float * x,
          float beta, gsl_vector_float * y)
 -- Function: int gsl_blas_dgemv (CBLAS_TRANSPOSE_t TransA,
          double alpha, const gsl_matrix * A, const gsl_vector * x,
          double beta, gsl_vector * y)
 -- Function: int gsl_blas_cgemv (CBLAS_TRANSPOSE_t TransA, const
          gsl_complex_float alpha, const gsl_matrix_complex_float * A,
          const gsl_vector_complex_float * x, const
          gsl_complex_float beta, gsl_vector_complex_float * y)
 -- Function: int gsl_blas_zgemv (CBLAS_TRANSPOSE_t TransA, const
          gsl_complex alpha, const gsl_matrix_complex * A, const
          gsl_vector_complex * x, const gsl_complex beta,
          gsl_vector_complex * y)

     These functions compute the matrix-vector product and sum y =
     \alpha op(A) x + \beta y, where op(A) = A, A^T, A^H for ‘TransA’ =
     ‘CblasNoTrans’, ‘CblasTrans’, ‘CblasConjTrans’.

 -- Function: int gsl_blas_strmv (CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const
          gsl_matrix_float * A, gsl_vector_float * x)
 -- Function: int gsl_blas_dtrmv (CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const gsl_matrix
          * A, gsl_vector * x)
 -- Function: int gsl_blas_ctrmv (CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const
          gsl_matrix_complex_float * A, gsl_vector_complex_float * x)
 -- Function: int gsl_blas_ztrmv (CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const
          gsl_matrix_complex * A, gsl_vector_complex * x)

     These functions compute the matrix-vector product x = op(A) x for
     the triangular matrix ‘A’, where op(A) = A, A^T, A^H for ‘TransA’ =
     ‘CblasNoTrans’, ‘CblasTrans’, ‘CblasConjTrans’.  When ‘Uplo’ is
     ‘CblasUpper’ then the upper triangle of ‘A’ is used, and when
     ‘Uplo’ is ‘CblasLower’ then the lower triangle of ‘A’ is used.  If
     ‘Diag’ is ‘CblasNonUnit’ then the diagonal of the matrix is used,
     but if ‘Diag’ is ‘CblasUnit’ then the diagonal elements of the
     matrix ‘A’ are taken as unity and are not referenced.

 -- Function: int gsl_blas_strsv (CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const
          gsl_matrix_float * A, gsl_vector_float * x)
 -- Function: int gsl_blas_dtrsv (CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const gsl_matrix
          * A, gsl_vector * x)
 -- Function: int gsl_blas_ctrsv (CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const
          gsl_matrix_complex_float * A, gsl_vector_complex_float * x)
 -- Function: int gsl_blas_ztrsv (CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const
          gsl_matrix_complex * A, gsl_vector_complex * x)

     These functions compute inv(op(A)) x for ‘x’, where op(A) = A, A^T,
     A^H for ‘TransA’ = ‘CblasNoTrans’, ‘CblasTrans’, ‘CblasConjTrans’.
     When ‘Uplo’ is ‘CblasUpper’ then the upper triangle of ‘A’ is used,
     and when ‘Uplo’ is ‘CblasLower’ then the lower triangle of ‘A’ is
     used.  If ‘Diag’ is ‘CblasNonUnit’ then the diagonal of the matrix
     is used, but if ‘Diag’ is ‘CblasUnit’ then the diagonal elements of
     the matrix ‘A’ are taken as unity and are not referenced.

 -- Function: int gsl_blas_ssymv (CBLAS_UPLO_t Uplo, float alpha, const
          gsl_matrix_float * A, const gsl_vector_float * x, float beta,
          gsl_vector_float * y)
 -- Function: int gsl_blas_dsymv (CBLAS_UPLO_t Uplo, double alpha, const
          gsl_matrix * A, const gsl_vector * x, double beta, gsl_vector
          * y)

     These functions compute the matrix-vector product and sum y =
     \alpha A x + \beta y for the symmetric matrix ‘A’.  Since the
     matrix ‘A’ is symmetric only its upper half or lower half need to
     be stored.  When ‘Uplo’ is ‘CblasUpper’ then the upper triangle and
     diagonal of ‘A’ are used, and when ‘Uplo’ is ‘CblasLower’ then the
     lower triangle and diagonal of ‘A’ are used.

 -- Function: int gsl_blas_chemv (CBLAS_UPLO_t Uplo, const
          gsl_complex_float alpha, const gsl_matrix_complex_float * A,
          const gsl_vector_complex_float * x, const
          gsl_complex_float beta, gsl_vector_complex_float * y)
 -- Function: int gsl_blas_zhemv (CBLAS_UPLO_t Uplo, const
          gsl_complex alpha, const gsl_matrix_complex * A, const
          gsl_vector_complex * x, const gsl_complex beta,
          gsl_vector_complex * y)

     These functions compute the matrix-vector product and sum y =
     \alpha A x + \beta y for the hermitian matrix ‘A’.  Since the
     matrix ‘A’ is hermitian only its upper half or lower half need to
     be stored.  When ‘Uplo’ is ‘CblasUpper’ then the upper triangle and
     diagonal of ‘A’ are used, and when ‘Uplo’ is ‘CblasLower’ then the
     lower triangle and diagonal of ‘A’ are used.  The imaginary
     elements of the diagonal are automatically assumed to be zero and
     are not referenced.

 -- Function: int gsl_blas_sger (float alpha, const gsl_vector_float
          * x, const gsl_vector_float * y, gsl_matrix_float * A)
 -- Function: int gsl_blas_dger (double alpha, const gsl_vector * x,
          const gsl_vector * y, gsl_matrix * A)
 -- Function: int gsl_blas_cgeru (const gsl_complex_float alpha, const
          gsl_vector_complex_float * x, const gsl_vector_complex_float
          * y, gsl_matrix_complex_float * A)
 -- Function: int gsl_blas_zgeru (const gsl_complex alpha, const
          gsl_vector_complex * x, const gsl_vector_complex * y,
          gsl_matrix_complex * A)

     These functions compute the rank-1 update A = \alpha x y^T + A of
     the matrix ‘A’.

 -- Function: int gsl_blas_cgerc (const gsl_complex_float alpha, const
          gsl_vector_complex_float * x, const gsl_vector_complex_float
          * y, gsl_matrix_complex_float * A)
 -- Function: int gsl_blas_zgerc (const gsl_complex alpha, const
          gsl_vector_complex * x, const gsl_vector_complex * y,
          gsl_matrix_complex * A)

     These functions compute the conjugate rank-1 update A = \alpha x
     y^H + A of the matrix ‘A’.

 -- Function: int gsl_blas_ssyr (CBLAS_UPLO_t Uplo, float alpha, const
          gsl_vector_float * x, gsl_matrix_float * A)
 -- Function: int gsl_blas_dsyr (CBLAS_UPLO_t Uplo, double alpha, const
          gsl_vector * x, gsl_matrix * A)

     These functions compute the symmetric rank-1 update A = \alpha x
     x^T + A of the symmetric matrix ‘A’.  Since the matrix ‘A’ is
     symmetric only its upper half or lower half need to be stored.
     When ‘Uplo’ is ‘CblasUpper’ then the upper triangle and diagonal of
     ‘A’ are used, and when ‘Uplo’ is ‘CblasLower’ then the lower
     triangle and diagonal of ‘A’ are used.

 -- Function: int gsl_blas_cher (CBLAS_UPLO_t Uplo, float alpha, const
          gsl_vector_complex_float * x, gsl_matrix_complex_float * A)
 -- Function: int gsl_blas_zher (CBLAS_UPLO_t Uplo, double alpha, const
          gsl_vector_complex * x, gsl_matrix_complex * A)

     These functions compute the hermitian rank-1 update A = \alpha x
     x^H + A of the hermitian matrix ‘A’.  Since the matrix ‘A’ is
     hermitian only its upper half or lower half need to be stored.
     When ‘Uplo’ is ‘CblasUpper’ then the upper triangle and diagonal of
     ‘A’ are used, and when ‘Uplo’ is ‘CblasLower’ then the lower
     triangle and diagonal of ‘A’ are used.  The imaginary elements of
     the diagonal are automatically set to zero.

 -- Function: int gsl_blas_ssyr2 (CBLAS_UPLO_t Uplo, float alpha, const
          gsl_vector_float * x, const gsl_vector_float * y,
          gsl_matrix_float * A)
 -- Function: int gsl_blas_dsyr2 (CBLAS_UPLO_t Uplo, double alpha, const
          gsl_vector * x, const gsl_vector * y, gsl_matrix * A)

     These functions compute the symmetric rank-2 update A = \alpha x
     y^T + \alpha y x^T + A of the symmetric matrix ‘A’.  Since the
     matrix ‘A’ is symmetric only its upper half or lower half need to
     be stored.  When ‘Uplo’ is ‘CblasUpper’ then the upper triangle and
     diagonal of ‘A’ are used, and when ‘Uplo’ is ‘CblasLower’ then the
     lower triangle and diagonal of ‘A’ are used.

 -- Function: int gsl_blas_cher2 (CBLAS_UPLO_t Uplo, const
          gsl_complex_float alpha, const gsl_vector_complex_float * x,
          const gsl_vector_complex_float * y, gsl_matrix_complex_float
          * A)
 -- Function: int gsl_blas_zher2 (CBLAS_UPLO_t Uplo, const
          gsl_complex alpha, const gsl_vector_complex * x, const
          gsl_vector_complex * y, gsl_matrix_complex * A)

     These functions compute the hermitian rank-2 update A = \alpha x
     y^H + \alpha^* y x^H + A of the hermitian matrix ‘A’.  Since the
     matrix ‘A’ is hermitian only its upper half or lower half need to
     be stored.  When ‘Uplo’ is ‘CblasUpper’ then the upper triangle and
     diagonal of ‘A’ are used, and when ‘Uplo’ is ‘CblasLower’ then the
     lower triangle and diagonal of ‘A’ are used.  The imaginary
     elements of the diagonal are automatically set to zero.


File: gsl-ref.info,  Node: Level 3,  Prev: Level 2,  Up: GSL BLAS Interface

13.1.3 Level 3
--------------

 -- Function: int gsl_blas_sgemm (CBLAS_TRANSPOSE_t TransA,
          CBLAS_TRANSPOSE_t TransB, float alpha, const gsl_matrix_float
          * A, const gsl_matrix_float * B, float beta, gsl_matrix_float
          * C)
 -- Function: int gsl_blas_dgemm (CBLAS_TRANSPOSE_t TransA,
          CBLAS_TRANSPOSE_t TransB, double alpha, const gsl_matrix * A,
          const gsl_matrix * B, double beta, gsl_matrix * C)
 -- Function: int gsl_blas_cgemm (CBLAS_TRANSPOSE_t TransA,
          CBLAS_TRANSPOSE_t TransB, const gsl_complex_float alpha, const
          gsl_matrix_complex_float * A, const gsl_matrix_complex_float
          * B, const gsl_complex_float beta, gsl_matrix_complex_float
          * C)
 -- Function: int gsl_blas_zgemm (CBLAS_TRANSPOSE_t TransA,
          CBLAS_TRANSPOSE_t TransB, const gsl_complex alpha, const
          gsl_matrix_complex * A, const gsl_matrix_complex * B, const
          gsl_complex beta, gsl_matrix_complex * C)

     These functions compute the matrix-matrix product and sum C =
     \alpha op(A) op(B) + \beta C where op(A) = A, A^T, A^H for ‘TransA’
     = ‘CblasNoTrans’, ‘CblasTrans’, ‘CblasConjTrans’ and similarly for
     the parameter ‘TransB’.

 -- Function: int gsl_blas_ssymm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo,
          float alpha, const gsl_matrix_float * A, const
          gsl_matrix_float * B, float beta, gsl_matrix_float * C)
 -- Function: int gsl_blas_dsymm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo,
          double alpha, const gsl_matrix * A, const gsl_matrix * B,
          double beta, gsl_matrix * C)
 -- Function: int gsl_blas_csymm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo,
          const gsl_complex_float alpha, const gsl_matrix_complex_float
          * A, const gsl_matrix_complex_float * B, const
          gsl_complex_float beta, gsl_matrix_complex_float * C)
 -- Function: int gsl_blas_zsymm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo,
          const gsl_complex alpha, const gsl_matrix_complex * A, const
          gsl_matrix_complex * B, const gsl_complex beta,
          gsl_matrix_complex * C)

     These functions compute the matrix-matrix product and sum C =
     \alpha A B + \beta C for ‘Side’ is ‘CblasLeft’ and C = \alpha B A +
     \beta C for ‘Side’ is ‘CblasRight’, where the matrix ‘A’ is
     symmetric.  When ‘Uplo’ is ‘CblasUpper’ then the upper triangle and
     diagonal of ‘A’ are used, and when ‘Uplo’ is ‘CblasLower’ then the
     lower triangle and diagonal of ‘A’ are used.

 -- Function: int gsl_blas_chemm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo,
          const gsl_complex_float alpha, const gsl_matrix_complex_float
          * A, const gsl_matrix_complex_float * B, const
          gsl_complex_float beta, gsl_matrix_complex_float * C)
 -- Function: int gsl_blas_zhemm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo,
          const gsl_complex alpha, const gsl_matrix_complex * A, const
          gsl_matrix_complex * B, const gsl_complex beta,
          gsl_matrix_complex * C)

     These functions compute the matrix-matrix product and sum C =
     \alpha A B + \beta C for ‘Side’ is ‘CblasLeft’ and C = \alpha B A +
     \beta C for ‘Side’ is ‘CblasRight’, where the matrix ‘A’ is
     hermitian.  When ‘Uplo’ is ‘CblasUpper’ then the upper triangle and
     diagonal of ‘A’ are used, and when ‘Uplo’ is ‘CblasLower’ then the
     lower triangle and diagonal of ‘A’ are used.  The imaginary
     elements of the diagonal are automatically set to zero.

 -- Function: int gsl_blas_strmm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, float alpha,
          const gsl_matrix_float * A, gsl_matrix_float * B)
 -- Function: int gsl_blas_dtrmm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, double alpha,
          const gsl_matrix * A, gsl_matrix * B)
 -- Function: int gsl_blas_ctrmm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const
          gsl_complex_float alpha, const gsl_matrix_complex_float * A,
          gsl_matrix_complex_float * B)
 -- Function: int gsl_blas_ztrmm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const
          gsl_complex alpha, const gsl_matrix_complex * A,
          gsl_matrix_complex * B)

     These functions compute the matrix-matrix product B = \alpha op(A)
     B for ‘Side’ is ‘CblasLeft’ and B = \alpha B op(A) for ‘Side’ is
     ‘CblasRight’.  The matrix ‘A’ is triangular and op(A) = A, A^T, A^H
     for ‘TransA’ = ‘CblasNoTrans’, ‘CblasTrans’, ‘CblasConjTrans’.
     When ‘Uplo’ is ‘CblasUpper’ then the upper triangle of ‘A’ is used,
     and when ‘Uplo’ is ‘CblasLower’ then the lower triangle of ‘A’ is
     used.  If ‘Diag’ is ‘CblasNonUnit’ then the diagonal of ‘A’ is
     used, but if ‘Diag’ is ‘CblasUnit’ then the diagonal elements of
     the matrix ‘A’ are taken as unity and are not referenced.

 -- Function: int gsl_blas_strsm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, float alpha,
          const gsl_matrix_float * A, gsl_matrix_float * B)
 -- Function: int gsl_blas_dtrsm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, double alpha,
          const gsl_matrix * A, gsl_matrix * B)
 -- Function: int gsl_blas_ctrsm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const
          gsl_complex_float alpha, const gsl_matrix_complex_float * A,
          gsl_matrix_complex_float * B)
 -- Function: int gsl_blas_ztrsm (CBLAS_SIDE_t Side, CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t TransA, CBLAS_DIAG_t Diag, const
          gsl_complex alpha, const gsl_matrix_complex * A,
          gsl_matrix_complex * B)

     These functions compute the inverse-matrix matrix product B =
     \alpha op(inv(A))B for ‘Side’ is ‘CblasLeft’ and B = \alpha B
     op(inv(A)) for ‘Side’ is ‘CblasRight’.  The matrix ‘A’ is
     triangular and op(A) = A, A^T, A^H for ‘TransA’ = ‘CblasNoTrans’,
     ‘CblasTrans’, ‘CblasConjTrans’.  When ‘Uplo’ is ‘CblasUpper’ then
     the upper triangle of ‘A’ is used, and when ‘Uplo’ is ‘CblasLower’
     then the lower triangle of ‘A’ is used.  If ‘Diag’ is
     ‘CblasNonUnit’ then the diagonal of ‘A’ is used, but if ‘Diag’ is
     ‘CblasUnit’ then the diagonal elements of the matrix ‘A’ are taken
     as unity and are not referenced.

 -- Function: int gsl_blas_ssyrk (CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t Trans, float alpha, const gsl_matrix_float
          * A, float beta, gsl_matrix_float * C)
 -- Function: int gsl_blas_dsyrk (CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t Trans, double alpha, const gsl_matrix * A,
          double beta, gsl_matrix * C)
 -- Function: int gsl_blas_csyrk (CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t Trans, const gsl_complex_float alpha, const
          gsl_matrix_complex_float * A, const gsl_complex_float beta,
          gsl_matrix_complex_float * C)
 -- Function: int gsl_blas_zsyrk (CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t Trans, const gsl_complex alpha, const
          gsl_matrix_complex * A, const gsl_complex beta,
          gsl_matrix_complex * C)

     These functions compute a rank-k update of the symmetric matrix
     ‘C’, C = \alpha A A^T + \beta C when ‘Trans’ is ‘CblasNoTrans’ and
     C = \alpha A^T A + \beta C when ‘Trans’ is ‘CblasTrans’.  Since the
     matrix ‘C’ is symmetric only its upper half or lower half need to
     be stored.  When ‘Uplo’ is ‘CblasUpper’ then the upper triangle and
     diagonal of ‘C’ are used, and when ‘Uplo’ is ‘CblasLower’ then the
     lower triangle and diagonal of ‘C’ are used.

 -- Function: int gsl_blas_cherk (CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t Trans, float alpha, const
          gsl_matrix_complex_float * A, float beta,
          gsl_matrix_complex_float * C)
 -- Function: int gsl_blas_zherk (CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t Trans, double alpha, const
          gsl_matrix_complex * A, double beta, gsl_matrix_complex * C)

     These functions compute a rank-k update of the hermitian matrix
     ‘C’, C = \alpha A A^H + \beta C when ‘Trans’ is ‘CblasNoTrans’ and
     C = \alpha A^H A + \beta C when ‘Trans’ is ‘CblasConjTrans’.  Since
     the matrix ‘C’ is hermitian only its upper half or lower half need
     to be stored.  When ‘Uplo’ is ‘CblasUpper’ then the upper triangle
     and diagonal of ‘C’ are used, and when ‘Uplo’ is ‘CblasLower’ then
     the lower triangle and diagonal of ‘C’ are used.  The imaginary
     elements of the diagonal are automatically set to zero.

 -- Function: int gsl_blas_ssyr2k (CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t Trans, float alpha, const gsl_matrix_float
          * A, const gsl_matrix_float * B, float beta, gsl_matrix_float
          * C)
 -- Function: int gsl_blas_dsyr2k (CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t Trans, double alpha, const gsl_matrix * A,
          const gsl_matrix * B, double beta, gsl_matrix * C)
 -- Function: int gsl_blas_csyr2k (CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t Trans, const gsl_complex_float alpha, const
          gsl_matrix_complex_float * A, const gsl_matrix_complex_float
          * B, const gsl_complex_float beta, gsl_matrix_complex_float
          * C)
 -- Function: int gsl_blas_zsyr2k (CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t Trans, const gsl_complex alpha, const
          gsl_matrix_complex * A, const gsl_matrix_complex * B, const
          gsl_complex beta, gsl_matrix_complex * C)

     These functions compute a rank-2k update of the symmetric matrix
     ‘C’, C = \alpha A B^T + \alpha B A^T + \beta C when ‘Trans’ is
     ‘CblasNoTrans’ and C = \alpha A^T B + \alpha B^T A + \beta C when
     ‘Trans’ is ‘CblasTrans’.  Since the matrix ‘C’ is symmetric only
     its upper half or lower half need to be stored.  When ‘Uplo’ is
     ‘CblasUpper’ then the upper triangle and diagonal of ‘C’ are used,
     and when ‘Uplo’ is ‘CblasLower’ then the lower triangle and
     diagonal of ‘C’ are used.

 -- Function: int gsl_blas_cher2k (CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t Trans, const gsl_complex_float alpha, const
          gsl_matrix_complex_float * A, const gsl_matrix_complex_float
          * B, float beta, gsl_matrix_complex_float * C)
 -- Function: int gsl_blas_zher2k (CBLAS_UPLO_t Uplo,
          CBLAS_TRANSPOSE_t Trans, const gsl_complex alpha, const
          gsl_matrix_complex * A, const gsl_matrix_complex * B,
          double beta, gsl_matrix_complex * C)

     These functions compute a rank-2k update of the hermitian matrix
     ‘C’, C = \alpha A B^H + \alpha^* B A^H + \beta C when ‘Trans’ is
     ‘CblasNoTrans’ and C = \alpha A^H B + \alpha^* B^H A + \beta C when
     ‘Trans’ is ‘CblasConjTrans’.  Since the matrix ‘C’ is hermitian
     only its upper half or lower half need to be stored.  When ‘Uplo’
     is ‘CblasUpper’ then the upper triangle and diagonal of ‘C’ are
     used, and when ‘Uplo’ is ‘CblasLower’ then the lower triangle and
     diagonal of ‘C’ are used.  The imaginary elements of the diagonal
     are automatically set to zero.


File: gsl-ref.info,  Node: Examples<8>,  Next: References and Further Reading<8>,  Prev: GSL BLAS Interface,  Up: BLAS Support

13.2 Examples
=============

The following program computes the product of two matrices using the
Level-3 BLAS function DGEMM,

     [ 0.11 0.12 0.13 ]  [ 1011 1012 ]     [ 367.76 368.12 ]
     [ 0.21 0.22 0.23 ]  [ 1021 1022 ]  =  [ 674.06 674.72 ]
                         [ 1031 1032 ]

The matrices are stored in row major order, according to the C
convention for arrays.

     #include <stdio.h>
     #include <gsl/gsl_blas.h>

     int
     main (void)
     {
       double a[] = { 0.11, 0.12, 0.13,
                      0.21, 0.22, 0.23 };

       double b[] = { 1011, 1012,
                      1021, 1022,
                      1031, 1032 };

       double c[] = { 0.00, 0.00,
                      0.00, 0.00 };

       gsl_matrix_view A = gsl_matrix_view_array(a, 2, 3);
       gsl_matrix_view B = gsl_matrix_view_array(b, 3, 2);
       gsl_matrix_view C = gsl_matrix_view_array(c, 2, 2);

       /* Compute C = A B */

       gsl_blas_dgemm (CblasNoTrans, CblasNoTrans,
                       1.0, &A.matrix, &B.matrix,
                       0.0, &C.matrix);

       printf ("[ %g, %g\n", c[0], c[1]);
       printf ("  %g, %g ]\n", c[2], c[3]);

       return 0;
     }

Here is the output from the program,

     [ 367.76, 368.12
       674.06, 674.72 ]


File: gsl-ref.info,  Node: References and Further Reading<8>,  Prev: Examples<8>,  Up: BLAS Support

13.3 References and Further Reading
===================================

Information on the BLAS standards, including both the legacy and updated
interface standards, is available online from the BLAS Homepage and BLAS
Technical Forum web-site.

   * BLAS Homepage, ‘http://www.netlib.org/blas/’

   * BLAS Technical Forum, ‘http://www.netlib.org/blas/blast-forum/’

The following papers contain the specifications for Level 1, Level 2 and
Level 3 BLAS.

   * C. Lawson, R. Hanson, D. Kincaid, F. Krogh, “Basic Linear Algebra
     Subprograms for Fortran Usage”, ACM Transactions on Mathematical
     Software, Vol.: 5 (1979), Pages 308–325.

   * J.J. Dongarra, J. DuCroz, S. Hammarling, R. Hanson, “An Extended
     Set of Fortran Basic Linear Algebra Subprograms”, ACM Transactions
     on Mathematical Software, Vol.: 14, No.: 1 (1988), Pages 1–32.

   * J.J. Dongarra, I. Duff, J. DuCroz, S. Hammarling, “A Set of Level 3
     Basic Linear Algebra Subprograms”, ACM Transactions on Mathematical
     Software, Vol.: 16 (1990), Pages 1–28.

Postscript versions of the latter two papers are available from
‘http://www.netlib.org/blas/’.  A CBLAS wrapper for Fortran BLAS
libraries is available from the same location.


File: gsl-ref.info,  Node: Linear Algebra,  Next: Eigensystems,  Prev: BLAS Support,  Up: Top

14 Linear Algebra
*****************

This chapter describes functions for solving linear systems.  The
library provides linear algebra operations which operate directly on the
*note gsl_vector: 35f. and *note gsl_matrix: 3a1. objects.  These
routines use the standard algorithms from Golub & Van Loan’s `Matrix
Computations' with Level-1 and Level-2 BLAS calls for efficiency.

The functions described in this chapter are declared in the header file
‘gsl_linalg.h’.

* Menu:

* LU Decomposition::
* QR Decomposition::
* QR Decomposition with Column Pivoting::
* LQ Decomposition::
* Complete Orthogonal Decomposition::
* Singular Value Decomposition::
* Cholesky Decomposition::
* Pivoted Cholesky Decomposition::
* Modified Cholesky Decomposition::
* LDLT Decomposition::
* Tridiagonal Decomposition of Real Symmetric Matrices::
* Tridiagonal Decomposition of Hermitian Matrices::
* Hessenberg Decomposition of Real Matrices::
* Hessenberg-Triangular Decomposition of Real Matrices::
* Bidiagonalization::
* Givens Rotations::
* Householder Transformations::
* Householder solver for linear systems::
* Tridiagonal Systems::
* Triangular Systems::
* Banded Systems::
* Balancing::
* Examples: Examples<9>.
* References and Further Reading: References and Further Reading<9>.


File: gsl-ref.info,  Node: LU Decomposition,  Next: QR Decomposition,  Up: Linear Algebra

14.1 LU Decomposition
=====================

A general M-by-N matrix A has an LU decomposition

     P A = L U

where P is an M-by-M permutation matrix, L is M-by-\min(M,N) and U is
\min(M,N)-by-N. For square matrices, L is a lower unit triangular matrix
and U is upper triangular.  For M > N, L is a unit lower trapezoidal
matrix of size M-by-N. For M < N, U is upper trapezoidal of size M-by-N.
For square matrices this decomposition can be used to convert the linear
system A x = b into a pair of triangular systems (L y = P b, U x = y),
which can be solved by forward and back-substitution.  Note that the LU
decomposition is valid for singular matrices.

 -- Function: int gsl_linalg_LU_decomp (gsl_matrix * A, gsl_permutation
          * p, int * signum)
 -- Function: int gsl_linalg_complex_LU_decomp (gsl_matrix_complex * A,
          gsl_permutation * p, int * signum)

     These functions factorize the matrix ‘A’ into the LU decomposition
     PA = LU. On output the diagonal and upper triangular (or
     trapezoidal) part of the input matrix ‘A’ contain the matrix U. The
     lower triangular (or trapezoidal) part of the input matrix
     (excluding the diagonal) contains L. The diagonal elements of L are
     unity, and are not stored.

     The permutation matrix P is encoded in the permutation ‘p’ on
     output.  The j-th column of the matrix P is given by the k-th
     column of the identity matrix, where k = p_j the j-th element of
     the permutation vector.  The sign of the permutation is given by
     ‘signum’.  It has the value (-1)^n, where n is the number of
     interchanges in the permutation.

     The algorithm used in the decomposition is Gaussian Elimination
     with partial pivoting (Golub & Van Loan, `Matrix Computations',
     Algorithm 3.4.1), combined with a recursive algorithm based on
     Level 3 BLAS (Peise and Bientinesi, 2016).

 -- Function: int gsl_linalg_LU_solve (const gsl_matrix * LU, const
          gsl_permutation * p, const gsl_vector * b, gsl_vector * x)
 -- Function: int gsl_linalg_complex_LU_solve (const gsl_matrix_complex
          * LU, const gsl_permutation * p, const gsl_vector_complex * b,
          gsl_vector_complex * x)

     These functions solve the square system A x = b using the LU
     decomposition of A into (‘LU’, ‘p’) given by *note
     gsl_linalg_LU_decomp(): 4e7. or *note
     gsl_linalg_complex_LU_decomp(): 4e8. as input.

 -- Function: int gsl_linalg_LU_svx (const gsl_matrix * LU, const
          gsl_permutation * p, gsl_vector * x)
 -- Function: int gsl_linalg_complex_LU_svx (const gsl_matrix_complex
          * LU, const gsl_permutation * p, gsl_vector_complex * x)

     These functions solve the square system A x = b in-place using the
     precomputed LU decomposition of A into (‘LU’, ‘p’).  On input ‘x’
     should contain the right-hand side b, which is replaced by the
     solution on output.

 -- Function: int gsl_linalg_LU_refine (const gsl_matrix * A, const
          gsl_matrix * LU, const gsl_permutation * p, const gsl_vector
          * b, gsl_vector * x, gsl_vector * work)
 -- Function: int gsl_linalg_complex_LU_refine (const gsl_matrix_complex
          * A, const gsl_matrix_complex * LU, const gsl_permutation * p,
          const gsl_vector_complex * b, gsl_vector_complex * x,
          gsl_vector_complex * work)

     These functions apply an iterative improvement to ‘x’, the solution
     of A x = b, from the precomputed LU decomposition of A into (‘LU’,
     ‘p’).  Additional workspace of length ‘N’ is required in ‘work’.

 -- Function: int gsl_linalg_LU_invert (const gsl_matrix * LU, const
          gsl_permutation * p, gsl_matrix * inverse)
 -- Function: int gsl_linalg_complex_LU_invert (const gsl_matrix_complex
          * LU, const gsl_permutation * p, gsl_matrix_complex * inverse)

     These functions compute the inverse of a matrix A from its LU
     decomposition (‘LU’, ‘p’), storing the result in the matrix
     ‘inverse’.  The inverse is computed by computing the inverses
     U^{-1}, L^{-1} and finally forming the product A^{-1} = U^{-1}
     L^{-1} P. Each step is based on Level 3 BLAS calls.

     It is preferable to avoid direct use of the inverse whenever
     possible, as the linear solver functions can obtain the same result
     more efficiently and reliably (consult any introductory textbook on
     numerical linear algebra for details).

 -- Function: int gsl_linalg_LU_invx (gsl_matrix * LU, const
          gsl_permutation * p)
 -- Function: int gsl_linalg_complex_LU_invx (gsl_matrix_complex * LU,
          const gsl_permutation * p)

     These functions compute the inverse of a matrix A from its LU
     decomposition (‘LU’, ‘p’), storing the result in-place in the
     matrix ‘LU’.  The inverse is computed by computing the inverses
     U^{-1}, L^{-1} and finally forming the product A^{-1} = U^{-1}
     L^{-1} P. Each step is based on Level 3 BLAS calls.

     It is preferable to avoid direct use of the inverse whenever
     possible, as the linear solver functions can obtain the same result
     more efficiently and reliably (consult any introductory textbook on
     numerical linear algebra for details).

 -- Function: double gsl_linalg_LU_det (gsl_matrix * LU, int signum)
 -- Function: gsl_complex gsl_linalg_complex_LU_det (gsl_matrix_complex
          * LU, int signum)

     These functions compute the determinant of a matrix A from its LU
     decomposition, ‘LU’.  The determinant is computed as the product of
     the diagonal elements of U and the sign of the row permutation
     ‘signum’.

 -- Function: double gsl_linalg_LU_lndet (gsl_matrix * LU)
 -- Function: double gsl_linalg_complex_LU_lndet (gsl_matrix_complex
          * LU)

     These functions compute the logarithm of the absolute value of the
     determinant of a matrix A, \ln|\det(A)|, from its LU decomposition,
     ‘LU’.  This function may be useful if the direct computation of the
     determinant would overflow or underflow.

 -- Function: int gsl_linalg_LU_sgndet (gsl_matrix * LU, int signum)
 -- Function: gsl_complex gsl_linalg_complex_LU_sgndet
          (gsl_matrix_complex * LU, int signum)

     These functions compute the sign or phase factor of the determinant
     of a matrix A, \det(A)/|\det(A)|, from its LU decomposition, ‘LU’.


File: gsl-ref.info,  Node: QR Decomposition,  Next: QR Decomposition with Column Pivoting,  Prev: LU Decomposition,  Up: Linear Algebra

14.2 QR Decomposition
=====================

A general rectangular M-by-N matrix A has a QR decomposition into the
product of an orthogonal M-by-M square matrix Q (where Q^T Q = I) and an
M-by-N right-triangular matrix R,

     A = Q R

This decomposition can be used to convert the square linear system A x =
b into the triangular system R x = Q^T b, which can be solved by
back-substitution.  Another use of the QR decomposition is to compute an
orthonormal basis for a set of vectors.  The first N columns of Q form
an orthonormal basis for the range of A, ran(A), when A has full column
rank.

When M > N, the bottom M - N rows of R are zero, and so A can be
naturally partioned as

     A = [ Q_1 Q_2 ] [ R_1  ] = Q_1 R_1
                     [  0   ]

where R_1 is N-by-N upper triangular, Q_1 is M-by-N, and Q_2 is
M-by-(M-N). Q_1 R_1 is sometimes called the `thin' or `reduced' QR
decomposition.  The solution of the least squares problem \min_x || b -
A x ||^2 when A has full rank is:

     x = R_1^{-1} c_1

where c_1 is the first N elements of Q^T b.  If A is rank deficient, see
*note QR Decomposition with Column Pivoting: 4fb. and *note Complete
Orthogonal Decomposition: 4fc.

GSL offers two interfaces for the QR decomposition.  The first proceeds
by zeroing out columns below the diagonal of A, one column at a time
using Householder transforms.  In this method, the factor Q is
represented as a product of Householder reflectors:

     Q = H_n \cdots H_2 H_1

where each H_i = I - \tau_i v_i v_i^T for a scalar \tau_i and column
vector v_i.  In this method, functions which compute the full matrix Q
or apply Q^T to a right hand side vector operate by applying the
Householder matrices one at a time using Level 2 BLAS.

The second interface is based on a Level 3 BLAS block recursive
algorithm developed by Elmroth and Gustavson.  In this case, Q is
written in block form as

     Q = I - V T V^T

where V is an M-by-N matrix of the column vectors v_i and T is an N-by-N
upper triangular matrix, whose diagonal elements are the \tau_i.
Computing the full T, while requiring more flops than the Level 2
approach, offers the advantage that all standard operations can take
advantage of cache efficient Level 3 BLAS operations, and so this method
often performs faster than the Level 2 approach.  The functions for the
recursive block algorithm have a ‘_r’ suffix, and it is recommended to
use this interface for performance critical applications.

 -- Function: int gsl_linalg_QR_decomp (gsl_matrix * A, gsl_vector
          * tau)

     This function factorizes the M-by-N matrix ‘A’ into the QR
     decomposition A = Q R. On output the diagonal and upper triangular
     part of the input matrix contain the matrix R. The vector ‘tau’ and
     the columns of the lower triangular part of the matrix ‘A’ contain
     the Householder coefficients and Householder vectors which encode
     the orthogonal matrix ‘Q’.  The vector ‘tau’ must be of length
     k=\min(M,N). The matrix Q is related to these components by, Q =
     H_k ... H_2 H_1 where H_i = I - \tau_i v_i v_i^T and v_i is the
     Householder vector v_i = (0,...,1,A(i+1,i),A(i+2,i),...,A(m,i)).
     This is the same storage scheme as used by LAPACK.

     The algorithm used to perform the decomposition is Householder QR
     (Golub & Van Loan, “Matrix Computations”, Algorithm 5.2.1).  It is
     recommended to use this function only for small to modest sized
     matrices, or if M < N.

 -- Function: int gsl_linalg_QR_decomp_r (gsl_matrix * A, gsl_matrix
          * T)

     This function factors the M-by-N matrix ‘A’ into the QR
     decomposition A = Q R using the recursive Level 3 BLAS algorithm of
     Elmroth and Gustavson.  On output the diagonal and upper triangular
     part of ‘A’ contain the matrix R. The N-by-N matrix ‘T’ stores the
     upper triangular factor appearing in Q. The matrix Q is given by Q
     = I - V T V^T, where the elements below the diagonal of ‘A’ contain
     the columns of V on output.

     This algorithm requires M \ge N and performs best for “tall-skinny”
     matrices, i.e.  M \gg N.

 -- Function: int gsl_linalg_QR_solve (const gsl_matrix * QR, const
          gsl_vector * tau, const gsl_vector * b, gsl_vector * x)
 -- Function: int gsl_linalg_QR_solve_r (const gsl_matrix * QR, const
          gsl_matrix * T, const gsl_vector * b, gsl_vector * x)

     These functions solve the square system A x = b using the QR
     decomposition of A held in (‘QR’, ‘tau’) or (‘QR’, ‘T’) which must
     have been computed previously with *note gsl_linalg_QR_decomp():
     4fd. or *note gsl_linalg_QR_decomp_r(): 4fe.  The least-squares
     solution for rectangular systems can be found using *note
     gsl_linalg_QR_lssolve(): 501. or *note gsl_linalg_QR_lssolve_r():
     502.

 -- Function: int gsl_linalg_QR_svx (const gsl_matrix * QR, const
          gsl_vector * tau, gsl_vector * x)

     This function solves the square system A x = b in-place using the
     QR decomposition of A held in (‘QR’, ‘tau’) which must have been
     computed previously by *note gsl_linalg_QR_decomp(): 4fd.  On input
     ‘x’ should contain the right-hand side b, which is replaced by the
     solution on output.

 -- Function: int gsl_linalg_QR_lssolve (const gsl_matrix * QR, const
          gsl_vector * tau, const gsl_vector * b, gsl_vector * x,
          gsl_vector * residual)

     This function finds the least squares solution to the
     overdetermined system A x = b where the matrix ‘A’ has more rows
     than columns.  The least squares solution minimizes the Euclidean
     norm of the residual, ||Ax - b||.The routine requires as input the
     QR decomposition of A into (‘QR’, ‘tau’) given by *note
     gsl_linalg_QR_decomp(): 4fd.  The solution is returned in ‘x’.  The
     residual is computed as a by-product and stored in ‘residual’.

 -- Function: int gsl_linalg_QR_lssolve_r (const gsl_matrix * QR, const
          gsl_matrix * T, const gsl_vector * b, gsl_vector * x,
          gsl_vector * work)

     This function finds the least squares solution to the
     overdetermined system A x = b where the matrix ‘A’ has more rows
     than columns.  The least squares solution minimizes the Euclidean
     norm of the residual, ||b - Ax||.The routine requires as input the
     QR decomposition of A into (‘QR’, ‘T’) given by *note
     gsl_linalg_QR_decomp_r(): 4fe.  The parameter ‘x’ is of length M.
     The solution x is returned in the first N rows of ‘x’, i.e.  x =
     ‘x[0], x[1], ..., x[N-1]’.  The last M - N rows of ‘x’ contain a
     vector whose norm is equal to the residual norm || b - A x ||.
     This similar to the behavior of LAPACK DGELS. Additional workspace
     of length N is required in ‘work’.

 -- Function: int gsl_linalg_QR_QTvec (const gsl_matrix * QR, const
          gsl_vector * tau, gsl_vector * v)
 -- Function: int gsl_linalg_QR_QTvec_r (const gsl_matrix * QR, const
          gsl_matrix * T, gsl_vector * v, gsl_vector * work)

     These functions apply the matrix Q^T encoded in the decomposition
     (‘QR’, ‘tau’) or (‘QR’, ‘T’) to the vector ‘v’, storing the result
     Q^T v in ‘v’.  The matrix multiplication is carried out directly
     using the encoding of the Householder vectors without needing to
     form the full matrix Q^T.

     The recursive variant ‘QTvec_r’ requires additional workspace of
     size N in ‘work’.

 -- Function: int gsl_linalg_QR_Qvec (const gsl_matrix * QR, const
          gsl_vector * tau, gsl_vector * v)

     This function applies the matrix Q encoded in the decomposition
     (‘QR’, ‘tau’) to the vector ‘v’, storing the result Q v in ‘v’.
     The matrix multiplication is carried out directly using the
     encoding of the Householder vectors without needing to form the
     full matrix Q.

 -- Function: int gsl_linalg_QR_QTmat (const gsl_matrix * QR, const
          gsl_vector * tau, gsl_matrix * B)
 -- Function: int gsl_linalg_QR_QTmat_r (const gsl_matrix * QR, const
          gsl_matrix * T, gsl_matrix * B, gsl_matrix * work)

     This function applies the matrix Q^T encoded in the decomposition
     (‘QR’, ‘tau’) or (‘QR’, ‘T’) to the M-by-K matrix ‘B’, storing the
     result Q^T B in ‘B’.  The matrix multiplication is carried out
     directly using the encoding of the Householder vectors without
     needing to form the full matrix Q^T.

     The recursive variant ‘QTmat_r’ requires additional workspace of
     size N-by-K in ‘work’.

 -- Function: int gsl_linalg_QR_Rsolve (const gsl_matrix * QR, const
          gsl_vector * b, gsl_vector * x)

     This function solves the triangular system R x = b for ‘x’.  It may
     be useful if the product b' = Q^T b has already been computed using
     *note gsl_linalg_QR_QTvec(): 504.

 -- Function: int gsl_linalg_QR_Rsvx (const gsl_matrix * QR, gsl_vector
          * x)

     This function solves the triangular system R x = b for ‘x’
     in-place.  On input ‘x’ should contain the right-hand side b and is
     replaced by the solution on output.  This function may be useful if
     the product b' = Q^T b has already been computed using *note
     gsl_linalg_QR_QTvec(): 504.

 -- Function: int gsl_linalg_QR_unpack (const gsl_matrix * QR, const
          gsl_vector * tau, gsl_matrix * Q, gsl_matrix * R)

     This function unpacks the encoded QR decomposition (‘QR’, ‘tau’)
     into the matrices ‘Q’ and ‘R’, where ‘Q’ is M-by-M and ‘R’ is
     M-by-N.

 -- Function: int gsl_linalg_QR_unpack_r (const gsl_matrix * QR, const
          gsl_matrix * T, gsl_matrix * Q, gsl_matrix * R)

     This function unpacks the encoded QR decomposition (‘QR’, ‘T’) as
     output from *note gsl_linalg_QR_decomp_r(): 4fe. into the matrices
     ‘Q’ and ‘R’, where ‘Q’ is M-by-M and ‘R’ is N-by-N. Note that the
     full R matrix is M-by-N, however the lower trapezoidal portion is
     zero, so only the upper triangular factor is stored.

 -- Function: int gsl_linalg_QR_QRsolve (gsl_matrix * Q, gsl_matrix * R,
          const gsl_vector * b, gsl_vector * x)

     This function solves the system R x = Q^T b for ‘x’.  It can be
     used when the QR decomposition of a matrix is available in unpacked
     form as (‘Q’, ‘R’).

 -- Function: int gsl_linalg_QR_update (gsl_matrix * Q, gsl_matrix * R,
          gsl_vector * w, const gsl_vector * v)

     This function performs a rank-1 update w v^T of the QR
     decomposition (‘Q’, ‘R’).  The update is given by Q'R' = Q (R + w
     v^T) where the output matrices Q and R are also orthogonal and
     right triangular.  Note that ‘w’ is destroyed by the update.

 -- Function: int gsl_linalg_R_solve (const gsl_matrix * R, const
          gsl_vector * b, gsl_vector * x)

     This function solves the triangular system R x = b for the N-by-N
     matrix ‘R’.

 -- Function: int gsl_linalg_R_svx (const gsl_matrix * R, gsl_vector
          * x)

     This function solves the triangular system R x = b in-place.  On
     input ‘x’ should contain the right-hand side b, which is replaced
     by the solution on output.

 -- Function: int gsl_linalg_QR_rcond (const gsl_matrix * QR, double
          * rcond, gsl_vector * work)

     This function estimates the reciprocal condition number (using the
     1-norm) of the R factor, stored in the upper triangle of ‘QR’.  The
     reciprocal condition number estimate, defined as 1 / (||R||_1 \cdot
     ||R^{-1}||_1), is stored in ‘rcond’.  Additional workspace of size
     3 N is required in ‘work’.

* Menu:

* Triangle on Top of Rectangle::


File: gsl-ref.info,  Node: Triangle on Top of Rectangle,  Up: QR Decomposition

14.2.1 Triangle on Top of Rectangle
-----------------------------------

This section provides routines for computing the QR decomposition of the
specialized matrix

     [ S ] = Q R
     [ A ]

where S is an N-by-N upper triangular matrix, and A is an M-by-N dense
matrix.  This type of matrix arises, for example, in the sequential TSQR
algorithm.  The Elmroth and Gustavson algorithm is used to efficiently
factor this matrix.  Due to the upper triangular factor, the Q matrix
takes the form

     Q = I - V T V^T

with

     V = [ I ]
         [ Y ]

and Y is dense and of the same dimensions as A.

 -- Function: int gsl_linalg_QR_TR_decomp (gsl_matrix * S, gsl_matrix
          * A, gsl_matrix * T)

     This function computes the QR decomposition of the matrix (S ; A),
     where S is N-by-N upper triangular and A is M-by-N dense.  On
     output, S is replaced by the R factor, and A is replaced by Y.


File: gsl-ref.info,  Node: QR Decomposition with Column Pivoting,  Next: LQ Decomposition,  Prev: QR Decomposition,  Up: Linear Algebra

14.3 QR Decomposition with Column Pivoting
==========================================

The QR decomposition of an M-by-N matrix A can be extended to the rank
deficient case by introducing a column permutation P,

     A P = Q R

The first r columns of Q form an orthonormal basis for the range of A
for a matrix with column rank r.  This decomposition can also be used to
convert the square linear system A x = b into the triangular system R y
= Q^T b, x = P y, which can be solved by back-substitution and
permutation.  We denote the QR decomposition with column pivoting by
QRP^T since A = Q R P^T. When A is rank deficient with r = {\rm
rank}(A), the matrix R can be partitioned as

     R = [ R11 R12 ] =~ [ R11 R12 ]
         [  0  R22 ]    [  0   0  ]

where R_{11} is r-by-r and nonsingular.  In this case, a `basic' least
squares solution for the overdetermined system A x = b can be obtained
as

     x = P [ R11^-1 c1 ]
           [     0     ]

where c_1 consists of the first r elements of Q^T b.  This basic
solution is not guaranteed to be the minimum norm solution unless R_{12}
= 0 (see *note Complete Orthogonal Decomposition: 4fc.).

 -- Function: int gsl_linalg_QRPT_decomp (gsl_matrix * A, gsl_vector
          * tau, gsl_permutation * p, int * signum, gsl_vector * norm)

     This function factorizes the M-by-N matrix ‘A’ into the QRP^T
     decomposition A = Q R P^T. On output the diagonal and upper
     triangular part of the input matrix contain the matrix R. The
     permutation matrix P is stored in the permutation ‘p’.  The sign of
     the permutation is given by ‘signum’.  It has the value (-1)^n,
     where n is the number of interchanges in the permutation.  The
     vector ‘tau’ and the columns of the lower triangular part of the
     matrix ‘A’ contain the Householder coefficients and vectors which
     encode the orthogonal matrix ‘Q’.  The vector ‘tau’ must be of
     length k=\min(M,N). The matrix Q is related to these components by,
     Q = Q_k ... Q_2 Q_1 where Q_i = I - \tau_i v_i v_i^T and v_i is the
     Householder vector

          v_i = (0,...,1,A(i+1,i),A(i+2,i),...,A(m,i))

     This is the same storage scheme as used by LAPACK. The vector
     ‘norm’ is a workspace of length ‘N’ used for column pivoting.

     The algorithm used to perform the decomposition is Householder QR
     with column pivoting (Golub & Van Loan, “Matrix Computations”,
     Algorithm 5.4.1).

 -- Function: int gsl_linalg_QRPT_decomp2 (const gsl_matrix * A,
          gsl_matrix * q, gsl_matrix * r, gsl_vector * tau,
          gsl_permutation * p, int * signum, gsl_vector * norm)

     This function factorizes the matrix ‘A’ into the decomposition A =
     Q R P^T without modifying ‘A’ itself and storing the output in the
     separate matrices ‘q’ and ‘r’.

 -- Function: int gsl_linalg_QRPT_solve (const gsl_matrix * QR, const
          gsl_vector * tau, const gsl_permutation * p, const gsl_vector
          * b, gsl_vector * x)

     This function solves the square system A x = b using the QRP^T
     decomposition of A held in (‘QR’, ‘tau’, ‘p’) which must have been
     computed previously by *note gsl_linalg_QRPT_decomp(): 515.

 -- Function: int gsl_linalg_QRPT_svx (const gsl_matrix * QR, const
          gsl_vector * tau, const gsl_permutation * p, gsl_vector * x)

     This function solves the square system A x = b in-place using the
     QRP^T decomposition of A held in (‘QR’, ‘tau’, ‘p’).  On input ‘x’
     should contain the right-hand side b, which is replaced by the
     solution on output.

 -- Function: int gsl_linalg_QRPT_lssolve (const gsl_matrix * QR, const
          gsl_vector * tau, const gsl_permutation * p, const gsl_vector
          * b, gsl_vector * x, gsl_vector * residual)

     This function finds the least squares solution to the
     overdetermined system A x = b where the matrix ‘A’ has more rows
     than columns and is assumed to have full rank.  The least squares
     solution minimizes the Euclidean norm of the residual, ||b - A x||.
     The routine requires as input the QR decomposition of A into (‘QR’,
     ‘tau’, ‘p’) given by *note gsl_linalg_QRPT_decomp(): 515.  The
     solution is returned in ‘x’.  The residual is computed as a
     by-product and stored in ‘residual’.  For rank deficient matrices,
     *note gsl_linalg_QRPT_lssolve2(): 51a. should be used instead.

 -- Function: int gsl_linalg_QRPT_lssolve2 (const gsl_matrix * QR, const
          gsl_vector * tau, const gsl_permutation * p, const gsl_vector
          * b, const size_t rank, gsl_vector * x, gsl_vector * residual)

     This function finds the least squares solution to the
     overdetermined system A x = b where the matrix ‘A’ has more rows
     than columns and has rank given by the input ‘rank’.  If the user
     does not know the rank of A, the routine *note
     gsl_linalg_QRPT_rank(): 51b. can be called to estimate it.  The
     least squares solution is the so-called “basic” solution discussed
     above and may not be the minimum norm solution.  The routine
     requires as input the QR decomposition of A into (‘QR’, ‘tau’, ‘p’)
     given by *note gsl_linalg_QRPT_decomp(): 515.  The solution is
     returned in ‘x’.  The residual is computed as a by-product and
     stored in ‘residual’.

 -- Function: int gsl_linalg_QRPT_QRsolve (const gsl_matrix * Q, const
          gsl_matrix * R, const gsl_permutation * p, const gsl_vector
          * b, gsl_vector * x)

     This function solves the square system R P^T x = Q^T b for ‘x’.  It
     can be used when the QR decomposition of a matrix is available in
     unpacked form as (‘Q’, ‘R’).

 -- Function: int gsl_linalg_QRPT_update (gsl_matrix * Q, gsl_matrix
          * R, const gsl_permutation * p, gsl_vector * w, const
          gsl_vector * v)

     This function performs a rank-1 update w v^T of the QRP^T
     decomposition (‘Q’, ‘R’, ‘p’).  The update is given by Q'R' = Q (R
     + w v^T P) where the output matrices Q' and R' are also orthogonal
     and right triangular.  Note that ‘w’ is destroyed by the update.
     The permutation ‘p’ is not changed.

 -- Function: int gsl_linalg_QRPT_Rsolve (const gsl_matrix * QR, const
          gsl_permutation * p, const gsl_vector * b, gsl_vector * x)

     This function solves the triangular system R P^T x = b for the
     N-by-N matrix R contained in ‘QR’.

 -- Function: int gsl_linalg_QRPT_Rsvx (const gsl_matrix * QR, const
          gsl_permutation * p, gsl_vector * x)

     This function solves the triangular system R P^T x = b in-place for
     the N-by-N matrix R contained in ‘QR’.  On input ‘x’ should contain
     the right-hand side b, which is replaced by the solution on output.

 -- Function: size_t gsl_linalg_QRPT_rank (const gsl_matrix * QR, const
          double tol)

     This function estimates the rank of the triangular matrix R
     contained in ‘QR’.  The algorithm simply counts the number of
     diagonal elements of R whose absolute value is greater than the
     specified tolerance ‘tol’.  If the input ‘tol’ is negative, a
     default value of 20 (M + N) eps(max(|diag(R)|)) is used.

 -- Function: int gsl_linalg_QRPT_rcond (const gsl_matrix * QR, double
          * rcond, gsl_vector * work)

     This function estimates the reciprocal condition number (using the
     1-norm) of the R factor, stored in the upper triangle of ‘QR’.  The
     reciprocal condition number estimate, defined as 1 / (||R||_1 \cdot
     ||R^{-1}||_1), is stored in ‘rcond’.  Additional workspace of size
     3 N is required in ‘work’.


File: gsl-ref.info,  Node: LQ Decomposition,  Next: Complete Orthogonal Decomposition,  Prev: QR Decomposition with Column Pivoting,  Up: Linear Algebra

14.4 LQ Decomposition
=====================

A general rectangular M-by-N matrix A has a LQ decomposition into the
product of a lower trapezoidal M-by-N matrix L and an orthogonal N-by-N
square matrix Q:

     A = L Q

If M \le N, then L can be written as L = (L_1 \quad 0) where L_1 is
M-by-M lower triangular, and

     A = \begin{pmatrix} L_1 & 0 \end{pmatrix} \begin{pmatrix} Q_1 \\ Q_2 \end{pmatrix} = L_1 Q_1

where Q_1 consists of the first M rows of Q, and Q_2 contains the
remaining N - M rows.  The LQ factorization of A is essentially the same
as the *note QR factorization: 4f9. of A^T.

The LQ factorization may be used to find the minimum norm solution of an
underdetermined system of equations A x = b, where A is M-by-N and M \le
N. The solution is

     x = Q^T \begin{pmatrix} L_1^{-1} b \\ 0 \end{pmatrix}

 -- Function: int gsl_linalg_LQ_decomp (gsl_matrix * A, gsl_vector
          * tau)

     This function factorizes the M-by-N matrix ‘A’ into the LQ
     decomposition A = L Q. On output the diagonal and lower trapezoidal
     part of the input matrix contain the matrix L. The vector ‘tau’ and
     the elements above the diagonal of the matrix ‘A’ contain the
     Householder coefficients and Householder vectors which encode the
     orthogonal matrix ‘Q’.  The vector ‘tau’ must be of length
     k=\min(M,N). The matrix Q is related to these components by, Q =
     Q_k ... Q_2 Q_1 where Q_i = I - \tau_i v_i v_i^T and v_i is the
     Householder vector v_i = (0,...,1,A(i,i+1),A(i,i+2),...,A(i,N)).
     This is the same storage scheme as used by LAPACK.

 -- Function: int gsl_linalg_LQ_lssolve (const gsl_matrix * LQ, const
          gsl_vector * tau, const gsl_vector * b, gsl_vector * x,
          gsl_vector * residual)

     This function finds the minimum norm least squares solution to the
     underdetermined system A x = b, where the M-by-N matrix ‘A’ has M
     \le N. The routine requires as input the LQ decomposition of A into
     (‘LQ’, ‘tau’) given by *note gsl_linalg_LQ_decomp(): 522.  The
     solution is returned in ‘x’.  The residual, b - Ax, is computed as
     a by-product and stored in ‘residual’.

 -- Function: int gsl_linalg_LQ_unpack (const gsl_matrix * LQ, const
          gsl_vector * tau, gsl_matrix * Q, gsl_matrix * L)

     This function unpacks the encoded LQ decomposition (‘LQ’, ‘tau’)
     into the matrices ‘Q’ and ‘L’, where ‘Q’ is N-by-N and ‘L’ is
     M-by-N.

 -- Function: int gsl_linalg_LQ_QTvec (const gsl_matrix * LQ, const
          gsl_vector * tau, gsl_vector * v)

     This function applies Q^T to the vector ‘v’, storing the result Q^T
     v in ‘v’ on output.


File: gsl-ref.info,  Node: Complete Orthogonal Decomposition,  Next: Singular Value Decomposition,  Prev: LQ Decomposition,  Up: Linear Algebra

14.5 Complete Orthogonal Decomposition
======================================

The complete orthogonal decomposition of a M-by-N matrix A is a
generalization of the QR decomposition with column pivoting, given by

     A P = Q [ R11 0 ] Z^T
             [  0  0 ]

where P is a N-by-N permutation matrix, Q is M-by-M orthogonal, R_{11}
is r-by-r upper triangular, with r = {\rm rank}(A), and Z is N-by-N
orthogonal.  If A has full rank, then R_{11} = R, Z = I and this reduces
to the QR decomposition with column pivoting.

For a rank deficient least squares problem, \min_x{|| b - Ax||^2}, the
solution vector x is not unique.  However if we further require that
||x||^2 is minimized, then the complete orthogonal decomposition gives
us the ability to compute the unique minimum norm solution, which is
given by

     x = P Z [ R11^-1 c1 ]
             [     0     ]

and the vector c_1 is the first r elements of Q^T b.

The COD also enables a straightforward solution of regularized least
squares problems in Tikhonov standard form, written as

     \min_x ||b - A x||^2 + \lambda^2 ||x||^2

where \lambda > 0 is a regularization parameter which represents a
tradeoff between minimizing the residual norm ||b - A x|| and the
solution norm ||x||.  For this system, the solution is given by

     x = P Z [ y1 ]
             [ 0  ]

where y_1 is a vector of length r which is found by solving

     [     R11     ] y_1 = [ c_1 ]
     [ \lambda I_r ]       [  0  ]

and c_1 is defined above.  The equation above can be solved efficiently
for different values of \lambda using QR factorizations of the left hand
side matrix.

 -- Function: int gsl_linalg_COD_decomp (gsl_matrix * A, gsl_vector
          * tau_Q, gsl_vector * tau_Z, gsl_permutation * p, size_t
          * rank, gsl_vector * work)
 -- Function: int gsl_linalg_COD_decomp_e (gsl_matrix * A, gsl_vector
          * tau_Q, gsl_vector * tau_Z, gsl_permutation * p, double tol,
          size_t * rank, gsl_vector * work)

     These functions factor the M-by-N matrix ‘A’ into the decomposition
     A = Q R Z P^T. The rank of ‘A’ is computed as the number of
     diagonal elements of R greater than the tolerance ‘tol’ and output
     in ‘rank’.  If ‘tol’ is not specified, a default value is used (see
     *note gsl_linalg_QRPT_rank(): 51b.).  On output, the permutation
     matrix P is stored in ‘p’.  The matrix R_{11} is stored in the
     upper ‘rank’-by-‘rank’ block of ‘A’.  The matrices Q and Z are
     encoded in packed storage in ‘A’ on output.  The vectors ‘tau_Q’
     and ‘tau_Z’ contain the Householder scalars corresponding to the
     matrices Q and Z respectively and must be of length k = \min(M,N).
     The vector ‘work’ is additional workspace of length N.

 -- Function: int gsl_linalg_COD_lssolve (const gsl_matrix * QRZT, const
          gsl_vector * tau_Q, const gsl_vector * tau_Z, const
          gsl_permutation * p, const size_t rank, const gsl_vector * b,
          gsl_vector * x, gsl_vector * residual)

     This function finds the unique minimum norm least squares solution
     to the overdetermined system A x = b where the matrix ‘A’ has more
     rows than columns.  The least squares solution minimizes the
     Euclidean norm of the residual, ||b - A x|| as well as the norm of
     the solution ||x||.  The routine requires as input the QRZT
     decomposition of A into (‘QRZT’, ‘tau_Q’, ‘tau_Z’, ‘p’, ‘rank’)
     given by *note gsl_linalg_COD_decomp(): 527.  The solution is
     returned in ‘x’.  The residual, b - Ax, is computed as a by-product
     and stored in ‘residual’.

 -- Function: int gsl_linalg_COD_lssolve2 (const double lambda, const
          gsl_matrix * QRZT, const gsl_vector * tau_Q, const gsl_vector
          * tau_Z, const gsl_permutation * p, const size_t rank, const
          gsl_vector * b, gsl_vector * x, gsl_vector * residual,
          gsl_matrix * S, gsl_vector * work)

     This function finds the solution to the regularized least squares
     problem in Tikhonov standard form, \min_x ||b - Ax||^2 + \lambda^2
     ||x||^2.  The routine requires as input the QRZT decomposition of A
     into (‘QRZT’, ‘tau_Q’, ‘tau_Z’, ‘p’, ‘rank’) given by *note
     gsl_linalg_COD_decomp(): 527.  The parameter \lambda is supplied in
     ‘lambda’.  The solution is returned in ‘x’.  The residual, b - Ax,
     is stored in ‘residual’ on output.  ‘S’ is additional workspace of
     size ‘rank’-by-‘rank’.  ‘work’ is additional workspace of length
     ‘rank’.

 -- Function: int gsl_linalg_COD_unpack (const gsl_matrix * QRZT, const
          gsl_vector * tau_Q, const gsl_vector * tau_Z, const
          size_t rank, gsl_matrix * Q, gsl_matrix * R, gsl_matrix * Z)

     This function unpacks the encoded QRZT decomposition (‘QRZT’,
     ‘tau_Q’, ‘tau_Z’, ‘rank’) into the matrices ‘Q’, ‘R’, and ‘Z’,
     where ‘Q’ is M-by-M, ‘R’ is M-by-N, and ‘Z’ is N-by-N.

 -- Function: int gsl_linalg_COD_matZ (const gsl_matrix * QRZT, const
          gsl_vector * tau_Z, const size_t rank, gsl_matrix * A,
          gsl_vector * work)

     This function multiplies the input matrix ‘A’ on the right by ‘Z’,
     A' = A Z using the encoded QRZT decomposition (‘QRZT’, ‘tau_Z’,
     ‘rank’).  ‘A’ must have N columns but may have any number of rows.
     Additional workspace of length M is provided in ‘work’.


File: gsl-ref.info,  Node: Singular Value Decomposition,  Next: Cholesky Decomposition,  Prev: Complete Orthogonal Decomposition,  Up: Linear Algebra

14.6 Singular Value Decomposition
=================================

A general rectangular M-by-N matrix A has a singular value decomposition
(SVD) into the product of an M-by-N orthogonal matrix U, an N-by-N
diagonal matrix of singular values S and the transpose of an N-by-N
orthogonal square matrix V,

     A = U S V^T

The singular values \sigma_i = S_{ii} are all non-negative and are
generally chosen to form a non-increasing sequence

     \sigma_1 >= \sigma_2 >= ... >= \sigma_N >= 0

The singular value decomposition of a matrix has many practical uses.
The condition number of the matrix is given by the ratio of the largest
singular value to the smallest singular value.  The presence of a zero
singular value indicates that the matrix is singular.  The number of
non-zero singular values indicates the rank of the matrix.  In practice
singular value decomposition of a rank-deficient matrix will not produce
exact zeroes for singular values, due to finite numerical precision.
Small singular values should be edited by choosing a suitable tolerance.

For a rank-deficient matrix, the null space of A is given by the columns
of V corresponding to the zero singular values.  Similarly, the range of
A is given by columns of U corresponding to the non-zero singular
values.

Note that the routines here compute the “thin” version of the SVD with U
as M-by-N orthogonal matrix.  This allows in-place computation and is
the most commonly-used form in practice.  Mathematically, the “full” SVD
is defined with U as an M-by-M orthogonal matrix and S as an M-by-N
diagonal matrix (with additional rows of zeros).

 -- Function: int gsl_linalg_SV_decomp (gsl_matrix * A, gsl_matrix * V,
          gsl_vector * S, gsl_vector * work)

     This function factorizes the M-by-N matrix ‘A’ into the singular
     value decomposition A = U S V^T for M \ge N. On output the matrix
     ‘A’ is replaced by U. The diagonal elements of the singular value
     matrix S are stored in the vector ‘S’.  The singular values are
     non-negative and form a non-increasing sequence from S_1 to S_N.
     The matrix ‘V’ contains the elements of V in untransposed form.  To
     form the product U S V^T it is necessary to take the transpose of
     ‘V’.  A workspace of length ‘N’ is required in ‘work’.

     This routine uses the Golub-Reinsch SVD algorithm.

 -- Function: int gsl_linalg_SV_decomp_mod (gsl_matrix * A, gsl_matrix
          * X, gsl_matrix * V, gsl_vector * S, gsl_vector * work)

     This function computes the SVD using the modified Golub-Reinsch
     algorithm, which is faster for M \gg N. It requires the vector
     ‘work’ of length ‘N’ and the N-by-N matrix ‘X’ as additional
     working space.

 -- Function: int gsl_linalg_SV_decomp_jacobi (gsl_matrix * A,
          gsl_matrix * V, gsl_vector * S)

     This function computes the SVD of the M-by-N matrix ‘A’ using
     one-sided Jacobi orthogonalization for M \ge N. The Jacobi method
     can compute singular values to higher relative accuracy than
     Golub-Reinsch algorithms (see references for details).

 -- Function: int gsl_linalg_SV_solve (const gsl_matrix * U, const
          gsl_matrix * V, const gsl_vector * S, const gsl_vector * b,
          gsl_vector * x)

     This function solves the system A x = b using the singular value
     decomposition (‘U’, ‘S’, ‘V’) of A which must have been computed
     previously with *note gsl_linalg_SV_decomp(): 52e.

     Only non-zero singular values are used in computing the solution.
     The parts of the solution corresponding to singular values of zero
     are ignored.  Other singular values can be edited out by setting
     them to zero before calling this function.

     In the over-determined case where ‘A’ has more rows than columns
     the system is solved in the least squares sense, returning the
     solution ‘x’ which minimizes ||A x - b||_2.

 -- Function: int gsl_linalg_SV_leverage (const gsl_matrix * U,
          gsl_vector * h)

     This function computes the statistical leverage values h_i of a
     matrix A using its singular value decomposition (‘U’, ‘S’, ‘V’)
     previously computed with *note gsl_linalg_SV_decomp(): 52e.  h_i
     are the diagonal values of the matrix A (A^T A)^{-1} A^T and depend
     only on the matrix ‘U’ which is the input to this function.


File: gsl-ref.info,  Node: Cholesky Decomposition,  Next: Pivoted Cholesky Decomposition,  Prev: Singular Value Decomposition,  Up: Linear Algebra

14.7 Cholesky Decomposition
===========================

A symmetric, positive definite square matrix A has a Cholesky
decomposition into a product of a lower triangular matrix L and its
transpose L^T,

     A = L L^T

This is sometimes referred to as taking the square-root of a matrix.
The Cholesky decomposition can only be carried out when all the
eigenvalues of the matrix are positive.  This decomposition can be used
to convert the linear system A x = b into a pair of triangular systems
(L y = b, L^T x = y), which can be solved by forward and
back-substitution.

If the matrix A is near singular, it is sometimes possible to reduce the
condition number and recover a more accurate solution vector x by
scaling as

     ( S A S ) ( S^(-1) x ) = S b

where S is a diagonal matrix whose elements are given by S_{ii} =
1/\sqrt{A_{ii}}.  This scaling is also known as Jacobi preconditioning.
There are routines below to solve both the scaled and unscaled systems.

 -- Function: int gsl_linalg_cholesky_decomp1 (gsl_matrix * A)
 -- Function: int gsl_linalg_complex_cholesky_decomp (gsl_matrix_complex
          * A)

     These functions factorize the symmetric, positive-definite square
     matrix ‘A’ into the Cholesky decomposition A = L L^T (or A = L
     L^{\dagger} for the complex case).  On input, the values from the
     diagonal and lower-triangular part of the matrix ‘A’ are used (the
     upper triangular part is ignored).  On output the diagonal and
     lower triangular part of the input matrix ‘A’ contain the matrix L,
     while the upper triangular part contains the original matrix.  If
     the matrix is not positive-definite then the decomposition will
     fail, returning the error code *note GSL_EDOM: 28.

     When testing whether a matrix is positive-definite, disable the
     error handler first to avoid triggering an error.  These functions
     use Level 3 BLAS to compute the Cholesky factorization (Peise and
     Bientinesi, 2016).

 -- Function: int gsl_linalg_cholesky_decomp (gsl_matrix * A)

     This function is now deprecated and is provided only for backward
     compatibility.

 -- Function: int gsl_linalg_cholesky_solve (const gsl_matrix
          * cholesky, const gsl_vector * b, gsl_vector * x)
 -- Function: int gsl_linalg_complex_cholesky_solve (const
          gsl_matrix_complex * cholesky, const gsl_vector_complex * b,
          gsl_vector_complex * x)

     These functions solve the system A x = b using the Cholesky
     decomposition of A held in the matrix ‘cholesky’ which must have
     been previously computed by *note gsl_linalg_cholesky_decomp():
     536. or *note gsl_linalg_complex_cholesky_decomp(): 535.

 -- Function: int gsl_linalg_cholesky_svx (const gsl_matrix * cholesky,
          gsl_vector * x)
 -- Function: int gsl_linalg_complex_cholesky_svx (const
          gsl_matrix_complex * cholesky, gsl_vector_complex * x)

     These functions solve the system A x = b in-place using the
     Cholesky decomposition of A held in the matrix ‘cholesky’ which
     must have been previously computed by *note
     gsl_linalg_cholesky_decomp(): 536. or *note
     gsl_linalg_complex_cholesky_decomp(): 535.  On input ‘x’ should
     contain the right-hand side b, which is replaced by the solution on
     output.

 -- Function: int gsl_linalg_cholesky_invert (gsl_matrix * cholesky)
 -- Function: int gsl_linalg_complex_cholesky_invert (gsl_matrix_complex
          * cholesky)

     These functions compute the inverse of a matrix from its Cholesky
     decomposition ‘cholesky’, which must have been previously computed
     by *note gsl_linalg_cholesky_decomp(): 536. or *note
     gsl_linalg_complex_cholesky_decomp(): 535.  On output, the inverse
     is stored in-place in ‘cholesky’.

 -- Function: int gsl_linalg_cholesky_decomp2 (gsl_matrix * A,
          gsl_vector * S)

     This function calculates a diagonal scaling transformation S for
     the symmetric, positive-definite square matrix ‘A’, and then
     computes the Cholesky decomposition S A S = L L^T. On input, the
     values from the diagonal and lower-triangular part of the matrix
     ‘A’ are used (the upper triangular part is ignored).  On output the
     diagonal and lower triangular part of the input matrix ‘A’ contain
     the matrix L, while the upper triangular part of the input matrix
     is overwritten with L^T (the diagonal terms being identical for
     both L and L^T). If the matrix is not positive-definite then the
     decomposition will fail, returning the error code *note GSL_EDOM:
     28.  The diagonal scale factors are stored in ‘S’ on output.

     When testing whether a matrix is positive-definite, disable the
     error handler first to avoid triggering an error.

 -- Function: int gsl_linalg_cholesky_solve2 (const gsl_matrix
          * cholesky, const gsl_vector * S, const gsl_vector * b,
          gsl_vector * x)

     This function solves the system (S A S) (S^{-1} x) = S b using the
     Cholesky decomposition of S A S held in the matrix ‘cholesky’ which
     must have been previously computed by *note
     gsl_linalg_cholesky_decomp2(): 53d.

 -- Function: int gsl_linalg_cholesky_svx2 (const gsl_matrix * cholesky,
          const gsl_vector * S, gsl_vector * x)

     This function solves the system (S A S) (S^{-1} x) = S b in-place
     using the Cholesky decomposition of S A S held in the matrix
     ‘cholesky’ which must have been previously computed by *note
     gsl_linalg_cholesky_decomp2(): 53d.  On input ‘x’ should contain
     the right-hand side b, which is replaced by the solution on output.

 -- Function: int gsl_linalg_cholesky_scale (const gsl_matrix * A,
          gsl_vector * S)

     This function calculates a diagonal scaling transformation of the
     symmetric, positive definite matrix ‘A’, such that S A S has a
     condition number within a factor of N of the matrix of smallest
     possible condition number over all possible diagonal scalings.  On
     output, ‘S’ contains the scale factors, given by S_i =
     1/\sqrt{A_{ii}}.  For any A_{ii} \le 0, the corresponding scale
     factor S_i is set to 1.

 -- Function: int gsl_linalg_cholesky_scale_apply (gsl_matrix * A, const
          gsl_vector * S)

     This function applies the scaling transformation ‘S’ to the matrix
     ‘A’.  On output, ‘A’ is replaced by S A S.

 -- Function: int gsl_linalg_cholesky_rcond (const gsl_matrix
          * cholesky, double * rcond, gsl_vector * work)

     This function estimates the reciprocal condition number (using the
     1-norm) of the symmetric positive definite matrix A, using its
     Cholesky decomposition provided in ‘cholesky’.  The reciprocal
     condition number estimate, defined as 1 / (||A||_1 \cdot
     ||A^{-1}||_1), is stored in ‘rcond’.  Additional workspace of size
     3 N is required in ‘work’.


File: gsl-ref.info,  Node: Pivoted Cholesky Decomposition,  Next: Modified Cholesky Decomposition,  Prev: Cholesky Decomposition,  Up: Linear Algebra

14.8 Pivoted Cholesky Decomposition
===================================

A symmetric positive semi-definite square matrix A has an alternate
Cholesky decomposition into a product of a lower unit triangular matrix
L, a diagonal matrix D and L^T, given by L D L^T. For postive definite
matrices, this is equivalent to the Cholesky formulation discussed
above, with the standard Cholesky lower triangular factor given by L
D^{1 \over 2}.  For ill-conditioned matrices, it can help to use a
pivoting strategy to prevent the entries of D and L from growing too
large, and also ensure D_1 \ge D_2 \ge \cdots \ge D_n > 0, where D_i are
the diagonal entries of D. The final decomposition is given by

     P A P^T = L D L^T

where P is a permutation matrix.

 -- Function: int gsl_linalg_pcholesky_decomp (gsl_matrix * A,
          gsl_permutation * p)

     This function factors the symmetric, positive-definite square
     matrix ‘A’ into the Pivoted Cholesky decomposition P A P^T = L D
     L^T. On input, the values from the diagonal and lower-triangular
     part of the matrix ‘A’ are used to construct the factorization.  On
     output the diagonal of the input matrix ‘A’ stores the diagonal
     elements of D, and the lower triangular portion of ‘A’ contains the
     matrix L. Since L has ones on its diagonal these do not need to be
     explicitely stored.  The upper triangular portion of ‘A’ is
     unmodified.  The permutation matrix P is stored in ‘p’ on output.

 -- Function: int gsl_linalg_pcholesky_solve (const gsl_matrix * LDLT,
          const gsl_permutation * p, const gsl_vector * b, gsl_vector
          * x)

     This function solves the system A x = b using the Pivoted Cholesky
     decomposition of A held in the matrix ‘LDLT’ and permutation ‘p’
     which must have been previously computed by *note
     gsl_linalg_pcholesky_decomp(): 544.

 -- Function: int gsl_linalg_pcholesky_svx (const gsl_matrix * LDLT,
          const gsl_permutation * p, gsl_vector * x)

     This function solves the system A x = b in-place using the Pivoted
     Cholesky decomposition of A held in the matrix ‘LDLT’ and
     permutation ‘p’ which must have been previously computed by *note
     gsl_linalg_pcholesky_decomp(): 544.  On input, ‘x’ contains the
     right hand side vector b which is replaced by the solution vector
     on output.

 -- Function: int gsl_linalg_pcholesky_decomp2 (gsl_matrix * A,
          gsl_permutation * p, gsl_vector * S)

     This function computes the pivoted Cholesky factorization of the
     matrix S A S, where the input matrix ‘A’ is symmetric and positive
     definite, and the diagonal scaling matrix ‘S’ is computed to reduce
     the condition number of ‘A’ as much as possible.  See *note
     Cholesky Decomposition: 3f1. for more information on the matrix
     ‘S’.  The Pivoted Cholesky decomposition satisfies P S A S P^T = L
     D L^T. On input, the values from the diagonal and lower-triangular
     part of the matrix ‘A’ are used to construct the factorization.  On
     output the diagonal of the input matrix ‘A’ stores the diagonal
     elements of D, and the lower triangular portion of ‘A’ contains the
     matrix L. Since L has ones on its diagonal these do not need to be
     explicitely stored.  The upper triangular portion of ‘A’ is
     unmodified.  The permutation matrix P is stored in ‘p’ on output.
     The diagonal scaling transformation is stored in ‘S’ on output.

 -- Function: int gsl_linalg_pcholesky_solve2 (const gsl_matrix * LDLT,
          const gsl_permutation * p, const gsl_vector * S, const
          gsl_vector * b, gsl_vector * x)

     This function solves the system (S A S) (S^{-1} x) = S b using the
     Pivoted Cholesky decomposition of S A S held in the matrix ‘LDLT’,
     permutation ‘p’, and vector ‘S’, which must have been previously
     computed by *note gsl_linalg_pcholesky_decomp2(): 547.

 -- Function: int gsl_linalg_pcholesky_svx2 (const gsl_matrix * LDLT,
          const gsl_permutation * p, const gsl_vector * S, gsl_vector
          * x)

     This function solves the system (S A S) (S^{-1} x) = S b in-place
     using the Pivoted Cholesky decomposition of S A S held in the
     matrix ‘LDLT’, permutation ‘p’ and vector ‘S’, which must have been
     previously computed by *note gsl_linalg_pcholesky_decomp2(): 547.
     On input, ‘x’ contains the right hand side vector b which is
     replaced by the solution vector on output.

 -- Function: int gsl_linalg_pcholesky_invert (const gsl_matrix * LDLT,
          const gsl_permutation * p, gsl_matrix * Ainv)

     This function computes the inverse of the matrix A, using the
     Pivoted Cholesky decomposition stored in ‘LDLT’ and ‘p’.  On
     output, the matrix ‘Ainv’ contains A^{-1}.

 -- Function: int gsl_linalg_pcholesky_rcond (const gsl_matrix * LDLT,
          const gsl_permutation * p, double * rcond, gsl_vector * work)

     This function estimates the reciprocal condition number (using the
     1-norm) of the symmetric positive definite matrix A, using its
     pivoted Cholesky decomposition provided in ‘LDLT’.  The reciprocal
     condition number estimate, defined as 1 / (||A||_1 \cdot
     ||A^{-1}||_1), is stored in ‘rcond’.  Additional workspace of size
     3 N is required in ‘work’.


File: gsl-ref.info,  Node: Modified Cholesky Decomposition,  Next: LDLT Decomposition,  Prev: Pivoted Cholesky Decomposition,  Up: Linear Algebra

14.9 Modified Cholesky Decomposition
====================================

The modified Cholesky decomposition is suitable for solving systems A x
= b where A is a symmetric indefinite matrix.  Such matrices arise in
nonlinear optimization algorithms.  The standard Cholesky decomposition
requires a positive definite matrix and would fail in this case.
Instead of resorting to a method like QR or SVD, which do not take into
account the symmetry of the matrix, we can instead introduce a small
perturbation to the matrix A to make it positive definite, and then use
a Cholesky decomposition on the perturbed matrix.  The resulting
decomposition satisfies

     P (A + E) P^T = L D L^T

where P is a permutation matrix, E is a diagonal perturbation matrix, L
is unit lower triangular, and D is diagonal.  If A is sufficiently
positive definite, then the perturbation matrix E will be zero and this
method is equivalent to the pivoted Cholesky algorithm.  For indefinite
matrices, the perturbation matrix E is computed to ensure that A + E is
positive definite and well conditioned.

 -- Function: int gsl_linalg_mcholesky_decomp (gsl_matrix * A,
          gsl_permutation * p, gsl_vector * E)

     This function factors the symmetric, indefinite square matrix ‘A’
     into the Modified Cholesky decomposition P (A + E) P^T = L D L^T.
     On input, the values from the diagonal and lower-triangular part of
     the matrix ‘A’ are used to construct the factorization.  On output
     the diagonal of the input matrix ‘A’ stores the diagonal elements
     of D, and the lower triangular portion of ‘A’ contains the matrix
     L. Since L has ones on its diagonal these do not need to be
     explicitely stored.  The upper triangular portion of ‘A’ is
     unmodified.  The permutation matrix P is stored in ‘p’ on output.
     The diagonal perturbation matrix is stored in ‘E’ on output.  The
     parameter ‘E’ may be set to NULL if it is not required.

 -- Function: int gsl_linalg_mcholesky_solve (const gsl_matrix * LDLT,
          const gsl_permutation * p, const gsl_vector * b, gsl_vector
          * x)

     This function solves the perturbed system (A + E) x = b using the
     Cholesky decomposition of A + E held in the matrix ‘LDLT’ and
     permutation ‘p’ which must have been previously computed by *note
     gsl_linalg_mcholesky_decomp(): 54d.

 -- Function: int gsl_linalg_mcholesky_svx (const gsl_matrix * LDLT,
          const gsl_permutation * p, gsl_vector * x)

     This function solves the perturbed system (A + E) x = b in-place
     using the Cholesky decomposition of A + E held in the matrix ‘LDLT’
     and permutation ‘p’ which must have been previously computed by
     *note gsl_linalg_mcholesky_decomp(): 54d.  On input, ‘x’ contains
     the right hand side vector b which is replaced by the solution
     vector on output.

 -- Function: int gsl_linalg_mcholesky_rcond (const gsl_matrix * LDLT,
          const gsl_permutation * p, double * rcond, gsl_vector * work)

     This function estimates the reciprocal condition number (using the
     1-norm) of the perturbed matrix A + E, using its pivoted Cholesky
     decomposition provided in ‘LDLT’.  The reciprocal condition number
     estimate, defined as 1 / (||A + E||_1 \cdot ||(A + E)^{-1}||_1), is
     stored in ‘rcond’.  Additional workspace of size 3 N is required in
     ‘work’.


File: gsl-ref.info,  Node: LDLT Decomposition,  Next: Tridiagonal Decomposition of Real Symmetric Matrices,  Prev: Modified Cholesky Decomposition,  Up: Linear Algebra

14.10 LDLT Decomposition
========================

If A is a symmetric, nonsingular square matrix, then it has a unique
factorization of the form

     A = L D L^T

where L is a unit lower triangular matrix and D is diagonal.  If A is
positive definite, then this factorization is equivalent to the Cholesky
factorization, where the lower triangular Cholesky factor is L
D^{\frac{1}{2}}.  Some indefinite matrices for which no Cholesky
decomposition exists have an L D L^T decomposition with negative entries
in D. The L D L^T algorithm is sometimes referred to as the `square root
free' Cholesky decomposition, as the algorithm does not require the
computation of square roots.  The algorithm is stable for positive
definite matrices, but is not guaranteed to be stable for indefinite
matrices.

 -- Function: int gsl_linalg_ldlt_decomp (gsl_matrix * A)

     This function factorizes the symmetric, non-singular square matrix
     ‘A’ into the decomposition A = L D L^T. On input, the values from
     the diagonal and lower-triangular part of the matrix ‘A’ are used.
     The upper triangle of ‘A’ is used as temporary workspace.  On
     output the diagonal of ‘A’ contains the matrix D and the lower
     triangle of ‘A’ contains the unit lower triangular matrix L. The
     matrix 1-norm, ||A||_1 is stored in the upper right corner on
     output, for later use by *note gsl_linalg_ldlt_rcond(): 554.

     If the matrix is detected to be singular, the function returns the
     error code *note GSL_EDOM: 28.

 -- Function: int gsl_linalg_ldlt_solve (const gsl_matrix * LDLT, const
          gsl_vector * b, gsl_vector * x)

     This function solves the system A x = b using the L D L^T
     decomposition of A held in the matrix ‘LDLT’ which must have been
     previously computed by *note gsl_linalg_ldlt_decomp(): 553.

 -- Function: int gsl_linalg_ldlt_svx (const gsl_matrix * LDLT,
          gsl_vector * x)

     This function solves the system A x = b in-place using the L D L^T
     decomposition of A held in the matrix ‘LDLT’ which must have been
     previously computed by *note gsl_linalg_ldlt_decomp(): 553.  On
     input ‘x’ should contain the right-hand side b, which is replaced
     by the solution on output.

 -- Function: int gsl_linalg_ldlt_rcond (const gsl_matrix * LDLT, double
          * rcond, gsl_vector * work)

     This function estimates the reciprocal condition number (using the
     1-norm) of the symmetric nonsingular matrix A, using its L D L^T
     decomposition provided in ‘LDLT’.  The reciprocal condition number
     estimate, defined as 1 / (||A||_1 \cdot ||A^{-1}||_1), is stored in
     ‘rcond’.  Additional workspace of size 3 N is required in ‘work’.


File: gsl-ref.info,  Node: Tridiagonal Decomposition of Real Symmetric Matrices,  Next: Tridiagonal Decomposition of Hermitian Matrices,  Prev: LDLT Decomposition,  Up: Linear Algebra

14.11 Tridiagonal Decomposition of Real Symmetric Matrices
==========================================================

A symmetric matrix A can be factorized by similarity transformations
into the form,

     A = Q T Q^T

where Q is an orthogonal matrix and T is a symmetric tridiagonal matrix.

 -- Function: int gsl_linalg_symmtd_decomp (gsl_matrix * A, gsl_vector
          * tau)

     This function factorizes the symmetric square matrix ‘A’ into the
     symmetric tridiagonal decomposition Q T Q^T. On output the diagonal
     and subdiagonal part of the input matrix ‘A’ contain the
     tridiagonal matrix T. The remaining lower triangular part of the
     input matrix contains the Householder vectors which, together with
     the Householder coefficients ‘tau’, encode the orthogonal matrix Q.
     This storage scheme is the same as used by LAPACK. The upper
     triangular part of ‘A’ is not referenced.

 -- Function: int gsl_linalg_symmtd_unpack (const gsl_matrix * A, const
          gsl_vector * tau, gsl_matrix * Q, gsl_vector * diag,
          gsl_vector * subdiag)

     This function unpacks the encoded symmetric tridiagonal
     decomposition (‘A’, ‘tau’) obtained from *note
     gsl_linalg_symmtd_decomp(): 558. into the orthogonal matrix ‘Q’,
     the vector of diagonal elements ‘diag’ and the vector of
     subdiagonal elements ‘subdiag’.

 -- Function: int gsl_linalg_symmtd_unpack_T (const gsl_matrix * A,
          gsl_vector * diag, gsl_vector * subdiag)

     This function unpacks the diagonal and subdiagonal of the encoded
     symmetric tridiagonal decomposition (‘A’, ‘tau’) obtained from
     *note gsl_linalg_symmtd_decomp(): 558. into the vectors ‘diag’ and
     ‘subdiag’.


File: gsl-ref.info,  Node: Tridiagonal Decomposition of Hermitian Matrices,  Next: Hessenberg Decomposition of Real Matrices,  Prev: Tridiagonal Decomposition of Real Symmetric Matrices,  Up: Linear Algebra

14.12 Tridiagonal Decomposition of Hermitian Matrices
=====================================================

A hermitian matrix A can be factorized by similarity transformations
into the form,

     A = U T U^T

where U is a unitary matrix and T is a real symmetric tridiagonal
matrix.

 -- Function: int gsl_linalg_hermtd_decomp (gsl_matrix_complex * A,
          gsl_vector_complex * tau)

     This function factorizes the hermitian matrix ‘A’ into the
     symmetric tridiagonal decomposition U T U^T. On output the real
     parts of the diagonal and subdiagonal part of the input matrix ‘A’
     contain the tridiagonal matrix T. The remaining lower triangular
     part of the input matrix contains the Householder vectors which,
     together with the Householder coefficients ‘tau’, encode the
     unitary matrix U. This storage scheme is the same as used by
     LAPACK. The upper triangular part of ‘A’ and imaginary parts of the
     diagonal are not referenced.

 -- Function: int gsl_linalg_hermtd_unpack (const gsl_matrix_complex
          * A, const gsl_vector_complex * tau, gsl_matrix_complex * U,
          gsl_vector * diag, gsl_vector * subdiag)

     This function unpacks the encoded tridiagonal decomposition (‘A’,
     ‘tau’) obtained from *note gsl_linalg_hermtd_decomp(): 55c. into
     the unitary matrix ‘U’, the real vector of diagonal elements ‘diag’
     and the real vector of subdiagonal elements ‘subdiag’.

 -- Function: int gsl_linalg_hermtd_unpack_T (const gsl_matrix_complex
          * A, gsl_vector * diag, gsl_vector * subdiag)

     This function unpacks the diagonal and subdiagonal of the encoded
     tridiagonal decomposition (‘A’, ‘tau’) obtained from the *note
     gsl_linalg_hermtd_decomp(): 55c. into the real vectors ‘diag’ and
     ‘subdiag’.


File: gsl-ref.info,  Node: Hessenberg Decomposition of Real Matrices,  Next: Hessenberg-Triangular Decomposition of Real Matrices,  Prev: Tridiagonal Decomposition of Hermitian Matrices,  Up: Linear Algebra

14.13 Hessenberg Decomposition of Real Matrices
===============================================

A general real matrix A can be decomposed by orthogonal similarity
transformations into the form

     A = U H U^T

where U is orthogonal and H is an upper Hessenberg matrix, meaning that
it has zeros below the first subdiagonal.  The Hessenberg reduction is
the first step in the Schur decomposition for the nonsymmetric
eigenvalue problem, but has applications in other areas as well.

 -- Function: int gsl_linalg_hessenberg_decomp (gsl_matrix * A,
          gsl_vector * tau)

     This function computes the Hessenberg decomposition of the matrix
     ‘A’ by applying the similarity transformation H = U^T A U. On
     output, H is stored in the upper portion of ‘A’.  The information
     required to construct the matrix U is stored in the lower
     triangular portion of ‘A’.  U is a product of N - 2 Householder
     matrices.  The Householder vectors are stored in the lower portion
     of ‘A’ (below the subdiagonal) and the Householder coefficients are
     stored in the vector ‘tau’.  ‘tau’ must be of length ‘N’.

 -- Function: int gsl_linalg_hessenberg_unpack (gsl_matrix * H,
          gsl_vector * tau, gsl_matrix * U)

     This function constructs the orthogonal matrix U from the
     information stored in the Hessenberg matrix ‘H’ along with the
     vector ‘tau’.  ‘H’ and ‘tau’ are outputs from *note
     gsl_linalg_hessenberg_decomp(): 560.

 -- Function: int gsl_linalg_hessenberg_unpack_accum (gsl_matrix * H,
          gsl_vector * tau, gsl_matrix * V)

     This function is similar to *note gsl_linalg_hessenberg_unpack():
     561, except it accumulates the matrix ‘U’ into ‘V’, so that V' =
     VU. The matrix ‘V’ must be initialized prior to calling this
     function.  Setting ‘V’ to the identity matrix provides the same
     result as *note gsl_linalg_hessenberg_unpack(): 561.  If ‘H’ is
     order ‘N’, then ‘V’ must have ‘N’ columns but may have any number
     of rows.

 -- Function: int gsl_linalg_hessenberg_set_zero (gsl_matrix * H)

     This function sets the lower triangular portion of ‘H’, below the
     subdiagonal, to zero.  It is useful for clearing out the
     Householder vectors after calling *note
     gsl_linalg_hessenberg_decomp(): 560.


File: gsl-ref.info,  Node: Hessenberg-Triangular Decomposition of Real Matrices,  Next: Bidiagonalization,  Prev: Hessenberg Decomposition of Real Matrices,  Up: Linear Algebra

14.14 Hessenberg-Triangular Decomposition of Real Matrices
==========================================================

A general real matrix pair (A, B) can be decomposed by orthogonal
similarity transformations into the form

     A = U H V^T
     B = U R V^T

where U and V are orthogonal, H is an upper Hessenberg matrix, and R is
upper triangular.  The Hessenberg-Triangular reduction is the first step
in the generalized Schur decomposition for the generalized eigenvalue
problem.

 -- Function: int gsl_linalg_hesstri_decomp (gsl_matrix * A, gsl_matrix
          * B, gsl_matrix * U, gsl_matrix * V, gsl_vector * work)

     This function computes the Hessenberg-Triangular decomposition of
     the matrix pair (‘A’, ‘B’).  On output, H is stored in ‘A’, and R
     is stored in ‘B’.  If ‘U’ and ‘V’ are provided (they may be null),
     the similarity transformations are stored in them.  Additional
     workspace of length N is needed in ‘work’.


File: gsl-ref.info,  Node: Bidiagonalization,  Next: Givens Rotations,  Prev: Hessenberg-Triangular Decomposition of Real Matrices,  Up: Linear Algebra

14.15 Bidiagonalization
=======================

A general matrix A can be factorized by similarity transformations into
the form,

     A = U B V^T

where U and V are orthogonal matrices and B is a N-by-N bidiagonal
matrix with non-zero entries only on the diagonal and superdiagonal.
The size of ‘U’ is M-by-N and the size of ‘V’ is N-by-N.

 -- Function: int gsl_linalg_bidiag_decomp (gsl_matrix * A, gsl_vector
          * tau_U, gsl_vector * tau_V)

     This function factorizes the M-by-N matrix ‘A’ into bidiagonal form
     U B V^T. The diagonal and superdiagonal of the matrix B are stored
     in the diagonal and superdiagonal of ‘A’.  The orthogonal matrices
     U and ‘V’ are stored as compressed Householder vectors in the
     remaining elements of ‘A’.  The Householder coefficients are stored
     in the vectors ‘tau_U’ and ‘tau_V’.  The length of ‘tau_U’ must
     equal the number of elements in the diagonal of ‘A’ and the length
     of ‘tau_V’ should be one element shorter.

 -- Function: int gsl_linalg_bidiag_unpack (const gsl_matrix * A, const
          gsl_vector * tau_U, gsl_matrix * U, const gsl_vector * tau_V,
          gsl_matrix * V, gsl_vector * diag, gsl_vector * superdiag)

     This function unpacks the bidiagonal decomposition of ‘A’ produced
     by *note gsl_linalg_bidiag_decomp(): 567, (‘A’, ‘tau_U’, ‘tau_V’)
     into the separate orthogonal matrices ‘U’, ‘V’ and the diagonal
     vector ‘diag’ and superdiagonal ‘superdiag’.  Note that ‘U’ is
     stored as a compact M-by-N orthogonal matrix satisfying U^T U = I
     for efficiency.

 -- Function: int gsl_linalg_bidiag_unpack2 (gsl_matrix * A, gsl_vector
          * tau_U, gsl_vector * tau_V, gsl_matrix * V)

     This function unpacks the bidiagonal decomposition of ‘A’ produced
     by *note gsl_linalg_bidiag_decomp(): 567, (‘A’, ‘tau_U’, ‘tau_V’)
     into the separate orthogonal matrices ‘U’, ‘V’ and the diagonal
     vector ‘diag’ and superdiagonal ‘superdiag’.  The matrix ‘U’ is
     stored in-place in ‘A’.

 -- Function: int gsl_linalg_bidiag_unpack_B (const gsl_matrix * A,
          gsl_vector * diag, gsl_vector * superdiag)

     This function unpacks the diagonal and superdiagonal of the
     bidiagonal decomposition of ‘A’ from *note
     gsl_linalg_bidiag_decomp(): 567, into the diagonal vector ‘diag’
     and superdiagonal vector ‘superdiag’.


File: gsl-ref.info,  Node: Givens Rotations,  Next: Householder Transformations,  Prev: Bidiagonalization,  Up: Linear Algebra

14.16 Givens Rotations
======================

A Givens rotation is a rotation in the plane acting on two elements of a
given vector.  It can be represented in matrix form as

where the \cos{\theta} and \sin{\theta} appear at the intersection of
the i-th and j-th rows and columns.  When acting on a vector x,
G(i,j,\theta) x performs a rotation of the (i,j) elements of x.  Givens
rotations are typically used to introduce zeros in vectors, such as
during the QR decomposition of a matrix.  In this case, it is typically
desired to find c and s such that

     [ c -s ] [ a ] = [ r ]
     [ s  c ] [ b ]   [ 0 ]

with r = \sqrt{a^2 + b^2}.

 -- Function: void gsl_linalg_givens (const double a, const double b,
          double * c, double * s)

     This function computes c = \cos{\theta} and s = \sin{\theta} so
     that the Givens matrix G(\theta) acting on the vector (a,b)
     produces (r, 0), with r = \sqrt{a^2 + b^2}.

 -- Function: void gsl_linalg_givens_gv (gsl_vector * v, const size_t i,
          const size_t j, const double c, const double s)

     This function applies the Givens rotation defined by c =
     \cos{\theta} and s = \sin{\theta} to the ‘i’ and ‘j’ elements of
     ‘v’.  On output, (v(i),v(j)) \leftarrow G(\theta) (v(i),v(j)).


File: gsl-ref.info,  Node: Householder Transformations,  Next: Householder solver for linear systems,  Prev: Givens Rotations,  Up: Linear Algebra

14.17 Householder Transformations
=================================

A Householder transformation is a rank-1 modification of the identity
matrix which can be used to zero out selected elements of a vector.  A
Householder matrix H takes the form,

     H = I - \tau v v^T

where v is a vector (called the `Householder vector') and \tau = 2/(v^T
v).  The functions described in this section use the rank-1 structure of
the Householder matrix to create and apply Householder transformations
efficiently.

 -- Function: double gsl_linalg_householder_transform (gsl_vector * w)
 -- Function: gsl_complex gsl_linalg_complex_householder_transform
          (gsl_vector_complex * w)

     This function prepares a Householder transformation H = I - \tau v
     v^T which can be used to zero all the elements of the input vector
     ‘w’ except the first.  On output the Householder vector ‘v’ is
     stored in ‘w’ and the scalar \tau is returned.  The householder
     vector ‘v’ is normalized so that ‘v[0] = 1’, however this 1 is not
     stored in the output vector.  Instead, ‘w[0]’ is set to the first
     element of the transformed vector, so that if u = H w, ‘w[0] =
     u[0]’ on output and the remainder of u is zero.

 -- Function: int gsl_linalg_householder_hm (double tau, const
          gsl_vector * v, gsl_matrix * A)
 -- Function: int gsl_linalg_complex_householder_hm (gsl_complex tau,
          const gsl_vector_complex * v, gsl_matrix_complex * A)

     This function applies the Householder matrix H defined by the
     scalar ‘tau’ and the vector ‘v’ to the left-hand side of the matrix
     ‘A’.  On output the result H A is stored in ‘A’.

 -- Function: int gsl_linalg_householder_mh (double tau, const
          gsl_vector * v, gsl_matrix * A)
 -- Function: int gsl_linalg_complex_householder_mh (gsl_complex tau,
          const gsl_vector_complex * v, gsl_matrix_complex * A)

     This function applies the Householder matrix H defined by the
     scalar ‘tau’ and the vector ‘v’ to the right-hand side of the
     matrix ‘A’.  On output the result A H is stored in ‘A’.

 -- Function: int gsl_linalg_householder_hv (double tau, const
          gsl_vector * v, gsl_vector * w)
 -- Function: int gsl_linalg_complex_householder_hv (gsl_complex tau,
          const gsl_vector_complex * v, gsl_vector_complex * w)

     This function applies the Householder transformation H defined by
     the scalar ‘tau’ and the vector ‘v’ to the vector ‘w’.  On output
     the result H w is stored in ‘w’.


File: gsl-ref.info,  Node: Householder solver for linear systems,  Next: Tridiagonal Systems,  Prev: Householder Transformations,  Up: Linear Algebra

14.18 Householder solver for linear systems
===========================================

 -- Function: int gsl_linalg_HH_solve (gsl_matrix * A, const gsl_vector
          * b, gsl_vector * x)

     This function solves the system A x = b directly using Householder
     transformations.  On output the solution is stored in ‘x’ and ‘b’
     is not modified.  The matrix ‘A’ is destroyed by the Householder
     transformations.

 -- Function: int gsl_linalg_HH_svx (gsl_matrix * A, gsl_vector * x)

     This function solves the system A x = b in-place using Householder
     transformations.  On input ‘x’ should contain the right-hand side
     b, which is replaced by the solution on output.  The matrix ‘A’ is
     destroyed by the Householder transformations.


File: gsl-ref.info,  Node: Tridiagonal Systems,  Next: Triangular Systems,  Prev: Householder solver for linear systems,  Up: Linear Algebra

14.19 Tridiagonal Systems
=========================

The functions described in this section efficiently solve symmetric,
non-symmetric and cyclic tridiagonal systems with minimal storage.  Note
that the current implementations of these functions use a variant of
Cholesky decomposition, so the tridiagonal matrix must be positive
definite.  For non-positive definite matrices, the functions return the
error code ‘GSL_ESING’.

 -- Function: int gsl_linalg_solve_tridiag (const gsl_vector * diag,
          const gsl_vector * e, const gsl_vector * f, const gsl_vector
          * b, gsl_vector * x)

     This function solves the general N-by-N system A x = b where ‘A’ is
     tridiagonal (N \geq 2).  The super-diagonal and sub-diagonal
     vectors ‘e’ and ‘f’ must be one element shorter than the diagonal
     vector ‘diag’.  The form of ‘A’ for the 4-by-4 case is shown below,

          A = ( d_0 e_0  0   0  )
              ( f_0 d_1 e_1  0  )
              (  0  f_1 d_2 e_2 )
              (  0   0  f_2 d_3 )

 -- Function: int gsl_linalg_solve_symm_tridiag (const gsl_vector
          * diag, const gsl_vector * e, const gsl_vector * b, gsl_vector
          * x)

     This function solves the general N-by-N system A x = b where ‘A’ is
     symmetric tridiagonal (N \geq 2).  The off-diagonal vector ‘e’ must
     be one element shorter than the diagonal vector ‘diag’.  The form
     of ‘A’ for the 4-by-4 case is shown below,

          A = ( d_0 e_0  0   0  )
              ( e_0 d_1 e_1  0  )
              (  0  e_1 d_2 e_2 )
              (  0   0  e_2 d_3 )

 -- Function: int gsl_linalg_solve_cyc_tridiag (const gsl_vector * diag,
          const gsl_vector * e, const gsl_vector * f, const gsl_vector
          * b, gsl_vector * x)

     This function solves the general N-by-N system A x = b where ‘A’ is
     cyclic tridiagonal (N \geq 3).  The cyclic super-diagonal and
     sub-diagonal vectors ‘e’ and ‘f’ must have the same number of
     elements as the diagonal vector ‘diag’.  The form of ‘A’ for the
     4-by-4 case is shown below,

          A = ( d_0 e_0  0  f_3 )
              ( f_0 d_1 e_1  0  )
              (  0  f_1 d_2 e_2 )
              ( e_3  0  f_2 d_3 )

 -- Function: int gsl_linalg_solve_symm_cyc_tridiag (const gsl_vector
          * diag, const gsl_vector * e, const gsl_vector * b, gsl_vector
          * x)

     This function solves the general N-by-N system A x = b where ‘A’ is
     symmetric cyclic tridiagonal (N \geq 3).  The cyclic off-diagonal
     vector ‘e’ must have the same number of elements as the diagonal
     vector ‘diag’.  The form of ‘A’ for the 4-by-4 case is shown below,

          A = ( d_0 e_0  0  e_3 )
              ( e_0 d_1 e_1  0  )
              (  0  e_1 d_2 e_2 )
              ( e_3  0  e_2 d_3 )


File: gsl-ref.info,  Node: Triangular Systems,  Next: Banded Systems,  Prev: Tridiagonal Systems,  Up: Linear Algebra

14.20 Triangular Systems
========================

 -- Function: int gsl_linalg_tri_invert (CBLAS_UPLO_t Uplo,
          CBLAS_DIAG_t Diag, gsl_matrix * T)
 -- Function: int gsl_linalg_complex_tri_invert (CBLAS_UPLO_t Uplo,
          CBLAS_DIAG_t Diag, gsl_matrix_complex * T)

     These functions compute the in-place inverse of the triangular
     matrix ‘T’, stored in the lower triangle when ‘Uplo’ = ‘CblasLower’
     and upper triangle when ‘Uplo’ = ‘CblasUpper’.  The parameter
     ‘Diag’ = ‘CblasUnit’, ‘CblasNonUnit’ specifies whether the matrix
     is unit triangular.

 -- Function: int gsl_linalg_tri_LTL (gsl_matrix * L)
 -- Function: int gsl_linalg_complex_tri_LHL (gsl_matrix_complex * L)

     These functions compute the product L^T L (or L^{\dagger} L)
     in-place and stores it in the lower triangle of ‘L’ on output.

 -- Function: int gsl_linalg_tri_UL (gsl_matrix * LU)
 -- Function: int gsl_linalg_complex_tri_UL (gsl_matrix_complex * LU)

     These functions compute the product U L where U is upper triangular
     and L is unit lower triangular, stored in ‘LU’, as computed by
     *note gsl_linalg_LU_decomp(): 4e7. or *note
     gsl_linalg_complex_LU_decomp(): 4e8.  The product is computed
     in-place using Level 3 BLAS.

 -- Function: int gsl_linalg_tri_rcond (CBLAS_UPLO_t Uplo, const
          gsl_matrix * A, double * rcond, gsl_vector * work)

     This function estimates the 1-norm reciprocal condition number of
     the triangular matrix ‘A’, using the lower triangle when ‘Uplo’ is
     ‘CblasLower’ and upper triangle when ‘Uplo’ is ‘CblasUpper’.  The
     reciprocal condition number 1 / \left( \left|\left| A
     \right|\right|_1 \left|\left| A^{-1} \right|\right|_1 \right) is
     stored in ‘rcond’ on output.  Additional workspace of size 3N is
     required in ‘work’.


File: gsl-ref.info,  Node: Banded Systems,  Next: Balancing,  Prev: Triangular Systems,  Up: Linear Algebra

14.21 Banded Systems
====================

Band matrices are sparse matrices whose non-zero entries are confined to
a diagonal `band'.  From a storage point of view, significant savings
can be achieved by storing only the non-zero diagonals of a banded
matrix.  Algorithms such as LU and Cholesky factorizations preserve the
band structure of these matrices.  Computationally, working with compact
banded matrices is preferable to working on the full dense matrix with
many zero entries.

* Menu:

* General Banded Format::
* Symmetric Banded Format::
* Banded Cholesky Decomposition::
* Banded LDLT Decomposition::


File: gsl-ref.info,  Node: General Banded Format,  Next: Symmetric Banded Format,  Up: Banded Systems

14.21.1 General Banded Format
-----------------------------

An example of a general banded matrix is given below.

     A = \begin{pmatrix} \alpha_1 & \beta_1 & \gamma_1 & 0 & 0 & 0 \\ \delta_1 & \alpha_2 & \beta_2 & \gamma_2 & 0 & 0 \\ 0 & \delta_2 & \alpha_3 & \beta_3 & \gamma_3 & 0 \\ 0 & 0 & \delta_3 & \alpha_4 & \beta_4 & \gamma_4 \\ 0 & 0 & 0 & \delta_4 & \alpha_5 & \beta_5 \\ 0 & 0 & 0 & 0 & \delta_5 & \alpha_6 \end{pmatrix}

This matrix has a `lower bandwidth' of 1 and an `upper bandwidth' of 2.
The lower bandwidth is the number of non-zero subdiagonals, and the
upper bandwidth is the number of non-zero superdiagonals.  A (p,q)
banded matrix has a lower bandwidth p and upper bandwidth q.  For
example, diagonal matrices are (0,0), tridiagonal matrices are (1,1),
and upper triangular matrices are (0,N-1) banded matrices.

The corresponding 6-by-4 packed banded matrix looks like

     AB = \begin{pmatrix} * & * & \alpha_1 & \delta_1 \\ * & \beta_1 & \alpha_2 & \delta_2 \\ \gamma_1 & \beta_2 & \alpha_3 & \delta_3 \\ \gamma_2 & \beta_3 & \alpha_4 & \delta_4 \\ \gamma_3 & \beta_4 & \alpha_5 & \delta_5 \\ \gamma_4 & \beta_5 & \alpha_6 & * \end{pmatrix}

where the superdiagonals are stored in columns, followed by the
diagonal, followed by the subdiagonals.  The entries marked by * are not
referenced by the banded routines.  With this format, each row of AB
corresponds to the non-zero entries of the corresponding column of A.
For an N-by-N matrix A, the dimension of AB will be N-by-(p+q+1).


File: gsl-ref.info,  Node: Symmetric Banded Format,  Next: Banded Cholesky Decomposition,  Prev: General Banded Format,  Up: Banded Systems

14.21.2 Symmetric Banded Format
-------------------------------

Symmetric banded matrices allow for additional storage savings.  As an
example, consider the following 6 \times 6 symmetric banded matrix with
lower bandwidth p = 2:

     A = \begin{pmatrix} \alpha_1 & \beta_1 & \gamma_1 & 0 & 0 & 0 \\ \beta_1 & \alpha_2 & \beta_2 & \gamma_2 & 0 & 0 \\ \gamma_1 & \beta_2 & \alpha_3 & \beta_3 & \gamma_3 & 0 \\ 0 & \gamma_2 & \beta_3 & \alpha_4 & \beta_4 & \gamma_4 \\ 0 & 0 & \gamma_3 & \beta_4 & \alpha_5 & \beta_5 \\ 0 & 0 & 0 & \gamma_4 & \beta_5 & \alpha_6 \end{pmatrix}

The packed symmetric banded 6 \times 3 matrix will look like:

     AB = \begin{pmatrix} \alpha_1 & \beta_1 & \gamma_1 \\ \alpha_2 & \beta_2 & \gamma_2 \\ \alpha_3 & \beta_3 & \gamma_3 \\ \alpha_4 & \beta_4 & \gamma_4 \\ \alpha_5 & \beta_5 & * \\ \alpha_6 & * & * \end{pmatrix}

The entries marked by * are not referenced by the symmetric banded
routines.

     Warning: Note that this format is the transpose of the symmetric
     banded format used by LAPACK. In order to develop efficient
     routines for symmetric banded matrices, it helps to have the
     nonzero elements in each column in contiguous memory locations.
     Since C uses row-major order, GSL stores the columns in the rows of
     the packed banded format, while LAPACK, written in Fortran, uses
     the transposed format.


File: gsl-ref.info,  Node: Banded Cholesky Decomposition,  Next: Banded LDLT Decomposition,  Prev: Symmetric Banded Format,  Up: Banded Systems

14.21.3 Banded Cholesky Decomposition
-------------------------------------

The routines in this section are designed to factor and solve N-by-N
linear systems of the form A x = b where A is a banded, symmetric, and
positive definite matrix with lower bandwidth p.  See *note Cholesky
Decomposition: 3f1. for more information on the factorization.  The
lower triangular factor of the Cholesky decomposition preserves the same
banded structure as the matrix A, enabling an efficient algorithm which
overwrites the original matrix with the L factor.

 -- Function: int gsl_linalg_cholesky_band_decomp (gsl_matrix * A)

     This function factorizes the symmetric, positive-definite square
     matrix ‘A’ into the Cholesky decomposition A = L L^T. The input
     matrix ‘A’ is given in *note symmetric banded format: 589, and has
     dimensions N-by-(p + 1), where p is the lower bandwidth of the
     matrix.  On output, the entries of ‘A’ are replaced by the entries
     of the matrix L in the same format.  In addition, the lower right
     element of ‘A’ is used to store the matrix 1-norm, used later by
     *note gsl_linalg_cholesky_band_rcond(): 58d. to calculate the
     reciprocal condition number.

     If the matrix is not positive-definite then the decomposition will
     fail, returning the error code *note GSL_EDOM: 28.  When testing
     whether a matrix is positive-definite, disable the error handler
     first to avoid triggering an error.

 -- Function: int gsl_linalg_cholesky_band_solve (const gsl_matrix
          * LLT, const gsl_vector * b, gsl_vector * x)

     This function solves the symmetric banded system A x = b using the
     Cholesky decomposition of A held in the matrix ‘LLT’ which must
     have been previously computed by *note
     gsl_linalg_cholesky_band_decomp(): 58c.

 -- Function: int gsl_linalg_cholesky_band_svx (const gsl_matrix * LLT,
          gsl_vector * x)

     This function solves the symmetric banded system A x = b in-place
     using the Cholesky decomposition of A held in the matrix ‘LLT’
     which must have been previously computed by *note
     gsl_linalg_cholesky_band_decomp(): 58c.  On input ‘x’ should
     contain the right-hand side b, which is replaced by the solution on
     output.

 -- Function: int gsl_linalg_cholesky_band_invert (const gsl_matrix
          * LLT, gsl_matrix * Ainv)

     This function computes the inverse of a symmetric banded matrix
     from its Cholesky decomposition ‘LLT’, which must have been
     previously computed by *note gsl_linalg_cholesky_band_decomp():
     58c.  On output, the inverse is stored in ‘Ainv’, using both the
     lower and upper portions.

 -- Function: int gsl_linalg_cholesky_band_unpack (const gsl_matrix
          * LLT, gsl_matrix * L)

     This function unpacks the lower triangular Cholesky factor from
     ‘LLT’ and stores it in the lower triangular portion of the N-by-N
     matrix ‘L’.  The upper triangular portion of ‘L’ is not referenced.

 -- Function: int gsl_linalg_cholesky_band_rcond (const gsl_matrix
          * LLT, double * rcond, gsl_vector * work)

     This function estimates the reciprocal condition number (using the
     1-norm) of the symmetric banded positive definite matrix A, using
     its Cholesky decomposition provided in ‘LLT’.  The reciprocal
     condition number estimate, defined as 1 / (||A||_1 \cdot
     ||A^{-1}||_1), is stored in ‘rcond’.  Additional workspace of size
     3 N is required in ‘work’.


File: gsl-ref.info,  Node: Banded LDLT Decomposition,  Prev: Banded Cholesky Decomposition,  Up: Banded Systems

14.21.4 Banded LDLT Decomposition
---------------------------------

The routines in this section are designed to factor and solve N-by-N
linear systems of the form A x = b where A is a banded, symmetric, and
non-singular matrix with lower bandwidth p.  See *note LDLT
Decomposition: 552. for more information on the factorization.  The
lower triangular factor of the L D L^T decomposition preserves the same
banded structure as the matrix A, enabling an efficient algorithm which
overwrites the original matrix with the L and D factors.

 -- Function: int gsl_linalg_ldlt_band_decomp (gsl_matrix * A)

     This function factorizes the symmetric, non-singular square matrix
     ‘A’ into the decomposition A = L D L^T. The input matrix ‘A’ is
     given in *note symmetric banded format: 589, and has dimensions
     N-by-(p + 1), where p is the lower bandwidth of the matrix.  On
     output, the entries of ‘A’ are replaced by the entries of the
     matrices D and L in the same format.

     If the matrix is singular then the decomposition will fail,
     returning the error code *note GSL_EDOM: 28.

 -- Function: int gsl_linalg_ldlt_band_solve (const gsl_matrix * LDLT,
          const gsl_vector * b, gsl_vector * x)

     This function solves the symmetric banded system A x = b using the
     L D L^T decomposition of A held in the matrix ‘LDLT’ which must
     have been previously computed by *note
     gsl_linalg_ldlt_band_decomp(): 593.

 -- Function: int gsl_linalg_ldlt_band_svx (const gsl_matrix * LDLT,
          gsl_vector * x)

     This function solves the symmetric banded system A x = b in-place
     using the L D L^T decomposition of A held in the matrix ‘LDLT’
     which must have been previously computed by *note
     gsl_linalg_ldlt_band_decomp(): 593.  On input ‘x’ should contain
     the right-hand side b, which is replaced by the solution on output.

 -- Function: gsl_linalg_ldlt_band_unpack (const gsl_matrix * LDLT,
          gsl_matrix * L, gsl_vector * D)

     This function unpacks the unit lower triangular factor L from
     ‘LDLT’ and stores it in the lower triangular portion of the N-by-N
     matrix ‘L’.  The upper triangular portion of ‘L’ is not referenced.
     The diagonal matrix D is stored in the vector ‘D’.

 -- Function: int gsl_linalg_ldlt_band_rcond (const gsl_matrix * LDLT,
          double * rcond, gsl_vector * work)

     This function estimates the reciprocal condition number (using the
     1-norm) of the symmetric banded nonsingular matrix A, using its L D
     L^T decomposition provided in ‘LDLT’.  The reciprocal condition
     number estimate, defined as 1 / (||A||_1 \cdot ||A^{-1}||_1), is
     stored in ‘rcond’.  Additional workspace of size 3 N is required in
     ‘work’.


File: gsl-ref.info,  Node: Balancing,  Next: Examples<9>,  Prev: Banded Systems,  Up: Linear Algebra

14.22 Balancing
===============

The process of balancing a matrix applies similarity transformations to
make the rows and columns have comparable norms.  This is useful, for
example, to reduce roundoff errors in the solution of eigenvalue
problems.  Balancing a matrix A consists of replacing A with a similar
matrix

     A' = D^{-1} A D

where D is a diagonal matrix whose entries are powers of the floating
point radix.

 -- Function: int gsl_linalg_balance_matrix (gsl_matrix * A, gsl_vector
          * D)

     This function replaces the matrix ‘A’ with its balanced counterpart
     and stores the diagonal elements of the similarity transformation
     into the vector ‘D’.


File: gsl-ref.info,  Node: Examples<9>,  Next: References and Further Reading<9>,  Prev: Balancing,  Up: Linear Algebra

14.23 Examples
==============

The following program solves the linear system A x = b.  The system to
be solved is,

     [ 0.18 0.60 0.57 0.96 ] [x0]   [1.0]
     [ 0.41 0.24 0.99 0.58 ] [x1] = [2.0]
     [ 0.14 0.30 0.97 0.66 ] [x2]   [3.0]
     [ 0.51 0.13 0.19 0.85 ] [x3]   [4.0]

and the solution is found using LU decomposition of the matrix A.

     #include <stdio.h>
     #include <gsl/gsl_linalg.h>

     int
     main (void)
     {
       double a_data[] = { 0.18, 0.60, 0.57, 0.96,
                           0.41, 0.24, 0.99, 0.58,
                           0.14, 0.30, 0.97, 0.66,
                           0.51, 0.13, 0.19, 0.85 };

       double b_data[] = { 1.0, 2.0, 3.0, 4.0 };

       gsl_matrix_view m
         = gsl_matrix_view_array (a_data, 4, 4);

       gsl_vector_view b
         = gsl_vector_view_array (b_data, 4);

       gsl_vector *x = gsl_vector_alloc (4);

       int s;

       gsl_permutation * p = gsl_permutation_alloc (4);

       gsl_linalg_LU_decomp (&m.matrix, p, &s);

       gsl_linalg_LU_solve (&m.matrix, p, &b.vector, x);

       printf ("x = \n");
       gsl_vector_fprintf (stdout, x, "%g");

       gsl_permutation_free (p);
       gsl_vector_free (x);
       return 0;
     }

Here is the output from the program,

     x =
     -4.05205
     -12.6056
     1.66091
     8.69377

This can be verified by multiplying the solution x by the original
matrix A using GNU octave,

     octave> A = [ 0.18, 0.60, 0.57, 0.96;
                   0.41, 0.24, 0.99, 0.58;
                   0.14, 0.30, 0.97, 0.66;
                   0.51, 0.13, 0.19, 0.85 ];

     octave> x = [ -4.05205; -12.6056; 1.66091; 8.69377];

     octave> A * x
     ans =
       1.0000
       2.0000
       3.0000
       4.0000

This reproduces the original right-hand side vector, b, in accordance
with the equation A x = b.


File: gsl-ref.info,  Node: References and Further Reading<9>,  Prev: Examples<9>,  Up: Linear Algebra

14.24 References and Further Reading
====================================

Further information on the algorithms described in this section can be
found in the following book,

   * G. H. Golub, C. F. Van Loan, “Matrix Computations” (3rd Ed, 1996),
     Johns Hopkins University Press, ISBN 0-8018-5414-8.

The LAPACK library is described in the following manual,

   * `LAPACK Users’ Guide' (Third Edition, 1999), Published by SIAM,
     ISBN 0-89871-447-8

The LAPACK source code can be found at ‘http://www.netlib.org/lapack’,
along with an online copy of the users guide.

Further information on recursive Level 3 BLAS algorithms may be found in
the following paper,

   * E. Peise and P. Bientinesi, “Recursive algorithms for dense linear
     algebra: the ReLAPACK collection”,
     ‘http://arxiv.org/abs/1602.06763’, 2016.

The recursive Level 3 BLAS QR decomposition is described in the
following paper,

   * E. Elmroth and F. G. Gustavson, 2000.  Applying recursion to serial
     and parallel QR factorization leads to better performance.  IBM
     Journal of Research and Development, 44(4), pp.605-624.

The Modified Golub-Reinsch algorithm is described in the following
paper,

   * T.F. Chan, “An Improved Algorithm for Computing the Singular Value
     Decomposition”, ACM Transactions on Mathematical Software, 8
     (1982), pp 72–83.

The Jacobi algorithm for singular value decomposition is described in
the following papers,

   * J.C. Nash, “A one-sided transformation method for the singular
     value decomposition and algebraic eigenproblem”, Computer Journal,
     Volume 18, Number 1 (1975), p 74–76

   * J.C. Nash and S. Shlien “Simple algorithms for the partial singular
     value decomposition”, Computer Journal, Volume 30 (1987), p
     268–275.

   * J. Demmel, K. Veselic, “Jacobi’s Method is more accurate than QR”,
     Lapack Working Note 15 (LAWN-15), October 1989.  Available from
     netlib, ‘http://www.netlib.org/lapack/’ in the ‘lawns’ or
     ‘lawnspdf’ directories.

The algorithm for estimating a matrix condition number is described in
the following paper,

   * N. J. Higham, “FORTRAN codes for estimating the one-norm of a real
     or complex matrix, with applications to condition estimation”, ACM
     Trans.  Math.  Soft., vol.  14, no.  4, pp.  381-396, December
     1988.


File: gsl-ref.info,  Node: Eigensystems,  Next: Fast Fourier Transforms FFTs,  Prev: Linear Algebra,  Up: Top

15 Eigensystems
***************

This chapter describes functions for computing eigenvalues and
eigenvectors of matrices.  There are routines for real symmetric, real
nonsymmetric, complex hermitian, real generalized symmetric-definite,
complex generalized hermitian-definite, and real generalized
nonsymmetric eigensystems.  Eigenvalues can be computed with or without
eigenvectors.  The hermitian and real symmetric matrix algorithms are
symmetric bidiagonalization followed by QR reduction.  The nonsymmetric
algorithm is the Francis QR double-shift.  The generalized nonsymmetric
algorithm is the QZ method due to Moler and Stewart.

The functions described in this chapter are declared in the header file
‘gsl_eigen.h’.

* Menu:

* Real Symmetric Matrices::
* Complex Hermitian Matrices::
* Real Nonsymmetric Matrices::
* Real Generalized Symmetric-Definite Eigensystems::
* Complex Generalized Hermitian-Definite Eigensystems::
* Real Generalized Nonsymmetric Eigensystems::
* Sorting Eigenvalues and Eigenvectors::
* Examples: Examples<10>.
* References and Further Reading: References and Further Reading<10>.


File: gsl-ref.info,  Node: Real Symmetric Matrices,  Next: Complex Hermitian Matrices,  Up: Eigensystems

15.1 Real Symmetric Matrices
============================

For real symmetric matrices, the library uses the symmetric
bidiagonalization and QR reduction method.  This is described in Golub &
van Loan, section 8.3.  The computed eigenvalues are accurate to an
absolute accuracy of \epsilon ||A||_2, where \epsilon is the machine
precision.

 -- Type: gsl_eigen_symm_workspace

     This workspace contains internal parameters used for solving
     symmetric eigenvalue problems.

 -- Function: gsl_eigen_symm_workspace * gsl_eigen_symm_alloc (const
          size_t n)

     This function allocates a workspace for computing eigenvalues of
     ‘n’-by-‘n’ real symmetric matrices.  The size of the workspace is
     O(2n).

 -- Function: void gsl_eigen_symm_free (gsl_eigen_symm_workspace * w)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: int gsl_eigen_symm (gsl_matrix * A, gsl_vector * eval,
          gsl_eigen_symm_workspace * w)

     This function computes the eigenvalues of the real symmetric matrix
     ‘A’.  Additional workspace of the appropriate size must be provided
     in ‘w’.  The diagonal and lower triangular part of ‘A’ are
     destroyed during the computation, but the strict upper triangular
     part is not referenced.  The eigenvalues are stored in the vector
     ‘eval’ and are unordered.

 -- Type: gsl_eigen_symmv_workspace

     This workspace contains internal parameters used for solving
     symmetric eigenvalue and eigenvector problems.

 -- Function: gsl_eigen_symmv_workspace * gsl_eigen_symmv_alloc (const
          size_t n)

     This function allocates a workspace for computing eigenvalues and
     eigenvectors of ‘n’-by-‘n’ real symmetric matrices.  The size of
     the workspace is O(4n).

 -- Function: void gsl_eigen_symmv_free (gsl_eigen_symmv_workspace * w)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: int gsl_eigen_symmv (gsl_matrix * A, gsl_vector * eval,
          gsl_matrix * evec, gsl_eigen_symmv_workspace * w)

     This function computes the eigenvalues and eigenvectors of the real
     symmetric matrix ‘A’.  Additional workspace of the appropriate size
     must be provided in ‘w’.  The diagonal and lower triangular part of
     ‘A’ are destroyed during the computation, but the strict upper
     triangular part is not referenced.  The eigenvalues are stored in
     the vector ‘eval’ and are unordered.  The corresponding
     eigenvectors are stored in the columns of the matrix ‘evec’.  For
     example, the eigenvector in the first column corresponds to the
     first eigenvalue.  The eigenvectors are guaranteed to be mutually
     orthogonal and normalised to unit magnitude.


File: gsl-ref.info,  Node: Complex Hermitian Matrices,  Next: Real Nonsymmetric Matrices,  Prev: Real Symmetric Matrices,  Up: Eigensystems

15.2 Complex Hermitian Matrices
===============================

For hermitian matrices, the library uses the complex form of the
symmetric bidiagonalization and QR reduction method.

 -- Type: gsl_eigen_herm_workspace

     This workspace contains internal parameters used for solving
     hermitian eigenvalue problems.

 -- Function: gsl_eigen_herm_workspace * gsl_eigen_herm_alloc (const
          size_t n)

     This function allocates a workspace for computing eigenvalues of
     ‘n’-by-‘n’ complex hermitian matrices.  The size of the workspace
     is O(3n).

 -- Function: void gsl_eigen_herm_free (gsl_eigen_herm_workspace * w)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: int gsl_eigen_herm (gsl_matrix_complex * A, gsl_vector
          * eval, gsl_eigen_herm_workspace * w)

     This function computes the eigenvalues of the complex hermitian
     matrix ‘A’.  Additional workspace of the appropriate size must be
     provided in ‘w’.  The diagonal and lower triangular part of ‘A’ are
     destroyed during the computation, but the strict upper triangular
     part is not referenced.  The imaginary parts of the diagonal are
     assumed to be zero and are not referenced.  The eigenvalues are
     stored in the vector ‘eval’ and are unordered.

 -- Type: gsl_eigen_hermv_workspace

     This workspace contains internal parameters used for solving
     hermitian eigenvalue and eigenvector problems.

 -- Function: gsl_eigen_hermv_workspace * gsl_eigen_hermv_alloc (const
          size_t n)

     This function allocates a workspace for computing eigenvalues and
     eigenvectors of ‘n’-by-‘n’ complex hermitian matrices.  The size of
     the workspace is O(5n).

 -- Function: void gsl_eigen_hermv_free (gsl_eigen_hermv_workspace * w)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: int gsl_eigen_hermv (gsl_matrix_complex * A, gsl_vector
          * eval, gsl_matrix_complex * evec, gsl_eigen_hermv_workspace
          * w)

     This function computes the eigenvalues and eigenvectors of the
     complex hermitian matrix ‘A’.  Additional workspace of the
     appropriate size must be provided in ‘w’.  The diagonal and lower
     triangular part of ‘A’ are destroyed during the computation, but
     the strict upper triangular part is not referenced.  The imaginary
     parts of the diagonal are assumed to be zero and are not
     referenced.  The eigenvalues are stored in the vector ‘eval’ and
     are unordered.  The corresponding complex eigenvectors are stored
     in the columns of the matrix ‘evec’.  For example, the eigenvector
     in the first column corresponds to the first eigenvalue.  The
     eigenvectors are guaranteed to be mutually orthogonal and
     normalised to unit magnitude.


File: gsl-ref.info,  Node: Real Nonsymmetric Matrices,  Next: Real Generalized Symmetric-Definite Eigensystems,  Prev: Complex Hermitian Matrices,  Up: Eigensystems

15.3 Real Nonsymmetric Matrices
===============================

The solution of the real nonsymmetric eigensystem problem for a matrix A
involves computing the Schur decomposition

     A = Z T Z^T

where Z is an orthogonal matrix of Schur vectors and T, the Schur form,
is quasi upper triangular with diagonal 1-by-1 blocks which are real
eigenvalues of A, and diagonal 2-by-2 blocks whose eigenvalues are
complex conjugate eigenvalues of A. The algorithm used is the
double-shift Francis method.

 -- Type: gsl_eigen_nonsymm_workspace

     This workspace contains internal parameters used for solving
     nonsymmetric eigenvalue problems.

 -- Function: gsl_eigen_nonsymm_workspace * gsl_eigen_nonsymm_alloc
          (const size_t n)

     This function allocates a workspace for computing eigenvalues of
     ‘n’-by-‘n’ real nonsymmetric matrices.  The size of the workspace
     is O(2n).

 -- Function: void gsl_eigen_nonsymm_free (gsl_eigen_nonsymm_workspace
          * w)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: void gsl_eigen_nonsymm_params (const int compute_t, const
          int balance, gsl_eigen_nonsymm_workspace * w)

     This function sets some parameters which determine how the
     eigenvalue problem is solved in subsequent calls to *note
     gsl_eigen_nonsymm(): 5b6.

     If ‘compute_t’ is set to 1, the full Schur form T will be computed
     by *note gsl_eigen_nonsymm(): 5b6.  If it is set to 0, T will not
     be computed (this is the default setting).  Computing the full
     Schur form T requires approximately 1.5–2 times the number of
     flops.

     If ‘balance’ is set to 1, a balancing transformation is applied to
     the matrix prior to computing eigenvalues.  This transformation is
     designed to make the rows and columns of the matrix have comparable
     norms, and can result in more accurate eigenvalues for matrices
     whose entries vary widely in magnitude.  See *note Balancing: 598.
     for more information.  Note that the balancing transformation does
     not preserve the orthogonality of the Schur vectors, so if you wish
     to compute the Schur vectors with *note gsl_eigen_nonsymm_Z(): 5b7.
     you will obtain the Schur vectors of the balanced matrix instead of
     the original matrix.  The relationship will be

          T = Q^T D^{-1} A D Q

     where ‘Q’ is the matrix of Schur vectors for the balanced matrix,
     and ‘D’ is the balancing transformation.  Then *note
     gsl_eigen_nonsymm_Z(): 5b7. will compute a matrix ‘Z’ which
     satisfies

          T = Z^{-1} A Z

     with Z = D Q. Note that ‘Z’ will not be orthogonal.  For this
     reason, balancing is not performed by default.

 -- Function: int gsl_eigen_nonsymm (gsl_matrix * A, gsl_vector_complex
          * eval, gsl_eigen_nonsymm_workspace * w)

     This function computes the eigenvalues of the real nonsymmetric
     matrix ‘A’ and stores them in the vector ‘eval’.  If T is desired,
     it is stored in the upper portion of ‘A’ on output.  Otherwise, on
     output, the diagonal of ‘A’ will contain the 1-by-1 real
     eigenvalues and 2-by-2 complex conjugate eigenvalue systems, and
     the rest of ‘A’ is destroyed.  In rare cases, this function may
     fail to find all eigenvalues.  If this happens, an error code is
     returned and the number of converged eigenvalues is stored in
     ‘w->n_evals’.  The converged eigenvalues are stored in the
     beginning of ‘eval’.

 -- Function: int gsl_eigen_nonsymm_Z (gsl_matrix * A,
          gsl_vector_complex * eval, gsl_matrix * Z,
          gsl_eigen_nonsymm_workspace * w)

     This function is identical to *note gsl_eigen_nonsymm(): 5b6.
     except that it also computes the Schur vectors and stores them into
     ‘Z’.

 -- Type: gsl_eigen_nonsymmv_workspace

     This workspace contains internal parameters used for solving
     nonsymmetric eigenvalue and eigenvector problems.

 -- Function: gsl_eigen_nonsymmv_workspace * gsl_eigen_nonsymmv_alloc
          (const size_t n)

     This function allocates a workspace for computing eigenvalues and
     eigenvectors of ‘n’-by-‘n’ real nonsymmetric matrices.  The size of
     the workspace is O(5n).

 -- Function: void gsl_eigen_nonsymmv_free (gsl_eigen_nonsymmv_workspace
          * w)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: void gsl_eigen_nonsymmv_params (const int balance,
          gsl_eigen_nonsymm_workspace * w)

     This function sets parameters which determine how the eigenvalue
     problem is solved in subsequent calls to *note
     gsl_eigen_nonsymmv(): 5bc.  If ‘balance’ is set to 1, a balancing
     transformation is applied to the matrix.  See *note
     gsl_eigen_nonsymm_params(): 5b5. for more information.  Balancing
     is turned off by default since it does not preserve the
     orthogonality of the Schur vectors.

 -- Function: int gsl_eigen_nonsymmv (gsl_matrix * A, gsl_vector_complex
          * eval, gsl_matrix_complex * evec,
          gsl_eigen_nonsymmv_workspace * w)

     This function computes eigenvalues and right eigenvectors of the
     ‘n’-by-‘n’ real nonsymmetric matrix ‘A’.  It first calls *note
     gsl_eigen_nonsymm(): 5b6. to compute the eigenvalues, Schur form T,
     and Schur vectors.  Then it finds eigenvectors of T and
     backtransforms them using the Schur vectors.  The Schur vectors are
     destroyed in the process, but can be saved by using *note
     gsl_eigen_nonsymmv_Z(): 5bd.  The computed eigenvectors are
     normalized to have unit magnitude.  On output, the upper portion of
     ‘A’ contains the Schur form T. If *note gsl_eigen_nonsymm(): 5b6.
     fails, no eigenvectors are computed, and an error code is returned.

 -- Function: int gsl_eigen_nonsymmv_Z (gsl_matrix * A,
          gsl_vector_complex * eval, gsl_matrix_complex * evec,
          gsl_matrix * Z, gsl_eigen_nonsymmv_workspace * w)

     This function is identical to *note gsl_eigen_nonsymmv(): 5bc.
     except that it also saves the Schur vectors into ‘Z’.


File: gsl-ref.info,  Node: Real Generalized Symmetric-Definite Eigensystems,  Next: Complex Generalized Hermitian-Definite Eigensystems,  Prev: Real Nonsymmetric Matrices,  Up: Eigensystems

15.4 Real Generalized Symmetric-Definite Eigensystems
=====================================================

The real generalized symmetric-definite eigenvalue problem is to find
eigenvalues \lambda and eigenvectors x such that

     A x = \lambda B x

where A and B are symmetric matrices, and B is positive-definite.  This
problem reduces to the standard symmetric eigenvalue problem by applying
the Cholesky decomposition to B:

     A x = \lambda B x
     A x = \lambda L L^T x
     ( L^{-1} A L^{-T} ) L^T x = \lambda L^T x

Therefore, the problem becomes C y = \lambda y where C = L^{-1} A L^{-T}
is symmetric, and y = L^T x.  The standard symmetric eigensolver can be
applied to the matrix C. The resulting eigenvectors are backtransformed
to find the vectors of the original problem.  The eigenvalues and
eigenvectors of the generalized symmetric-definite eigenproblem are
always real.

 -- Type: gsl_eigen_gensymm_workspace

     This workspace contains internal parameters used for solving
     generalized symmetric eigenvalue problems.

 -- Function: gsl_eigen_gensymm_workspace * gsl_eigen_gensymm_alloc
          (const size_t n)

     This function allocates a workspace for computing eigenvalues of
     ‘n’-by-‘n’ real generalized symmetric-definite eigensystems.  The
     size of the workspace is O(2n).

 -- Function: void gsl_eigen_gensymm_free (gsl_eigen_gensymm_workspace
          * w)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: int gsl_eigen_gensymm (gsl_matrix * A, gsl_matrix * B,
          gsl_vector * eval, gsl_eigen_gensymm_workspace * w)

     This function computes the eigenvalues of the real generalized
     symmetric-definite matrix pair (‘A’, ‘B’), and stores them in
     ‘eval’, using the method outlined above.  On output, ‘B’ contains
     its Cholesky decomposition and ‘A’ is destroyed.

 -- Type: gsl_eigen_gensymmv_workspace

     This workspace contains internal parameters used for solving
     generalized symmetric eigenvalue and eigenvector problems.

 -- Function: gsl_eigen_gensymmv_workspace * gsl_eigen_gensymmv_alloc
          (const size_t n)

     This function allocates a workspace for computing eigenvalues and
     eigenvectors of ‘n’-by-‘n’ real generalized symmetric-definite
     eigensystems.  The size of the workspace is O(4n).

 -- Function: void gsl_eigen_gensymmv_free (gsl_eigen_gensymmv_workspace
          * w)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: int gsl_eigen_gensymmv (gsl_matrix * A, gsl_matrix * B,
          gsl_vector * eval, gsl_matrix * evec,
          gsl_eigen_gensymmv_workspace * w)

     This function computes the eigenvalues and eigenvectors of the real
     generalized symmetric-definite matrix pair (‘A’, ‘B’), and stores
     them in ‘eval’ and ‘evec’ respectively.  The computed eigenvectors
     are normalized to have unit magnitude.  On output, ‘B’ contains its
     Cholesky decomposition and ‘A’ is destroyed.


File: gsl-ref.info,  Node: Complex Generalized Hermitian-Definite Eigensystems,  Next: Real Generalized Nonsymmetric Eigensystems,  Prev: Real Generalized Symmetric-Definite Eigensystems,  Up: Eigensystems

15.5 Complex Generalized Hermitian-Definite Eigensystems
========================================================

The complex generalized hermitian-definite eigenvalue problem is to find
eigenvalues \lambda and eigenvectors x such that

     A x = \lambda B x

where A and B are hermitian matrices, and B is positive-definite.
Similarly to the real case, this can be reduced to C y = \lambda y where
C = L^{-1} A L^{-\dagger} is hermitian, and y = L^{\dagger} x.  The
standard hermitian eigensolver can be applied to the matrix C. The
resulting eigenvectors are backtransformed to find the vectors of the
original problem.  The eigenvalues of the generalized hermitian-definite
eigenproblem are always real.

 -- Type: gsl_eigen_genherm_workspace

     This workspace contains internal parameters used for solving
     generalized hermitian eigenvalue problems.

 -- Function: gsl_eigen_genherm_workspace * gsl_eigen_genherm_alloc
          (const size_t n)

     This function allocates a workspace for computing eigenvalues of
     ‘n’-by-‘n’ complex generalized hermitian-definite eigensystems.
     The size of the workspace is O(3n).

 -- Function: void gsl_eigen_genherm_free (gsl_eigen_genherm_workspace
          * w)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: int gsl_eigen_genherm (gsl_matrix_complex * A,
          gsl_matrix_complex * B, gsl_vector * eval,
          gsl_eigen_genherm_workspace * w)

     This function computes the eigenvalues of the complex generalized
     hermitian-definite matrix pair (‘A’, ‘B’), and stores them in
     ‘eval’, using the method outlined above.  On output, ‘B’ contains
     its Cholesky decomposition and ‘A’ is destroyed.

 -- Type: gsl_eigen_genhermv_workspace

     This workspace contains internal parameters used for solving
     generalized hermitian eigenvalue and eigenvector problems.

 -- Function: gsl_eigen_genhermv_workspace * gsl_eigen_genhermv_alloc
          (const size_t n)

     This function allocates a workspace for computing eigenvalues and
     eigenvectors of ‘n’-by-‘n’ complex generalized hermitian-definite
     eigensystems.  The size of the workspace is O(5n).

 -- Function: void gsl_eigen_genhermv_free (gsl_eigen_genhermv_workspace
          * w)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: int gsl_eigen_genhermv (gsl_matrix_complex * A,
          gsl_matrix_complex * B, gsl_vector * eval, gsl_matrix_complex
          * evec, gsl_eigen_genhermv_workspace * w)

     This function computes the eigenvalues and eigenvectors of the
     complex generalized hermitian-definite matrix pair (‘A’, ‘B’), and
     stores them in ‘eval’ and ‘evec’ respectively.  The computed
     eigenvectors are normalized to have unit magnitude.  On output, ‘B’
     contains its Cholesky decomposition and ‘A’ is destroyed.


File: gsl-ref.info,  Node: Real Generalized Nonsymmetric Eigensystems,  Next: Sorting Eigenvalues and Eigenvectors,  Prev: Complex Generalized Hermitian-Definite Eigensystems,  Up: Eigensystems

15.6 Real Generalized Nonsymmetric Eigensystems
===============================================

Given two square matrices (A, B), the generalized nonsymmetric
eigenvalue problem is to find eigenvalues \lambda and eigenvectors x
such that

     A x = \lambda B x

We may also define the problem as finding eigenvalues \mu and
eigenvectors y such that

     \mu A y = B y

Note that these two problems are equivalent (with \lambda = 1/\mu) if
neither \lambda nor \mu is zero.  If say, \lambda is zero, then it is
still a well defined eigenproblem, but its alternate problem involving
\mu is not.  Therefore, to allow for zero (and infinite) eigenvalues,
the problem which is actually solved is

     \beta A x = \alpha B x

The eigensolver routines below will return two values \alpha and \beta
and leave it to the user to perform the divisions \lambda = \alpha /
\beta and \mu = \beta / \alpha.

If the determinant of the matrix pencil A - \lambda B is zero for all
\lambda, the problem is said to be singular; otherwise it is called
regular.  Singularity normally leads to some \alpha = \beta = 0 which
means the eigenproblem is ill-conditioned and generally does not have
well defined eigenvalue solutions.  The routines below are intended for
regular matrix pencils and could yield unpredictable results when
applied to singular pencils.

The solution of the real generalized nonsymmetric eigensystem problem
for a matrix pair (A, B) involves computing the generalized Schur
decomposition

     A = Q S Z^T
     B = Q T Z^T

where Q and Z are orthogonal matrices of left and right Schur vectors
respectively, and (S, T) is the generalized Schur form whose diagonal
elements give the \alpha and \beta values.  The algorithm used is the QZ
method due to Moler and Stewart (see references).

 -- Type: gsl_eigen_gen_workspace

     This workspace contains internal parameters used for solving
     generalized eigenvalue problems.

 -- Function: gsl_eigen_gen_workspace * gsl_eigen_gen_alloc (const
          size_t n)

     This function allocates a workspace for computing eigenvalues of
     ‘n’-by-‘n’ real generalized nonsymmetric eigensystems.  The size of
     the workspace is O(n).

 -- Function: void gsl_eigen_gen_free (gsl_eigen_gen_workspace * w)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: void gsl_eigen_gen_params (const int compute_s, const
          int compute_t, const int balance, gsl_eigen_gen_workspace * w)

     This function sets some parameters which determine how the
     eigenvalue problem is solved in subsequent calls to *note
     gsl_eigen_gen(): 5d5.

     If ‘compute_s’ is set to 1, the full Schur form S will be computed
     by *note gsl_eigen_gen(): 5d5.  If it is set to 0, S will not be
     computed (this is the default setting).  S is a quasi upper
     triangular matrix with 1-by-1 and 2-by-2 blocks on its diagonal.
     1-by-1 blocks correspond to real eigenvalues, and 2-by-2 blocks
     correspond to complex eigenvalues.

     If ‘compute_t’ is set to 1, the full Schur form T will be computed
     by *note gsl_eigen_gen(): 5d5.  If it is set to 0, T will not be
     computed (this is the default setting).  T is an upper triangular
     matrix with non-negative elements on its diagonal.  Any 2-by-2
     blocks in S will correspond to a 2-by-2 diagonal block in T.

     The ‘balance’ parameter is currently ignored, since generalized
     balancing is not yet implemented.

 -- Function: int gsl_eigen_gen (gsl_matrix * A, gsl_matrix * B,
          gsl_vector_complex * alpha, gsl_vector * beta,
          gsl_eigen_gen_workspace * w)

     This function computes the eigenvalues of the real generalized
     nonsymmetric matrix pair (‘A’, ‘B’), and stores them as pairs in
     (*note alpha: 49a, ‘beta’), where *note alpha: 49a. is complex and
     ‘beta’ is real.  If \beta_i is non-zero, then \lambda = \alpha_i /
     \beta_i is an eigenvalue.  Likewise, if \alpha_i is non-zero, then
     \mu = \beta_i / \alpha_i is an eigenvalue of the alternate problem
     \mu A y = B y.  The elements of ‘beta’ are normalized to be
     non-negative.

     If S is desired, it is stored in ‘A’ on output.  If T is desired,
     it is stored in ‘B’ on output.  The ordering of eigenvalues in
     (*note alpha: 49a, ‘beta’) follows the ordering of the diagonal
     blocks in the Schur forms S and T. In rare cases, this function may
     fail to find all eigenvalues.  If this occurs, an error code is
     returned.

 -- Function: int gsl_eigen_gen_QZ (gsl_matrix * A, gsl_matrix * B,
          gsl_vector_complex * alpha, gsl_vector * beta, gsl_matrix * Q,
          gsl_matrix * Z, gsl_eigen_gen_workspace * w)

     This function is identical to *note gsl_eigen_gen(): 5d5. except
     that it also computes the left and right Schur vectors and stores
     them into ‘Q’ and ‘Z’ respectively.

 -- Type: gsl_eigen_genv_workspace

     This workspace contains internal parameters used for solving
     generalized eigenvalue and eigenvector problems.

 -- Function: gsl_eigen_genv_workspace * gsl_eigen_genv_alloc (const
          size_t n)

     This function allocates a workspace for computing eigenvalues and
     eigenvectors of ‘n’-by-‘n’ real generalized nonsymmetric
     eigensystems.  The size of the workspace is O(7n).

 -- Function: void gsl_eigen_genv_free (gsl_eigen_genv_workspace * w)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: int gsl_eigen_genv (gsl_matrix * A, gsl_matrix * B,
          gsl_vector_complex * alpha, gsl_vector * beta,
          gsl_matrix_complex * evec, gsl_eigen_genv_workspace * w)

     This function computes eigenvalues and right eigenvectors of the
     ‘n’-by-‘n’ real generalized nonsymmetric matrix pair (‘A’, ‘B’).
     The eigenvalues are stored in (*note alpha: 49a, ‘beta’) and the
     eigenvectors are stored in ‘evec’.  It first calls *note
     gsl_eigen_gen(): 5d5. to compute the eigenvalues, Schur forms, and
     Schur vectors.  Then it finds eigenvectors of the Schur forms and
     backtransforms them using the Schur vectors.  The Schur vectors are
     destroyed in the process, but can be saved by using *note
     gsl_eigen_genv_QZ(): 5db.  The computed eigenvectors are normalized
     to have unit magnitude.  On output, (‘A’, ‘B’) contains the
     generalized Schur form (S, T). If *note gsl_eigen_gen(): 5d5.
     fails, no eigenvectors are computed, and an error code is returned.

 -- Function: int gsl_eigen_genv_QZ (gsl_matrix * A, gsl_matrix * B,
          gsl_vector_complex * alpha, gsl_vector * beta,
          gsl_matrix_complex * evec, gsl_matrix * Q, gsl_matrix * Z,
          gsl_eigen_genv_workspace * w)

     This function is identical to *note gsl_eigen_genv(): 5da. except
     that it also computes the left and right Schur vectors and stores
     them into ‘Q’ and ‘Z’ respectively.


File: gsl-ref.info,  Node: Sorting Eigenvalues and Eigenvectors,  Next: Examples<10>,  Prev: Real Generalized Nonsymmetric Eigensystems,  Up: Eigensystems

15.7 Sorting Eigenvalues and Eigenvectors
=========================================

 -- Function: int gsl_eigen_symmv_sort (gsl_vector * eval, gsl_matrix
          * evec, gsl_eigen_sort_t sort_type)

     This function simultaneously sorts the eigenvalues stored in the
     vector ‘eval’ and the corresponding real eigenvectors stored in the
     columns of the matrix ‘evec’ into ascending or descending order
     according to the value of the parameter ‘sort_type’,

      -- Type: gsl_eigen_sort_t

          ‘GSL_EIGEN_SORT_VAL_ASC’              ascending order in numerical value
                                                
                                                
          ‘GSL_EIGEN_SORT_VAL_DESC’             descending order in numerical value
                                                
                                                
          ‘GSL_EIGEN_SORT_ABS_ASC’              ascending order in magnitude
                                                
                                                
          ‘GSL_EIGEN_SORT_ABS_DESC’             descending order in magnitude
                                                

 -- Function: int gsl_eigen_hermv_sort (gsl_vector * eval,
          gsl_matrix_complex * evec, gsl_eigen_sort_t sort_type)

     This function simultaneously sorts the eigenvalues stored in the
     vector ‘eval’ and the corresponding complex eigenvectors stored in
     the columns of the matrix ‘evec’ into ascending or descending order
     according to the value of the parameter ‘sort_type’ as shown above.

 -- Function: int gsl_eigen_nonsymmv_sort (gsl_vector_complex * eval,
          gsl_matrix_complex * evec, gsl_eigen_sort_t sort_type)

     This function simultaneously sorts the eigenvalues stored in the
     vector ‘eval’ and the corresponding complex eigenvectors stored in
     the columns of the matrix ‘evec’ into ascending or descending order
     according to the value of the parameter ‘sort_type’ as shown above.
     Only ‘GSL_EIGEN_SORT_ABS_ASC’ and ‘GSL_EIGEN_SORT_ABS_DESC’ are
     supported due to the eigenvalues being complex.

 -- Function: int gsl_eigen_gensymmv_sort (gsl_vector * eval, gsl_matrix
          * evec, gsl_eigen_sort_t sort_type)

     This function simultaneously sorts the eigenvalues stored in the
     vector ‘eval’ and the corresponding real eigenvectors stored in the
     columns of the matrix ‘evec’ into ascending or descending order
     according to the value of the parameter ‘sort_type’ as shown above.

 -- Function: int gsl_eigen_genhermv_sort (gsl_vector * eval,
          gsl_matrix_complex * evec, gsl_eigen_sort_t sort_type)

     This function simultaneously sorts the eigenvalues stored in the
     vector ‘eval’ and the corresponding complex eigenvectors stored in
     the columns of the matrix ‘evec’ into ascending or descending order
     according to the value of the parameter ‘sort_type’ as shown above.

 -- Function: int gsl_eigen_genv_sort (gsl_vector_complex * alpha,
          gsl_vector * beta, gsl_matrix_complex * evec,
          gsl_eigen_sort_t sort_type)

     This function simultaneously sorts the eigenvalues stored in the
     vectors (*note alpha: 49a, ‘beta’) and the corresponding complex
     eigenvectors stored in the columns of the matrix ‘evec’ into
     ascending or descending order according to the value of the
     parameter ‘sort_type’ as shown above.  Only
     ‘GSL_EIGEN_SORT_ABS_ASC’ and ‘GSL_EIGEN_SORT_ABS_DESC’ are
     supported due to the eigenvalues being complex.


File: gsl-ref.info,  Node: Examples<10>,  Next: References and Further Reading<10>,  Prev: Sorting Eigenvalues and Eigenvectors,  Up: Eigensystems

15.8 Examples
=============

The following program computes the eigenvalues and eigenvectors of the
4-th order Hilbert matrix, H(i,j) = 1/(i + j + 1).

     #include <stdio.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_eigen.h>

     int
     main (void)
     {
       double data[] = { 1.0  , 1/2.0, 1/3.0, 1/4.0,
                         1/2.0, 1/3.0, 1/4.0, 1/5.0,
                         1/3.0, 1/4.0, 1/5.0, 1/6.0,
                         1/4.0, 1/5.0, 1/6.0, 1/7.0 };

       gsl_matrix_view m
         = gsl_matrix_view_array (data, 4, 4);

       gsl_vector *eval = gsl_vector_alloc (4);
       gsl_matrix *evec = gsl_matrix_alloc (4, 4);

       gsl_eigen_symmv_workspace * w =
         gsl_eigen_symmv_alloc (4);

       gsl_eigen_symmv (&m.matrix, eval, evec, w);

       gsl_eigen_symmv_free (w);

       gsl_eigen_symmv_sort (eval, evec,
                             GSL_EIGEN_SORT_ABS_ASC);

       {
         int i;

         for (i = 0; i < 4; i++)
           {
             double eval_i
                = gsl_vector_get (eval, i);
             gsl_vector_view evec_i
                = gsl_matrix_column (evec, i);

             printf ("eigenvalue = %g\n", eval_i);
             printf ("eigenvector = \n");
             gsl_vector_fprintf (stdout,
                                 &evec_i.vector, "%g");
           }
       }

       gsl_vector_free (eval);
       gsl_matrix_free (evec);

       return 0;
     }

Here is the beginning of the output from the program:

     $ ./a.out
     eigenvalue = 9.67023e-05
     eigenvector =
     -0.0291933
     0.328712
     -0.791411
     0.514553
     ...

This can be compared with the corresponding output from GNU octave:

     octave> [v,d] = eig(hilb(4));
     octave> diag(d)
     ans =

        9.6702e-05
        6.7383e-03
        1.6914e-01
        1.5002e+00

     octave> v
     v =

        0.029193   0.179186  -0.582076   0.792608
       -0.328712  -0.741918   0.370502   0.451923
        0.791411   0.100228   0.509579   0.322416
       -0.514553   0.638283   0.514048   0.252161

Note that the eigenvectors can differ by a change of sign, since the
sign of an eigenvector is arbitrary.

The following program illustrates the use of the nonsymmetric
eigensolver, by computing the eigenvalues and eigenvectors of the
Vandermonde matrix V(x;i,j) = x_i^{n - j} with x = (-1,-2,3,4).

     #include <stdio.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_eigen.h>

     int
     main (void)
     {
       double data[] = { -1.0, 1.0, -1.0, 1.0,
                         -8.0, 4.0, -2.0, 1.0,
                         27.0, 9.0, 3.0, 1.0,
                         64.0, 16.0, 4.0, 1.0 };

       gsl_matrix_view m
         = gsl_matrix_view_array (data, 4, 4);

       gsl_vector_complex *eval = gsl_vector_complex_alloc (4);
       gsl_matrix_complex *evec = gsl_matrix_complex_alloc (4, 4);

       gsl_eigen_nonsymmv_workspace * w =
         gsl_eigen_nonsymmv_alloc (4);

       gsl_eigen_nonsymmv (&m.matrix, eval, evec, w);

       gsl_eigen_nonsymmv_free (w);

       gsl_eigen_nonsymmv_sort (eval, evec,
                                GSL_EIGEN_SORT_ABS_DESC);

       {
         int i, j;

         for (i = 0; i < 4; i++)
           {
             gsl_complex eval_i
                = gsl_vector_complex_get (eval, i);
             gsl_vector_complex_view evec_i
                = gsl_matrix_complex_column (evec, i);

             printf ("eigenvalue = %g + %gi\n",
                     GSL_REAL(eval_i), GSL_IMAG(eval_i));
             printf ("eigenvector = \n");
             for (j = 0; j < 4; ++j)
               {
                 gsl_complex z =
                   gsl_vector_complex_get(&evec_i.vector, j);
                 printf("%g + %gi\n", GSL_REAL(z), GSL_IMAG(z));
               }
           }
       }

       gsl_vector_complex_free(eval);
       gsl_matrix_complex_free(evec);

       return 0;
     }

Here is the beginning of the output from the program:

     $ ./a.out
     eigenvalue = -6.41391 + 0i
     eigenvector =
     -0.0998822 + 0i
     -0.111251 + 0i
     0.292501 + 0i
     0.944505 + 0i
     eigenvalue = 5.54555 + 3.08545i
     eigenvector =
     -0.043487 + -0.0076308i
     0.0642377 + -0.142127i
     -0.515253 + 0.0405118i
     -0.840592 + -0.00148565i
     ...

This can be compared with the corresponding output from GNU octave:

     octave> [v,d] = eig(vander([-1 -2 3 4]));
     octave> diag(d)
     ans =

       -6.4139 + 0.0000i
        5.5456 + 3.0854i
        5.5456 - 3.0854i
        2.3228 + 0.0000i

     octave> v
     v =

      Columns 1 through 3:

       -0.09988 + 0.00000i  -0.04350 - 0.00755i  -0.04350 + 0.00755i
       -0.11125 + 0.00000i   0.06399 - 0.14224i   0.06399 + 0.14224i
        0.29250 + 0.00000i  -0.51518 + 0.04142i  -0.51518 - 0.04142i
        0.94451 + 0.00000i  -0.84059 + 0.00000i  -0.84059 - 0.00000i

      Column 4:

       -0.14493 + 0.00000i
        0.35660 + 0.00000i
        0.91937 + 0.00000i
        0.08118 + 0.00000i

Note that the eigenvectors corresponding to the eigenvalue 5.54555 +
3.08545i differ by the multiplicative constant 0.9999984 + 0.0017674i
which is an arbitrary phase factor of magnitude 1.


File: gsl-ref.info,  Node: References and Further Reading<10>,  Prev: Examples<10>,  Up: Eigensystems

15.9 References and Further Reading
===================================

Further information on the algorithms described in this section can be
found in the following book,

   * G. H. Golub, C. F. Van Loan, “Matrix Computations” (3rd Ed, 1996),
     Johns Hopkins University Press, ISBN 0-8018-5414-8.

Further information on the generalized eigensystems QZ algorithm can be
found in this paper,

   * C. Moler, G. Stewart, “An Algorithm for Generalized Matrix
     Eigenvalue Problems”, SIAM J. Numer.  Anal., Vol 10, No 2, 1973.

Eigensystem routines for very large matrices can be found in the Fortran
library LAPACK. The LAPACK library is described in,

   * LAPACK Users’ Guide (Third Edition, 1999), Published by SIAM, ISBN
     0-89871-447-8.

The LAPACK source code can be found at the website
‘http://www.netlib.org/lapack’ along with an online copy of the users
guide.


File: gsl-ref.info,  Node: Fast Fourier Transforms FFTs,  Next: Numerical Integration,  Prev: Eigensystems,  Up: Top

16 Fast Fourier Transforms (FFTs)
*********************************

This chapter describes functions for performing Fast Fourier Transforms
(FFTs).  The library includes radix-2 routines (for lengths which are a
power of two) and mixed-radix routines (which work for any length).  For
efficiency there are separate versions of the routines for real data and
for complex data.  The mixed-radix routines are a reimplementation of
the FFTPACK library of Paul Swarztrauber.  Fortran code for FFTPACK is
available on Netlib (FFTPACK also includes some routines for sine and
cosine transforms but these are currently not available in GSL). For
details and derivations of the underlying algorithms consult the
document “GSL FFT Algorithms” (see *note References and Further Reading:
5e8.)

* Menu:

* Mathematical Definitions::
* Overview of complex data FFTs::
* Radix-2 FFT routines for complex data::
* Mixed-radix FFT routines for complex data::
* Overview of real data FFTs::
* Radix-2 FFT routines for real data::
* Mixed-radix FFT routines for real data::
* References and Further Reading: References and Further Reading<11>.


File: gsl-ref.info,  Node: Mathematical Definitions,  Next: Overview of complex data FFTs,  Up: Fast Fourier Transforms FFTs

16.1 Mathematical Definitions
=============================

Fast Fourier Transforms are efficient algorithms for calculating the
discrete Fourier transform (DFT),

     x_j = \sum_{k=0}^{n-1} z_k \exp(-2 \pi i j k / n)

The DFT usually arises as an approximation to the continuous Fourier
transform when functions are sampled at discrete intervals in space or
time.  The naive evaluation of the discrete Fourier transform is a
matrix-vector multiplication W\vec{z}.  A general matrix-vector
multiplication takes O(n^2) operations for n data-points.  Fast Fourier
transform algorithms use a divide-and-conquer strategy to factorize the
matrix W into smaller sub-matrices, corresponding to the integer factors
of the length n.  If n can be factorized into a product of integers f_1
f_2 \ldots f_m then the DFT can be computed in O(n \sum f_i) operations.
For a radix-2 FFT this gives an operation count of O(n \log_2 n).

All the FFT functions offer three types of transform: forwards, inverse
and backwards, based on the same mathematical definitions.  The
definition of the `forward Fourier transform', x = \hbox{FFT}(z), is,

     x_j = \sum_{k=0}^{n-1} z_k \exp(-2 \pi i j k / n)

and the definition of the `inverse Fourier transform', x =
\hbox{IFFT}(z), is,

     z_j = {1 \over n} \sum_{k=0}^{n-1} x_k \exp(2 \pi i j k / n).

The factor of 1/n makes this a true inverse.  For example, a call to
*note gsl_fft_complex_forward(): 5ea. followed by a call to *note
gsl_fft_complex_inverse(): 5eb. should return the original data (within
numerical errors).

In general there are two possible choices for the sign of the
exponential in the transform/ inverse-transform pair.  GSL follows the
same convention as FFTPACK, using a negative exponential for the forward
transform.  The advantage of this convention is that the inverse
transform recreates the original function with simple Fourier synthesis.
Numerical Recipes uses the opposite convention, a positive exponential
in the forward transform.

The `backwards FFT' is simply our terminology for an unscaled version of
the inverse FFT,

     z^{backwards}_j = \sum_{k=0}^{n-1} x_k \exp(2 \pi i j k / n)

When the overall scale of the result is unimportant it is often
convenient to use the backwards FFT instead of the inverse to save
unnecessary divisions.


File: gsl-ref.info,  Node: Overview of complex data FFTs,  Next: Radix-2 FFT routines for complex data,  Prev: Mathematical Definitions,  Up: Fast Fourier Transforms FFTs

16.2 Overview of complex data FFTs
==================================

The inputs and outputs for the complex FFT routines are `packed arrays'
of floating point numbers.  In a packed array the real and imaginary
parts of each complex number are placed in alternate neighboring
elements.  For example, the following definition of a packed array of
length 6:

     double x[3*2];
     gsl_complex_packed_array data = x;

can be used to hold an array of three complex numbers, ‘z[3]’, in the
following way:

     data[0] = Re(z[0])
     data[1] = Im(z[0])
     data[2] = Re(z[1])
     data[3] = Im(z[1])
     data[4] = Re(z[2])
     data[5] = Im(z[2])

The array indices for the data have the same ordering as those in the
definition of the DFT—i.e.  there are no index transformations or
permutations of the data.

A `stride' parameter allows the user to perform transforms on the
elements ‘z[stride*i]’ instead of ‘z[i]’.  A stride greater than 1 can
be used to take an in-place FFT of the column of a matrix.  A stride of
1 accesses the array without any additional spacing between elements.

To perform an FFT on a vector argument, such as ‘gsl_vector_complex *
v’, use the following definitions (or their equivalents) when calling
the functions described in this chapter:

     gsl_complex_packed_array data = v->data;
     size_t stride = v->stride;
     size_t n = v->size;

For physical applications it is important to remember that the index
appearing in the DFT does not correspond directly to a physical
frequency.  If the time-step of the DFT is \Delta then the
frequency-domain includes both positive and negative frequencies,
ranging from -1/(2\Delta) through 0 to +1/(2\Delta).  The positive
frequencies are stored from the beginning of the array up to the middle,
and the negative frequencies are stored backwards from the end of the
array.

Here is a table which shows the layout of the array ‘data’, and the
correspondence between the time-domain data z, and the frequency-domain
data x:

     index    z               x = FFT(z)

     0        z(t = 0)        x(f = 0)
     1        z(t = 1)        x(f = 1/(n Delta))
     2        z(t = 2)        x(f = 2/(n Delta))
     .        ........        ..................
     n/2      z(t = n/2)      x(f = +1/(2 Delta),
                                    -1/(2 Delta))
     .        ........        ..................
     n-3      z(t = n-3)      x(f = -3/(n Delta))
     n-2      z(t = n-2)      x(f = -2/(n Delta))
     n-1      z(t = n-1)      x(f = -1/(n Delta))

When n is even the location n/2 contains the most positive and negative
frequencies (+1/(2 \Delta), -1/(2 \Delta)) which are equivalent.  If n
is odd then general structure of the table above still applies, but n/2
does not appear.


File: gsl-ref.info,  Node: Radix-2 FFT routines for complex data,  Next: Mixed-radix FFT routines for complex data,  Prev: Overview of complex data FFTs,  Up: Fast Fourier Transforms FFTs

16.3 Radix-2 FFT routines for complex data
==========================================

The radix-2 algorithms described in this section are simple and compact,
although not necessarily the most efficient.  They use the Cooley-Tukey
algorithm to compute in-place complex FFTs for lengths which are a power
of 2—no additional storage is required.  The corresponding self-sorting
mixed-radix routines offer better performance at the expense of
requiring additional working space.

All the functions described in this section are declared in the header
file ‘gsl_fft_complex.h’.

 -- Function: int gsl_fft_complex_radix2_forward
          (gsl_complex_packed_array data, size_t stride, size_t n)
 -- Function: int gsl_fft_complex_radix2_transform
          (gsl_complex_packed_array data, size_t stride, size_t n,
          gsl_fft_direction sign)
 -- Function: int gsl_fft_complex_radix2_backward
          (gsl_complex_packed_array data, size_t stride, size_t n)
 -- Function: int gsl_fft_complex_radix2_inverse
          (gsl_complex_packed_array data, size_t stride, size_t n)

     These functions compute forward, backward and inverse FFTs of
     length ‘n’ with stride ‘stride’, on the packed complex array ‘data’
     using an in-place radix-2 decimation-in-time algorithm.  The length
     of the transform ‘n’ is restricted to powers of two.  For the
     ‘transform’ version of the function the ‘sign’ argument can be
     either ‘forward’ (-1) or ‘backward’ (+1).

     The functions return a value of ‘GSL_SUCCESS’ if no errors were
     detected, or *note GSL_EDOM: 28. if the length of the data ‘n’ is
     not a power of two.

 -- Function: int gsl_fft_complex_radix2_dif_forward
          (gsl_complex_packed_array data, size_t stride, size_t n)
 -- Function: int gsl_fft_complex_radix2_dif_transform
          (gsl_complex_packed_array data, size_t stride, size_t n,
          gsl_fft_direction sign)
 -- Function: int gsl_fft_complex_radix2_dif_backward
          (gsl_complex_packed_array data, size_t stride, size_t n)
 -- Function: int gsl_fft_complex_radix2_dif_inverse
          (gsl_complex_packed_array data, size_t stride, size_t n)

     These are decimation-in-frequency versions of the radix-2 FFT
     functions.

Here is an example program which computes the FFT of a short pulse in a
sample of length 128.  To make the resulting Fourier transform real the
pulse is defined for equal positive and negative times (-10 \dots 10),
where the negative times wrap around the end of the array.

     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_fft_complex.h>

     #define REAL(z,i) ((z)[2*(i)])
     #define IMAG(z,i) ((z)[2*(i)+1])

     int
     main (void)
     {
       int i; double data[2*128];

       for (i = 0; i < 128; i++)
         {
            REAL(data,i) = 0.0; IMAG(data,i) = 0.0;
         }

       REAL(data,0) = 1.0;

       for (i = 1; i <= 10; i++)
         {
            REAL(data,i) = REAL(data,128-i) = 1.0;
         }

       for (i = 0; i < 128; i++)
         {
           printf ("%d %e %e\n", i,
                   REAL(data,i), IMAG(data,i));
         }
       printf ("\n\n");

       gsl_fft_complex_radix2_forward (data, 1, 128);

       for (i = 0; i < 128; i++)
         {
           printf ("%d %e %e\n", i,
                   REAL(data,i)/sqrt(128),
                   IMAG(data,i)/sqrt(128));
         }

       return 0;
     }

Note that we have assumed that the program is using the default error
handler (which calls ‘abort()’ for any errors).  If you are not using a
safe error handler you would need to check the return status of *note
gsl_fft_complex_radix2_forward(): 35.

The transformed data is rescaled by 1/\sqrt n so that it fits on the
same plot as the input.  Only the real part is shown, by the choice of
the input data the imaginary part is zero.  Allowing for the wrap-around
of negative times at t=128, and working in units of k/n, the DFT
approximates the continuum Fourier transform, giving a modulated sine
function.

     \int_{-a}^{+a} e^{-2 \pi i k x} dx = {\sin(2\pi k a) \over\pi k}

The output of the example program is plotted in Fig.  %s.

[image src="fft-complex-radix2.png"]

Figure: A pulse and its discrete Fourier transform, output from the
example program.


File: gsl-ref.info,  Node: Mixed-radix FFT routines for complex data,  Next: Overview of real data FFTs,  Prev: Radix-2 FFT routines for complex data,  Up: Fast Fourier Transforms FFTs

16.4 Mixed-radix FFT routines for complex data
==============================================

This section describes mixed-radix FFT algorithms for complex data.  The
mixed-radix functions work for FFTs of any length.  They are a
reimplementation of Paul Swarztrauber’s Fortran FFTPACK library.  The
theory is explained in the review article “Self-sorting Mixed-radix
FFTs” by Clive Temperton.  The routines here use the same indexing
scheme and basic algorithms as FFTPACK.

The mixed-radix algorithm is based on sub-transform modules—highly
optimized small length FFTs which are combined to create larger FFTs.
There are efficient modules for factors of 2, 3, 4, 5, 6 and 7.  The
modules for the composite factors of 4 and 6 are faster than combining
the modules for 2*2 and 2*3.

For factors which are not implemented as modules there is a fall-back to
a general length-n module which uses Singleton’s method for efficiently
computing a DFT. This module is O(n^2), and slower than a dedicated
module would be but works for any length n.  Of course, lengths which
use the general length-n module will still be factorized as much as
possible.  For example, a length of 143 will be factorized into 11*13.
Large prime factors are the worst case scenario, e.g.  as found in
n=2*3*99991, and should be avoided because their O(n^2) scaling will
dominate the run-time (consult the document “GSL FFT Algorithms”
included in the GSL distribution if you encounter this problem).

The mixed-radix initialization function *note
gsl_fft_complex_wavetable_alloc(): 5f7. returns the list of factors
chosen by the library for a given length n.  It can be used to check how
well the length has been factorized, and estimate the run-time.  To a
first approximation the run-time scales as n \sum f_i, where the f_i are
the factors of n.  For programs under user control you may wish to issue
a warning that the transform will be slow when the length is poorly
factorized.  If you frequently encounter data lengths which cannot be
factorized using the existing small-prime modules consult “GSL FFT
Algorithms” for details on adding support for other factors.

All the functions described in this section are declared in the header
file ‘gsl_fft_complex.h’.

 -- Function: gsl_fft_complex_wavetable *
          gsl_fft_complex_wavetable_alloc (size_t n)

     This function prepares a trigonometric lookup table for a complex
     FFT of length ‘n’.  The function returns a pointer to the newly
     allocated *note gsl_fft_complex_wavetable: 5f8. if no errors were
     detected, and a null pointer in the case of error.  The length ‘n’
     is factorized into a product of subtransforms, and the factors and
     their trigonometric coefficients are stored in the wavetable.  The
     trigonometric coefficients are computed using direct calls to ‘sin’
     and ‘cos’, for accuracy.  Recursion relations could be used to
     compute the lookup table faster, but if an application performs
     many FFTs of the same length then this computation is a one-off
     overhead which does not affect the final throughput.

     The wavetable structure can be used repeatedly for any transform of
     the same length.  The table is not modified by calls to any of the
     other FFT functions.  The same wavetable can be used for both
     forward and backward (or inverse) transforms of a given length.

 -- Function: void gsl_fft_complex_wavetable_free
          (gsl_fft_complex_wavetable * wavetable)

     This function frees the memory associated with the wavetable
     ‘wavetable’.  The wavetable can be freed if no further FFTs of the
     same length will be needed.

These functions operate on a *note gsl_fft_complex_wavetable: 5f8.
structure which contains internal parameters for the FFT. It is not
necessary to set any of the components directly but it can sometimes be
useful to examine them.  For example, the chosen factorization of the
FFT length is given and can be used to provide an estimate of the
run-time or numerical error.  The wavetable structure is declared in the
header file ‘gsl_fft_complex.h’.

 -- Type: gsl_fft_complex_wavetable

     This is a structure that holds the factorization and trigonometric
     lookup tables for the mixed radix fft algorithm.  It has the
     following components:

     ‘size_t n’                            This is the number of complex data points
                                           
                                           
     ‘size_t nf’                           This is the number of factors that the length ‘n’ was decomposed into.
                                           
                                           
     ‘size_t factor[64]’                   This is the array of factors.  Only the first ‘nf’ elements are used.
                                           
                                           
     ‘gsl_complex * trig’                  This is a pointer to a preallocated trigonometric lookup table of ‘n’ complex elements.
                                           
                                           
     ‘gsl_complex * twiddle[64]’           This is an array of pointers into ‘trig’, giving the twiddle factors for each pass.
                                           

 -- Type: gsl_fft_complex_workspace

     The mixed radix algorithms require additional working space to hold
     the intermediate steps of the transform.

 -- Function: gsl_fft_complex_workspace *
          gsl_fft_complex_workspace_alloc (size_t n)

     This function allocates a workspace for a complex transform of
     length ‘n’.

 -- Function: void gsl_fft_complex_workspace_free
          (gsl_fft_complex_workspace * workspace)

     This function frees the memory associated with the workspace
     ‘workspace’.  The workspace can be freed if no further FFTs of the
     same length will be needed.

The following functions compute the transform,

 -- Function: int gsl_fft_complex_forward
          (gsl_complex_packed_array data, size_t stride, size_t n, const
          gsl_fft_complex_wavetable * wavetable,
          gsl_fft_complex_workspace * work)
 -- Function: int gsl_fft_complex_transform
          (gsl_complex_packed_array data, size_t stride, size_t n, const
          gsl_fft_complex_wavetable * wavetable,
          gsl_fft_complex_workspace * work, gsl_fft_direction sign)
 -- Function: int gsl_fft_complex_backward
          (gsl_complex_packed_array data, size_t stride, size_t n, const
          gsl_fft_complex_wavetable * wavetable,
          gsl_fft_complex_workspace * work)
 -- Function: int gsl_fft_complex_inverse
          (gsl_complex_packed_array data, size_t stride, size_t n, const
          gsl_fft_complex_wavetable * wavetable,
          gsl_fft_complex_workspace * work)

     These functions compute forward, backward and inverse FFTs of
     length ‘n’ with stride ‘stride’, on the packed complex array
     ‘data’, using a mixed radix decimation-in-frequency algorithm.
     There is no restriction on the length ‘n’.  Efficient modules are
     provided for subtransforms of length 2, 3, 4, 5, 6 and 7.  Any
     remaining factors are computed with a slow, O(n^2), general-n
     module.  The caller must supply a ‘wavetable’ containing the
     trigonometric lookup tables and a workspace ‘work’.  For the
     ‘transform’ version of the function the ‘sign’ argument can be
     either ‘forward’ (-1) or ‘backward’ (+1).

     The functions return a value of ‘0’ if no errors were detected.
     The following ‘gsl_errno’ conditions are defined for these
     functions:

     *note GSL_EDOM: 28.                     The length of the data ‘n’ is not a positive integer (i.e.  ‘n’ is zero).
                                             
                                             
     *note GSL_EINVAL: 2b.                   The length of the data ‘n’ and the length used to compute the given ‘wavetable’ do not match.
                                             

Here is an example program which computes the FFT of a short pulse in a
sample of length 630 (=2*3*3*5*7) using the mixed-radix algorithm.

     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_fft_complex.h>

     #define REAL(z,i) ((z)[2*(i)])
     #define IMAG(z,i) ((z)[2*(i)+1])

     int
     main (void)
     {
       int i;
       const int n = 630;
       double data[2*n];

       gsl_fft_complex_wavetable * wavetable;
       gsl_fft_complex_workspace * workspace;

       for (i = 0; i < n; i++)
         {
           REAL(data,i) = 0.0;
           IMAG(data,i) = 0.0;
         }

       data[0] = 1.0;

       for (i = 1; i <= 10; i++)
         {
           REAL(data,i) = REAL(data,n-i) = 1.0;
         }

       for (i = 0; i < n; i++)
         {
           printf ("%d: %e %e\n", i, REAL(data,i),
                                     IMAG(data,i));
         }
       printf ("\n");

       wavetable = gsl_fft_complex_wavetable_alloc (n);
       workspace = gsl_fft_complex_workspace_alloc (n);

       for (i = 0; i < (int) wavetable->nf; i++)
         {
            printf ("# factor %d: %zu\n", i,
                    wavetable->factor[i]);
         }

       gsl_fft_complex_forward (data, 1, n,
                                wavetable, workspace);

       for (i = 0; i < n; i++)
         {
           printf ("%d: %e %e\n", i, REAL(data,i),
                                     IMAG(data,i));
         }

       gsl_fft_complex_wavetable_free (wavetable);
       gsl_fft_complex_workspace_free (workspace);
       return 0;
     }

Note that we have assumed that the program is using the default ‘gsl’
error handler (which calls ‘abort()’ for any errors).  If you are not
using a safe error handler you would need to check the return status of
all the ‘gsl’ routines.


File: gsl-ref.info,  Node: Overview of real data FFTs,  Next: Radix-2 FFT routines for real data,  Prev: Mixed-radix FFT routines for complex data,  Up: Fast Fourier Transforms FFTs

16.5 Overview of real data FFTs
===============================

The functions for real data are similar to those for complex data.
However, there is an important difference between forward and inverse
transforms.  The Fourier transform of a real sequence is not real.  It
is a complex sequence with a special symmetry:

     z_k = z_{n-k}^*

A sequence with this symmetry is called `conjugate-complex' or
`half-complex'.  This different structure requires different storage
layouts for the forward transform (from real to half-complex) and
inverse transform (from half-complex back to real).  As a consequence
the routines are divided into two sets: functions in ‘gsl_fft_real’
which operate on real sequences and functions in ‘gsl_fft_halfcomplex’
which operate on half-complex sequences.

Functions in ‘gsl_fft_real’ compute the frequency coefficients of a real
sequence.  The half-complex coefficients c of a real sequence x are
given by Fourier analysis,

     c_k = \sum_{j=0}^{n-1} x_j \exp(-2 \pi i j k /n)

Functions in ‘gsl_fft_halfcomplex’ compute inverse or backwards
transforms.  They reconstruct real sequences by Fourier synthesis from
their half-complex frequency coefficients, c,

     x_j = {1 \over n} \sum_{k=0}^{n-1} c_k \exp(2 \pi i j k /n)

The symmetry of the half-complex sequence implies that only half of the
complex numbers in the output need to be stored.  The remaining half can
be reconstructed using the half-complex symmetry condition.  This works
for all lengths, even and odd—when the length is even the middle value
where k=n/2 is also real.  Thus only ‘n’ real numbers are required to
store the half-complex sequence, and the transform of a real sequence
can be stored in the same size array as the original data.

The precise storage arrangements depend on the algorithm, and are
different for radix-2 and mixed-radix routines.  The radix-2 function
operates in-place, which constrains the locations where each element can
be stored.  The restriction forces real and imaginary parts to be stored
far apart.  The mixed-radix algorithm does not have this restriction,
and it stores the real and imaginary parts of a given term in
neighboring locations (which is desirable for better locality of memory
accesses).


File: gsl-ref.info,  Node: Radix-2 FFT routines for real data,  Next: Mixed-radix FFT routines for real data,  Prev: Overview of real data FFTs,  Up: Fast Fourier Transforms FFTs

16.6 Radix-2 FFT routines for real data
=======================================

This section describes radix-2 FFT algorithms for real data.  They use
the Cooley-Tukey algorithm to compute in-place FFTs for lengths which
are a power of 2.

The radix-2 FFT functions for real data are declared in the header files
‘gsl_fft_real.h’

 -- Function: int gsl_fft_real_radix2_transform (double data[],
          size_t stride, size_t n)

     This function computes an in-place radix-2 FFT of length ‘n’ and
     stride ‘stride’ on the real array ‘data’.  The output is a
     half-complex sequence, which is stored in-place.  The arrangement
     of the half-complex terms uses the following scheme: for k < n/2
     the real part of the k-th term is stored in location k, and the
     corresponding imaginary part is stored in location n-k.  Terms with
     k > n/2 can be reconstructed using the symmetry z_k = z^*_{n-k}.
     The terms for k=0 and k=n/2 are both purely real, and count as a
     special case.  Their real parts are stored in locations 0 and n/2
     respectively, while their imaginary parts which are zero are not
     stored.

     The following table shows the correspondence between the output
     ‘data’ and the equivalent results obtained by considering the input
     data as a complex sequence with zero imaginary part (assuming
     ‘stride’ = 1}):

          complex[0].real    =    data[0]
          complex[0].imag    =    0
          complex[1].real    =    data[1]
          complex[1].imag    =    data[n-1]
          ...............         ................
          complex[k].real    =    data[k]
          complex[k].imag    =    data[n-k]
          ...............         ................
          complex[n/2].real  =    data[n/2]
          complex[n/2].imag  =    0
          ...............         ................
          complex[k'].real   =    data[k]        k' = n - k
          complex[k'].imag   =   -data[n-k]
          ...............         ................
          complex[n-1].real  =    data[1]
          complex[n-1].imag  =   -data[n-1]

     Note that the output data can be converted into the full complex
     sequence using the function *note
     gsl_fft_halfcomplex_radix2_unpack(): 602. described below.

The radix-2 FFT functions for halfcomplex data are declared in the
header file ‘gsl_fft_halfcomplex.h’.

 -- Function: int gsl_fft_halfcomplex_radix2_inverse (double data[],
          size_t stride, size_t n)
 -- Function: int gsl_fft_halfcomplex_radix2_backward (double data[],
          size_t stride, size_t n)

     These functions compute the inverse or backwards in-place radix-2
     FFT of length ‘n’ and stride ‘stride’ on the half-complex sequence
     ‘data’ stored according the output scheme used by
     ‘gsl_fft_real_radix2()’.  The result is a real array stored in
     natural order.

 -- Function: int gsl_fft_halfcomplex_radix2_unpack (const
          double halfcomplex_coefficient[],
          gsl_complex_packed_array complex_coefficient, size_t stride,
          size_t n)

     This function converts ‘halfcomplex_coefficient’, an array of
     half-complex coefficients as returned by *note
     gsl_fft_real_radix2_transform(): 601, into an ordinary complex
     array, ‘complex_coefficient’.  It fills in the complex array using
     the symmetry z_k = z_{n-k}^* to reconstruct the redundant elements.
     The algorithm for the conversion is:

          complex_coefficient[0].real = halfcomplex_coefficient[0];
          complex_coefficient[0].imag = 0.0;

          for (i = 1; i < n - i; i++)
            {
              double hc_real = halfcomplex_coefficient[i*stride];
              double hc_imag = halfcomplex_coefficient[(n-i)*stride];
              complex_coefficient[i*stride].real = hc_real;
              complex_coefficient[i*stride].imag = hc_imag;
              complex_coefficient[(n - i)*stride].real = hc_real;
              complex_coefficient[(n - i)*stride].imag = -hc_imag;
            }

          if (i == n - i)
            {
              complex_coefficient[i*stride].real = halfcomplex_coefficient[(n - 1)*stride];
              complex_coefficient[i*stride].imag = 0.0;
            }


File: gsl-ref.info,  Node: Mixed-radix FFT routines for real data,  Next: References and Further Reading<11>,  Prev: Radix-2 FFT routines for real data,  Up: Fast Fourier Transforms FFTs

16.7 Mixed-radix FFT routines for real data
===========================================

This section describes mixed-radix FFT algorithms for real data.  The
mixed-radix functions work for FFTs of any length.  They are a
reimplementation of the real-FFT routines in the Fortran FFTPACK library
by Paul Swarztrauber.  The theory behind the algorithm is explained in
the article “Fast Mixed-Radix Real Fourier Transforms” by Clive
Temperton.  The routines here use the same indexing scheme and basic
algorithms as FFTPACK.

The functions use the FFTPACK storage convention for half-complex
sequences.  In this convention the half-complex transform of a real
sequence is stored with frequencies in increasing order, starting at
zero, with the real and imaginary parts of each frequency in neighboring
locations.  When a value is known to be real the imaginary part is not
stored.  The imaginary part of the zero-frequency component is never
stored.  It is known to be zero (since the zero frequency component is
simply the sum of the input data (all real)).  For a sequence of even
length the imaginary part of the frequency n/2 is not stored either,
since the symmetry z_k = z_{n-k}^* implies that this is purely real too.

The storage scheme is best shown by some examples.  The table below
shows the output for an odd-length sequence, n=5.  The two columns give
the correspondence between the 5 values in the half-complex sequence
returned by *note gsl_fft_real_transform(): 606, ‘halfcomplex[]’ and the
values ‘complex[]’ that would be returned if the same real input
sequence were passed to *note gsl_fft_complex_backward(): 5fe. as a
complex sequence (with imaginary parts set to ‘0’):

     complex[0].real  =  halfcomplex[0]
     complex[0].imag  =  0
     complex[1].real  =  halfcomplex[1]
     complex[1].imag  =  halfcomplex[2]
     complex[2].real  =  halfcomplex[3]
     complex[2].imag  =  halfcomplex[4]
     complex[3].real  =  halfcomplex[3]
     complex[3].imag  = -halfcomplex[4]
     complex[4].real  =  halfcomplex[1]
     complex[4].imag  = -halfcomplex[2]

The upper elements of the ‘complex’ array, ‘complex[3]’ and ‘complex[4]’
are filled in using the symmetry condition.  The imaginary part of the
zero-frequency term ‘complex[0].imag’ is known to be zero by the
symmetry.

The next table shows the output for an even-length sequence, n=6.  In
the even case there are two values which are purely real:

     complex[0].real  =  halfcomplex[0]
     complex[0].imag  =  0
     complex[1].real  =  halfcomplex[1]
     complex[1].imag  =  halfcomplex[2]
     complex[2].real  =  halfcomplex[3]
     complex[2].imag  =  halfcomplex[4]
     complex[3].real  =  halfcomplex[5]
     complex[3].imag  =  0
     complex[4].real  =  halfcomplex[3]
     complex[4].imag  = -halfcomplex[4]
     complex[5].real  =  halfcomplex[1]
     complex[5].imag  = -halfcomplex[2]

The upper elements of the ‘complex’ array, ‘complex[4]’ and ‘complex[5]’
are filled in using the symmetry condition.  Both ‘complex[0].imag’ and
‘complex[3].imag’ are known to be zero.

All these functions are declared in the header files ‘gsl_fft_real.h’
and ‘gsl_fft_halfcomplex.h’.

 -- Type: gsl_fft_real_wavetable
 -- Type: gsl_fft_halfcomplex_wavetable

     These data structures contain lookup tables for an FFT of a fixed
     size.

 -- Function: gsl_fft_real_wavetable * gsl_fft_real_wavetable_alloc
          (size_t n)
 -- Function: gsl_fft_halfcomplex_wavetable *
          gsl_fft_halfcomplex_wavetable_alloc (size_t n)

     These functions prepare trigonometric lookup tables for an FFT of
     size n real elements.  The functions return a pointer to the newly
     allocated struct if no errors were detected, and a null pointer in
     the case of error.  The length ‘n’ is factorized into a product of
     subtransforms, and the factors and their trigonometric coefficients
     are stored in the wavetable.  The trigonometric coefficients are
     computed using direct calls to ‘sin’ and ‘cos’, for accuracy.
     Recursion relations could be used to compute the lookup table
     faster, but if an application performs many FFTs of the same length
     then computing the wavetable is a one-off overhead which does not
     affect the final throughput.

     The wavetable structure can be used repeatedly for any transform of
     the same length.  The table is not modified by calls to any of the
     other FFT functions.  The appropriate type of wavetable must be
     used for forward real or inverse half-complex transforms.

 -- Function: void gsl_fft_real_wavetable_free (gsl_fft_real_wavetable
          * wavetable)
 -- Function: void gsl_fft_halfcomplex_wavetable_free
          (gsl_fft_halfcomplex_wavetable * wavetable)

     These functions free the memory associated with the wavetable
     ‘wavetable’.  The wavetable can be freed if no further FFTs of the
     same length will be needed.

The mixed radix algorithms require additional working space to hold the
intermediate steps of the transform,

 -- Type: gsl_fft_real_workspace

     This workspace contains parameters needed to compute a real FFT.

 -- Function: gsl_fft_real_workspace * gsl_fft_real_workspace_alloc
          (size_t n)

     This function allocates a workspace for a real transform of length
     ‘n’.  The same workspace can be used for both forward real and
     inverse halfcomplex transforms.

 -- Function: void gsl_fft_real_workspace_free (gsl_fft_real_workspace
          * workspace)

     This function frees the memory associated with the workspace
     ‘workspace’.  The workspace can be freed if no further FFTs of the
     same length will be needed.

The following functions compute the transforms of real and half-complex
data,

 -- Function: int gsl_fft_real_transform (double data[], size_t stride,
          size_t n, const gsl_fft_real_wavetable * wavetable,
          gsl_fft_real_workspace * work)
 -- Function: int gsl_fft_halfcomplex_transform (double data[],
          size_t stride, size_t n, const gsl_fft_halfcomplex_wavetable
          * wavetable, gsl_fft_real_workspace * work)

     These functions compute the FFT of ‘data’, a real or half-complex
     array of length ‘n’, using a mixed radix decimation-in-frequency
     algorithm.  For *note gsl_fft_real_transform(): 606. ‘data’ is an
     array of time-ordered real data.  For *note
     gsl_fft_halfcomplex_transform(): 610. ‘data’ contains Fourier
     coefficients in the half-complex ordering described above.  There
     is no restriction on the length ‘n’.  Efficient modules are
     provided for subtransforms of length 2, 3, 4 and 5.  Any remaining
     factors are computed with a slow, O(n^2), general-n module.  The
     caller must supply a ‘wavetable’ containing trigonometric lookup
     tables and a workspace ‘work’.

 -- Function: int gsl_fft_real_unpack (const double real_coefficient[],
          gsl_complex_packed_array complex_coefficient, size_t stride,
          size_t n)

     This function converts a single real array, ‘real_coefficient’ into
     an equivalent complex array, ‘complex_coefficient’, (with imaginary
     part set to zero), suitable for ‘gsl_fft_complex’ routines.  The
     algorithm for the conversion is simply:

          for (i = 0; i < n; i++)
            {
              complex_coefficient[i*stride].real = real_coefficient[i*stride];
              complex_coefficient[i*stride].imag = 0.0;
            }

 -- Function: int gsl_fft_halfcomplex_unpack (const
          double halfcomplex_coefficient[],
          gsl_complex_packed_array complex_coefficient, size_t stride,
          size_t n)

     This function converts ‘halfcomplex_coefficient’, an array of
     half-complex coefficients as returned by *note
     gsl_fft_real_transform(): 606, into an ordinary complex array,
     ‘complex_coefficient’.  It fills in the complex array using the
     symmetry z_k = z_{n-k}^* to reconstruct the redundant elements.
     The algorithm for the conversion is:

          complex_coefficient[0].real = halfcomplex_coefficient[0];
          complex_coefficient[0].imag = 0.0;

          for (i = 1; i < n - i; i++)
            {
              double hc_real = halfcomplex_coefficient[(2 * i - 1)*stride];
              double hc_imag = halfcomplex_coefficient[(2 * i)*stride];
              complex_coefficient[i*stride].real = hc_real;
              complex_coefficient[i*stride].imag = hc_imag;
              complex_coefficient[(n - i)*stride].real = hc_real;
              complex_coefficient[(n - i)*stride].imag = -hc_imag;
            }

          if (i == n - i)
            {
              complex_coefficient[i*stride].real = halfcomplex_coefficient[(n - 1)*stride];
              complex_coefficient[i*stride].imag = 0.0;
            }

Here is an example program using *note gsl_fft_real_transform(): 606.
and ‘gsl_fft_halfcomplex_inverse()’.  It generates a real signal in the
shape of a square pulse.  The pulse is Fourier transformed to frequency
space, and all but the lowest ten frequency components are removed from
the array of Fourier coefficients returned by *note
gsl_fft_real_transform(): 606.

The remaining Fourier coefficients are transformed back to the
time-domain, to give a filtered version of the square pulse.  Since
Fourier coefficients are stored using the half-complex symmetry both
positive and negative frequencies are removed and the final filtered
signal is also real.

     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_fft_real.h>
     #include <gsl/gsl_fft_halfcomplex.h>

     int
     main (void)
     {
       int i, n = 100;
       double data[n];

       gsl_fft_real_wavetable * real;
       gsl_fft_halfcomplex_wavetable * hc;
       gsl_fft_real_workspace * work;

       for (i = 0; i < n; i++)
         {
           data[i] = 0.0;
         }

       for (i = n / 3; i < 2 * n / 3; i++)
         {
           data[i] = 1.0;
         }

       for (i = 0; i < n; i++)
         {
           printf ("%d: %e\n", i, data[i]);
         }
       printf ("\n");

       work = gsl_fft_real_workspace_alloc (n);
       real = gsl_fft_real_wavetable_alloc (n);

       gsl_fft_real_transform (data, 1, n,
                               real, work);

       gsl_fft_real_wavetable_free (real);

       for (i = 11; i < n; i++)
         {
           data[i] = 0;
         }

       hc = gsl_fft_halfcomplex_wavetable_alloc (n);

       gsl_fft_halfcomplex_inverse (data, 1, n,
                                    hc, work);
       gsl_fft_halfcomplex_wavetable_free (hc);

       for (i = 0; i < n; i++)
         {
           printf ("%d: %e\n", i, data[i]);
         }

       gsl_fft_real_workspace_free (work);
       return 0;
     }

The program output is shown in Fig.  %s.

[image src="fft-real-mixedradix.png"]

Figure: Low-pass filtered version of a real pulse, output from the
example program.


File: gsl-ref.info,  Node: References and Further Reading<11>,  Prev: Mixed-radix FFT routines for real data,  Up: Fast Fourier Transforms FFTs

16.8 References and Further Reading
===================================

A good starting point for learning more about the FFT is the following
review article,

   * P. Duhamel and M. Vetterli.  Fast Fourier transforms: A tutorial
     review and a state of the art.  Signal Processing, 19:259–299,
     1990.

To find out about the algorithms used in the GSL routines you may want
to consult the document “GSL FFT Algorithms” (it is included in GSL, as
‘doc/fftalgorithms.tex’).  This has general information on FFTs and
explicit derivations of the implementation for each routine.  There are
also references to the relevant literature.  For convenience some of the
more important references are reproduced below.

There are several introductory books on the FFT with example programs,
such as “The Fast Fourier Transform” by Brigham and “DFT/FFT and
Convolution Algorithms” by Burrus and Parks,

   * 
       5. Oran Brigham.  “The Fast Fourier Transform”.  Prentice Hall,
          1974.

   * C. S. Burrus and T. W. Parks.  “DFT/FFT and Convolution
     Algorithms”, Wiley, 1984.

Both these introductory books cover the radix-2 FFT in some detail.  The
mixed-radix algorithm at the heart of the FFTPACK routines is reviewed
in Clive Temperton’s paper,

   * Clive Temperton, Self-sorting mixed-radix fast Fourier transforms,
     Journal of Computational Physics, 52(1):1–23, 1983.

The derivation of FFTs for real-valued data is explained in the
following two articles,

   * Henrik V. Sorenson, Douglas L. Jones, Michael T. Heideman, and C.
     Sidney Burrus.  Real-valued fast Fourier transform algorithms.
     “IEEE Transactions on Acoustics, Speech, and Signal Processing”,
     ASSP-35(6):849–863, 1987.

   * Clive Temperton.  Fast mixed-radix real Fourier transforms.
     “Journal of Computational Physics”, 52:340–350, 1983.

In 1979 the IEEE published a compendium of carefully-reviewed Fortran
FFT programs in “Programs for Digital Signal Processing”.  It is a
useful reference for implementations of many different FFT algorithms,

   * Digital Signal Processing Committee and IEEE Acoustics, Speech, and
     Signal Processing Committee, editors.  Programs for Digital Signal
     Processing.  IEEE Press, 1979.

For large-scale FFT work we recommend the use of the dedicated FFTW
library by Frigo and Johnson.  The FFTW library is self-optimizing—it
automatically tunes itself for each hardware platform in order to
achieve maximum performance.  It is available under the GNU GPL.

   * FFTW Website, ‘http://www.fftw.org/’

The source code for FFTPACK is available from
‘http://www.netlib.org/fftpack/’


File: gsl-ref.info,  Node: Numerical Integration,  Next: Random Number Generation,  Prev: Fast Fourier Transforms FFTs,  Up: Top

17 Numerical Integration
************************

This chapter describes routines for performing numerical integration
(quadrature) of a function in one dimension.  There are routines for
adaptive and non-adaptive integration of general functions, with
specialised routines for specific cases.  These include integration over
infinite and semi-infinite ranges, singular integrals, including
logarithmic singularities, computation of Cauchy principal values and
oscillatory integrals.  The library reimplements the algorithms used in
QUADPACK, a numerical integration package written by Piessens, de
Doncker-Kapenga, Ueberhuber and Kahaner.  Fortran code for QUADPACK is
available on Netlib.  Also included are non-adaptive, fixed-order
Gauss-Legendre integration routines with high precision coefficients, as
well as fixed-order quadrature rules for a variety of weighting
functions from IQPACK.

The functions described in this chapter are declared in the header file
‘gsl_integration.h’.

* Menu:

* Introduction: Introduction<2>.
* QNG non-adaptive Gauss-Kronrod integration::
* QAG adaptive integration::
* QAGS adaptive integration with singularities::
* QAGP adaptive integration with known singular points::
* QAGI adaptive integration on infinite intervals::
* QAWC adaptive integration for Cauchy principal values::
* QAWS adaptive integration for singular functions::
* QAWO adaptive integration for oscillatory functions::
* QAWF adaptive integration for Fourier integrals::
* CQUAD doubly-adaptive integration::
* Romberg integration::
* Gauss-Legendre integration::
* Fixed point quadratures::
* Error codes::
* Examples: Examples<11>.
* References and Further Reading: References and Further Reading<12>.


File: gsl-ref.info,  Node: Introduction<2>,  Next: QNG non-adaptive Gauss-Kronrod integration,  Up: Numerical Integration

17.1 Introduction
=================

Each algorithm computes an approximation to a definite integral of the
form,

     I = \int_a^b f(x) w(x) dx

where w(x) is a weight function (for general integrands w(x) = 1).  The
user provides absolute and relative error bounds (epsabs, epsrel) which
specify the following accuracy requirement,

     |RESULT - I| <= max(epsabs, epsrel |I|)

where RESULT is the numerical approximation obtained by the algorithm.
The algorithms attempt to estimate the absolute error ABSERR = |RESULT -
I| in such a way that the following inequality holds,

     |RESULT - I| <= ABSERR <= max(epsabs, epsrel |I|)

In short, the routines return the first approximation which has an
absolute error smaller than epsabs or a relative error smaller than
epsrel.

Note that this is an `either-or' constraint, not simultaneous.  To
compute to a specified absolute error, set epsrel to zero.  To compute
to a specified relative error, set epsabs to zero.  The routines will
fail to converge if the error bounds are too stringent, but always
return the best approximation obtained up to that stage.

The algorithms in QUADPACK use a naming convention based on the
following letters:

     Q - quadrature routine

     N - non-adaptive integrator
     A - adaptive integrator

     G - general integrand (user-defined)
     W - weight function with integrand

     S - singularities can be more readily integrated
     P - points of special difficulty can be supplied
     I - infinite range of integration
     O - oscillatory weight function, cos or sin
     F - Fourier integral
     C - Cauchy principal value

The algorithms are built on pairs of quadrature rules, a higher order
rule and a lower order rule.  The higher order rule is used to compute
the best approximation to an integral over a small range.  The
difference between the results of the higher order rule and the lower
order rule gives an estimate of the error in the approximation.

* Menu:

* Integrands without weight functions::
* Integrands with weight functions::
* Integrands with singular weight functions::


File: gsl-ref.info,  Node: Integrands without weight functions,  Next: Integrands with weight functions,  Up: Introduction<2>

17.1.1 Integrands without weight functions
------------------------------------------

The algorithms for general functions (without a weight function) are
based on Gauss-Kronrod rules.

A Gauss-Kronrod rule begins with a classical Gaussian quadrature rule of
order m.  This is extended with additional points between each of the
abscissae to give a higher order Kronrod rule of order 2m + 1.  The
Kronrod rule is efficient because it reuses existing function
evaluations from the Gaussian rule.

The higher order Kronrod rule is used as the best approximation to the
integral, and the difference between the two rules is used as an
estimate of the error in the approximation.


File: gsl-ref.info,  Node: Integrands with weight functions,  Next: Integrands with singular weight functions,  Prev: Integrands without weight functions,  Up: Introduction<2>

17.1.2 Integrands with weight functions
---------------------------------------

For integrands with weight functions the algorithms use Clenshaw-Curtis
quadrature rules.

A Clenshaw-Curtis rule begins with an n-th order Chebyshev polynomial
approximation to the integrand.  This polynomial can be integrated
exactly to give an approximation to the integral of the original
function.  The Chebyshev expansion can be extended to higher orders to
improve the approximation and provide an estimate of the error.


File: gsl-ref.info,  Node: Integrands with singular weight functions,  Prev: Integrands with weight functions,  Up: Introduction<2>

17.1.3 Integrands with singular weight functions
------------------------------------------------

The presence of singularities (or other behavior) in the integrand can
cause slow convergence in the Chebyshev approximation.  The modified
Clenshaw-Curtis rules used in QUADPACK separate out several common
weight functions which cause slow convergence.

These weight functions are integrated analytically against the Chebyshev
polynomials to precompute `modified Chebyshev moments'.  Combining the
moments with the Chebyshev approximation to the function gives the
desired integral.  The use of analytic integration for the singular part
of the function allows exact cancellations and substantially improves
the overall convergence behavior of the integration.


File: gsl-ref.info,  Node: QNG non-adaptive Gauss-Kronrod integration,  Next: QAG adaptive integration,  Prev: Introduction<2>,  Up: Numerical Integration

17.2 QNG non-adaptive Gauss-Kronrod integration
===============================================

The QNG algorithm is a non-adaptive procedure which uses fixed
Gauss-Kronrod-Patterson abscissae to sample the integrand at a maximum
of 87 points.  It is provided for fast integration of smooth functions.

 -- Function: int gsl_integration_qng (const gsl_function * f, double a,
          double b, double epsabs, double epsrel, double * result,
          double * abserr, size_t * neval)

     This function applies the Gauss-Kronrod 10-point, 21-point,
     43-point and 87-point integration rules in succession until an
     estimate of the integral of f over (a,b) is achieved within the
     desired absolute and relative error limits, ‘epsabs’ and ‘epsrel’.
     The function returns the final approximation, ‘result’, an estimate
     of the absolute error, ‘abserr’ and the number of function
     evaluations used, ‘neval’.  The Gauss-Kronrod rules are designed in
     such a way that each rule uses all the results of its predecessors,
     in order to minimize the total number of function evaluations.


File: gsl-ref.info,  Node: QAG adaptive integration,  Next: QAGS adaptive integration with singularities,  Prev: QNG non-adaptive Gauss-Kronrod integration,  Up: Numerical Integration

17.3 QAG adaptive integration
=============================

The QAG algorithm is a simple adaptive integration procedure.  The
integration region is divided into subintervals, and on each iteration
the subinterval with the largest estimated error is bisected.  This
reduces the overall error rapidly, as the subintervals become
concentrated around local difficulties in the integrand.  These
subintervals are managed by the following struct,

 -- Type: gsl_integration_workspace

     This workspace handles the memory for the subinterval ranges,
     results and error estimates.

 -- Function: gsl_integration_workspace *
          gsl_integration_workspace_alloc (size_t n)

     This function allocates a workspace sufficient to hold ‘n’ double
     precision intervals, their integration results and error estimates.
     One workspace may be used multiple times as all necessary
     reinitialization is performed automatically by the integration
     routines.

 -- Function: void gsl_integration_workspace_free
          (gsl_integration_workspace * w)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: int gsl_integration_qag (const gsl_function * f, double a,
          double b, double epsabs, double epsrel, size_t limit, int key,
          gsl_integration_workspace * workspace, double * result, double
          * abserr)

     This function applies an integration rule adaptively until an
     estimate of the integral of f over (a,b) is achieved within the
     desired absolute and relative error limits, ‘epsabs’ and ‘epsrel’.
     The function returns the final approximation, ‘result’, and an
     estimate of the absolute error, ‘abserr’.  The integration rule is
     determined by the value of ‘key’, which should be chosen from the
     following symbolic names,

        *  -- Macro: GSL_INTEG_GAUSS15 (key = 1)

        *  -- Macro: GSL_INTEG_GAUSS21 (key = 2)

        *  -- Macro: GSL_INTEG_GAUSS31 (key = 3)

        *  -- Macro: GSL_INTEG_GAUSS41 (key = 4)

        *  -- Macro: GSL_INTEG_GAUSS51 (key = 5)

        *  -- Macro: GSL_INTEG_GAUSS61 (key = 6)

     corresponding to the 15, 21, 31, 41, 51 and 61 point Gauss-Kronrod
     rules.  The higher-order rules give better accuracy for smooth
     functions, while lower-order rules save time when the function
     contains local difficulties, such as discontinuities.

     On each iteration the adaptive integration strategy bisects the
     interval with the largest error estimate.  The subintervals and
     their results are stored in the memory provided by ‘workspace’.
     The maximum number of subintervals is given by ‘limit’, which may
     not exceed the allocated size of the workspace.


File: gsl-ref.info,  Node: QAGS adaptive integration with singularities,  Next: QAGP adaptive integration with known singular points,  Prev: QAG adaptive integration,  Up: Numerical Integration

17.4 QAGS adaptive integration with singularities
=================================================

The presence of an integrable singularity in the integration region
causes an adaptive routine to concentrate new subintervals around the
singularity.  As the subintervals decrease in size the successive
approximations to the integral converge in a limiting fashion.  This
approach to the limit can be accelerated using an extrapolation
procedure.  The QAGS algorithm combines adaptive bisection with the Wynn
epsilon-algorithm to speed up the integration of many types of
integrable singularities.

 -- Function: int gsl_integration_qags (const gsl_function * f,
          double a, double b, double epsabs, double epsrel,
          size_t limit, gsl_integration_workspace * workspace, double
          * result, double * abserr)

     This function applies the Gauss-Kronrod 21-point integration rule
     adaptively until an estimate of the integral of f over (a,b) is
     achieved within the desired absolute and relative error limits,
     ‘epsabs’ and ‘epsrel’.  The results are extrapolated using the
     epsilon-algorithm, which accelerates the convergence of the
     integral in the presence of discontinuities and integrable
     singularities.  The function returns the final approximation from
     the extrapolation, ‘result’, and an estimate of the absolute error,
     ‘abserr’.  The subintervals and their results are stored in the
     memory provided by ‘workspace’.  The maximum number of subintervals
     is given by ‘limit’, which may not exceed the allocated size of the
     workspace.


File: gsl-ref.info,  Node: QAGP adaptive integration with known singular points,  Next: QAGI adaptive integration on infinite intervals,  Prev: QAGS adaptive integration with singularities,  Up: Numerical Integration

17.5 QAGP adaptive integration with known singular points
=========================================================

 -- Function: int gsl_integration_qagp (const gsl_function * f, double
          * pts, size_t npts, double epsabs, double epsrel,
          size_t limit, gsl_integration_workspace * workspace, double
          * result, double * abserr)

     This function applies the adaptive integration algorithm QAGS
     taking account of the user-supplied locations of singular points.
     The array ‘pts’ of length ‘npts’ should contain the endpoints of
     the integration ranges defined by the integration region and
     locations of the singularities.  For example, to integrate over the
     region (a,b) with break-points at x_1, x_2, x_3 (where a < x_1 <
     x_2 < x_3 < b) the following ‘pts’ array should be used:

          pts[0] = a
          pts[1] = x_1
          pts[2] = x_2
          pts[3] = x_3
          pts[4] = b

     with ‘npts’ = 5.

     If you know the locations of the singular points in the integration
     region then this routine will be faster than *note
     gsl_integration_qags(): 629.


File: gsl-ref.info,  Node: QAGI adaptive integration on infinite intervals,  Next: QAWC adaptive integration for Cauchy principal values,  Prev: QAGP adaptive integration with known singular points,  Up: Numerical Integration

17.6 QAGI adaptive integration on infinite intervals
====================================================

 -- Function: int gsl_integration_qagi (gsl_function * f, double epsabs,
          double epsrel, size_t limit, gsl_integration_workspace
          * workspace, double * result, double * abserr)

     This function computes the integral of the function ‘f’ over the
     infinite interval (-\infty,+\infty).  The integral is mapped onto
     the semi-open interval (0,1] using the transformation x = (1-t)/t,

          \int_{-\infty}^{+\infty} dx f(x) = \int_0^1 dt (f((1-t)/t) + f(-(1-t)/t))/t^2.

     It is then integrated using the QAGS algorithm.  The normal
     21-point Gauss-Kronrod rule of QAGS is replaced by a 15-point rule,
     because the transformation can generate an integrable singularity
     at the origin.  In this case a lower-order rule is more efficient.

 -- Function: int gsl_integration_qagiu (gsl_function * f, double a,
          double epsabs, double epsrel, size_t limit,
          gsl_integration_workspace * workspace, double * result, double
          * abserr)

     This function computes the integral of the function ‘f’ over the
     semi-infinite interval (a,+\infty).  The integral is mapped onto
     the semi-open interval (0,1] using the transformation x = a +
     (1-t)/t,

          \int_{a}^{+\infty} dx f(x) = \int_0^1 dt f(a + (1-t)/t)/t^2

     and then integrated using the QAGS algorithm.

 -- Function: int gsl_integration_qagil (gsl_function * f, double b,
          double epsabs, double epsrel, size_t limit,
          gsl_integration_workspace * workspace, double * result, double
          * abserr)

     This function computes the integral of the function ‘f’ over the
     semi-infinite interval (-\infty,b).  The integral is mapped onto
     the semi-open interval (0,1] using the transformation x = b -
     (1-t)/t,

          \int_{-\infty}^{b} dx f(x) = \int_0^1 dt f(b - (1-t)/t)/t^2

     and then integrated using the QAGS algorithm.


File: gsl-ref.info,  Node: QAWC adaptive integration for Cauchy principal values,  Next: QAWS adaptive integration for singular functions,  Prev: QAGI adaptive integration on infinite intervals,  Up: Numerical Integration

17.7 QAWC adaptive integration for Cauchy principal values
==========================================================

 -- Function: int gsl_integration_qawc (gsl_function * f, double a,
          double b, double c, double epsabs, double epsrel,
          size_t limit, gsl_integration_workspace * workspace, double
          * result, double * abserr)

     This function computes the Cauchy principal value of the integral
     of f over (a,b), with a singularity at ‘c’,

          I = \int_a^b dx f(x) / (x - c)

     The adaptive bisection algorithm of QAG is used, with modifications
     to ensure that subdivisions do not occur at the singular point x =
     c.  When a subinterval contains the point x = c or is close to it
     then a special 25-point modified Clenshaw-Curtis rule is used to
     control the singularity.  Further away from the singularity the
     algorithm uses an ordinary 15-point Gauss-Kronrod integration rule.


File: gsl-ref.info,  Node: QAWS adaptive integration for singular functions,  Next: QAWO adaptive integration for oscillatory functions,  Prev: QAWC adaptive integration for Cauchy principal values,  Up: Numerical Integration

17.8 QAWS adaptive integration for singular functions
=====================================================

The QAWS algorithm is designed for integrands with algebraic-logarithmic
singularities at the end-points of an integration region.  In order to
work efficiently the algorithm requires a precomputed table of Chebyshev
moments.

 -- Type: gsl_integration_qaws_table

     This structure contains precomputed quantities for the QAWS
     algorithm.

 -- Function: gsl_integration_qaws_table *
          gsl_integration_qaws_table_alloc (double alpha, double beta,
          int mu, int nu)

     This function allocates space for a *note
     gsl_integration_qaws_table: 633. struct describing a singular
     weight function w(x) with the parameters (\alpha, \beta, \mu, \nu),

          w(x) = (x - a)^\alpha (b - x)^\beta \log^\mu (x - a) \log^\nu (b - x)

     where \alpha > -1, \beta > -1, and \mu = 0, 1, \nu = 0, 1.  The
     weight function can take four different forms depending on the
     values of \mu and \nu,

     Weight function w(x)                                             (\mu,\nu)
                                                                      
     ---------------------------------------------------------------------------------------
                                                                      
     (x - a)^\alpha (b - x)^\beta                                     (0,0)
                                                                      
                                                                      
     (x - a)^\alpha (b - x)^\beta \log{(x-a)}                         (1,0)
                                                                      
                                                                      
     (x - a)^\alpha (b - x)^\beta \log{(b-x)}                         (0,1)
                                                                      
                                                                      
     (x - a)^\alpha (b - x)^\beta \log{(x-a)} \log{(b-x)}             (1,1)
                                                                      

     The singular points (a,b) do not have to be specified until the
     integral is computed, where they are the endpoints of the
     integration range.

     The function returns a pointer to the newly allocated table *note
     gsl_integration_qaws_table: 633. if no errors were detected, and 0
     in the case of error.

 -- Function: int gsl_integration_qaws_table_set
          (gsl_integration_qaws_table * t, double alpha, double beta,
          int mu, int nu)

     This function modifies the parameters (\alpha, \beta, \mu, \nu) of
     an existing *note gsl_integration_qaws_table: 633. struct ‘t’.

 -- Function: void gsl_integration_qaws_table_free
          (gsl_integration_qaws_table * t)

     This function frees all the memory associated with the *note
     gsl_integration_qaws_table: 633. struct ‘t’.

 -- Function: int gsl_integration_qaws (gsl_function * f, const
          double a, const double b, gsl_integration_qaws_table * t,
          const double epsabs, const double epsrel, const size_t limit,
          gsl_integration_workspace * workspace, double * result, double
          * abserr)

     This function computes the integral of the function f(x) over the
     interval (a,b) with the singular weight function (x-a)^\alpha
     (b-x)^\beta \log^\mu (x-a) \log^\nu (b-x).  The parameters of the
     weight function (\alpha, \beta, \mu, \nu) are taken from the table
     ‘t’.  The integral is,

          I = \int_a^b dx f(x) (x - a)^\alpha (b - x)^\beta \log^\mu (x - a) \log^\nu (b - x).

     The adaptive bisection algorithm of QAG is used.  When a
     subinterval contains one of the endpoints then a special 25-point
     modified Clenshaw-Curtis rule is used to control the singularities.
     For subintervals which do not include the endpoints an ordinary
     15-point Gauss-Kronrod integration rule is used.


File: gsl-ref.info,  Node: QAWO adaptive integration for oscillatory functions,  Next: QAWF adaptive integration for Fourier integrals,  Prev: QAWS adaptive integration for singular functions,  Up: Numerical Integration

17.9 QAWO adaptive integration for oscillatory functions
========================================================

The QAWO algorithm is designed for integrands with an oscillatory
factor, \sin(\omega x) or \cos(\omega x).  In order to work efficiently
the algorithm requires a table of Chebyshev moments which must be
pre-computed with calls to the functions below.

 -- Function: gsl_integration_qawo_table *
          gsl_integration_qawo_table_alloc (double omega, double L, enum
          gsl_integration_qawo_enum sine, size_t n)

     This function allocates space for a ‘gsl_integration_qawo_table’
     struct and its associated workspace describing a sine or cosine
     weight function w(x) with the parameters (\omega, L),

          w(x) = sin(omega x) 
          w(x) = cos(omega x) 

     The parameter ‘L’ must be the length of the interval over which the
     function will be integrated L = b - a.  The choice of sine or
     cosine is made with the parameter ‘sine’ which should be chosen
     from one of the two following symbolic values:

      -- Macro: GSL_INTEG_COSINE

      -- Macro: GSL_INTEG_SINE

     The ‘gsl_integration_qawo_table’ is a table of the trigonometric
     coefficients required in the integration process.  The parameter
     ‘n’ determines the number of levels of coefficients that are
     computed.  Each level corresponds to one bisection of the interval
     L, so that ‘n’ levels are sufficient for subintervals down to the
     length L/2^n.  The integration routine *note
     gsl_integration_qawo(): 63c. returns the error ‘GSL_ETABLE’ if the
     number of levels is insufficient for the requested accuracy.

 -- Function: int gsl_integration_qawo_table_set
          (gsl_integration_qawo_table * t, double omega, double L, enum
          gsl_integration_qawo_enum sine)

     This function changes the parameters ‘omega’, ‘L’ and ‘sine’ of the
     existing workspace ‘t’.

 -- Function: int gsl_integration_qawo_table_set_length
          (gsl_integration_qawo_table * t, double L)

     This function allows the length parameter ‘L’ of the workspace ‘t’
     to be changed.

 -- Function: void gsl_integration_qawo_table_free
          (gsl_integration_qawo_table * t)

     This function frees all the memory associated with the workspace
     ‘t’.

 -- Function: int gsl_integration_qawo (gsl_function * f, const
          double a, const double epsabs, const double epsrel, const
          size_t limit, gsl_integration_workspace * workspace,
          gsl_integration_qawo_table * wf, double * result, double
          * abserr)

     This function uses an adaptive algorithm to compute the integral of
     f over (a,b) with the weight function \sin(\omega x) or \cos(\omega
     x) defined by the table ‘wf’,

          I = int_a^b dx f(x) sin(omega x) 
          I = int_a^b dx f(x) cos(omega x) 

     The results are extrapolated using the epsilon-algorithm to
     accelerate the convergence of the integral.  The function returns
     the final approximation from the extrapolation, ‘result’, and an
     estimate of the absolute error, ‘abserr’.  The subintervals and
     their results are stored in the memory provided by ‘workspace’.
     The maximum number of subintervals is given by ‘limit’, which may
     not exceed the allocated size of the workspace.

     Those subintervals with “large” widths d where d\omega > 4 are
     computed using a 25-point Clenshaw-Curtis integration rule, which
     handles the oscillatory behavior.  Subintervals with a “small”
     widths where d\omega < 4 are computed using a 15-point
     Gauss-Kronrod integration.


File: gsl-ref.info,  Node: QAWF adaptive integration for Fourier integrals,  Next: CQUAD doubly-adaptive integration,  Prev: QAWO adaptive integration for oscillatory functions,  Up: Numerical Integration

17.10 QAWF adaptive integration for Fourier integrals
=====================================================

 -- Function: int gsl_integration_qawf (gsl_function * f, const
          double a, const double epsabs, const size_t limit,
          gsl_integration_workspace * workspace,
          gsl_integration_workspace * cycle_workspace,
          gsl_integration_qawo_table * wf, double * result, double
          * abserr)

     This function attempts to compute a Fourier integral of the
     function ‘f’ over the semi-infinite interval [a,+\infty)

          I = \int_a^{+\infty} dx f(x) sin(omega x)
          I = \int_a^{+\infty} dx f(x) cos(omega x)

     The parameter \omega and choice of \sin or \cos is taken from the
     table ‘wf’ (the length ‘L’ can take any value, since it is
     overridden by this function to a value appropriate for the Fourier
     integration).  The integral is computed using the QAWO algorithm
     over each of the subintervals,

          C_1 = [a, a + c]
          C_2 = [a + c, a + 2 c]
          ... = ...
          C_k = [a + (k-1) c, a + k c]

     where c = (2 floor(|\omega|) + 1) \pi/|\omega|.  The width c is
     chosen to cover an odd number of periods so that the contributions
     from the intervals alternate in sign and are monotonically
     decreasing when ‘f’ is positive and monotonically decreasing.  The
     sum of this sequence of contributions is accelerated using the
     epsilon-algorithm.

     This function works to an overall absolute tolerance of ‘abserr’.
     The following strategy is used: on each interval C_k the algorithm
     tries to achieve the tolerance

          TOL_k = u_k abserr

     where u_k = (1 - p)p^{k-1} and p = 9/10.  The sum of the geometric
     series of contributions from each interval gives an overall
     tolerance of ‘abserr’.

     If the integration of a subinterval leads to difficulties then the
     accuracy requirement for subsequent intervals is relaxed,

          TOL_k = u_k max(abserr, max_{i<k}(E_i))

     where E_k is the estimated error on the interval C_k.

     The subintervals and their results are stored in the memory
     provided by ‘workspace’.  The maximum number of subintervals is
     given by ‘limit’, which may not exceed the allocated size of the
     workspace.  The integration over each subinterval uses the memory
     provided by ‘cycle_workspace’ as workspace for the QAWO algorithm.


File: gsl-ref.info,  Node: CQUAD doubly-adaptive integration,  Next: Romberg integration,  Prev: QAWF adaptive integration for Fourier integrals,  Up: Numerical Integration

17.11 CQUAD doubly-adaptive integration
=======================================

CQUAD is a new doubly-adaptive general-purpose quadrature routine which
can handle most types of singularities, non-numerical function values
such as ‘Inf’ or ‘NaN’, as well as some divergent integrals.  It
generally requires more function evaluations than the integration
routines in QUADPACK, yet fails less often for difficult integrands.

The underlying algorithm uses a doubly-adaptive scheme in which
Clenshaw-Curtis quadrature rules of increasing degree are used to
compute the integral in each interval.  The L_2-norm of the difference
between the underlying interpolatory polynomials of two successive rules
is used as an error estimate.  The interval is subdivided if the
difference between two successive rules is too large or a rule of
maximum degree has been reached.

 -- Function: gsl_integration_cquad_workspace *
          gsl_integration_cquad_workspace_alloc (size_t n)

     This function allocates a workspace sufficient to hold the data for
     ‘n’ intervals.  The number ‘n’ is not the maximum number of
     intervals that will be evaluated.  If the workspace is full,
     intervals with smaller error estimates will be discarded.  A
     minimum of 3 intervals is required and for most functions, a
     workspace of size 100 is sufficient.

 -- Function: void gsl_integration_cquad_workspace_free
          (gsl_integration_cquad_workspace * w)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: int gsl_integration_cquad (const gsl_function * f,
          double a, double b, double epsabs, double epsrel,
          gsl_integration_cquad_workspace * workspace, double * result,
          double * abserr, size_t * nevals)

     This function computes the integral of f over (a,b) within the
     desired absolute and relative error limits, ‘epsabs’ and ‘epsrel’
     using the CQUAD algorithm.  The function returns the final
     approximation, ‘result’, an estimate of the absolute error,
     ‘abserr’, and the number of function evaluations required,
     ‘nevals’.

     The CQUAD algorithm divides the integration region into
     subintervals, and in each iteration, the subinterval with the
     largest estimated error is processed.  The algorithm uses
     Clenshaw-Curtis quadrature rules of degree 4, 8, 16 and 32 over 5,
     9, 17 and 33 nodes respectively.  Each interval is initialized with
     the lowest-degree rule.  When an interval is processed, the
     next-higher degree rule is evaluated and an error estimate is
     computed based on the L_2-norm of the difference between the
     underlying interpolating polynomials of both rules.  If the
     highest-degree rule has already been used, or the interpolatory
     polynomials differ significantly, the interval is bisected.

     The subintervals and their results are stored in the memory
     provided by ‘workspace’.  If the error estimate or the number of
     function evaluations is not needed, the pointers ‘abserr’ and
     ‘nevals’ can be set to ‘NULL’.


File: gsl-ref.info,  Node: Romberg integration,  Next: Gauss-Legendre integration,  Prev: CQUAD doubly-adaptive integration,  Up: Numerical Integration

17.12 Romberg integration
=========================

The Romberg integration method estimates the definite integral

     I = \int_a^b f(x) dx

by applying Richardson extrapolation on the trapezoidal rule, using
equally spaced points with spacing

     h_k = (b - a) 2^{-k}

for k = 1, \dots, n.  For each k, Richardson extrapolation is used k-1
times on previous approximations to improve the order of accuracy as
much as possible.  Romberg integration typically works well (and
converges quickly) for smooth integrands with no singularities in the
interval or at the end points.

 -- Function: gsl_integration_romberg_workspace *
          gsl_integration_romberg_alloc (const size_t n)

     This function allocates a workspace for Romberg integration,
     specifying a maximum of n iterations, or divisions of the interval.
     Since the number of divisions is 2^n + 1, n can be kept relatively
     small (i.e.  10 or 20).  It is capped at a maximum value of 30 to
     prevent overflow.  The size of the workspace is O(2n).

 -- Function: void gsl_integration_romberg_free
          (gsl_integration_romberg_workspace * w)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: int gsl_integration_romberg (const gsl_function * f, const
          double a, const double b, const double epsabs, const
          double epsrel, double * result, size_t * neval,
          gsl_integration_romberg_workspace * w)

     This function integrates f(x), specified by ‘f’, from ‘a’ to ‘b’,
     storing the answer in ‘result’.  At each step in the iteration,
     convergence is tested by checking:

          | I_k - I_{k-1} | \le \textrm{max} \left( epsabs, epsrel \times |I_k| \right)

     where I_k is the current approximation and I_{k-1} is the
     approximation of the previous iteration.  If the method does not
     converge within the previously specified n iterations, the function
     stores the best current estimate in ‘result’ and returns
     ‘GSL_EMAXITER’.  If the method converges, the function returns
     ‘GSL_SUCCESS’.  The total number of function evaluations is
     returned in ‘neval’.


File: gsl-ref.info,  Node: Gauss-Legendre integration,  Next: Fixed point quadratures,  Prev: Romberg integration,  Up: Numerical Integration

17.13 Gauss-Legendre integration
================================

The fixed-order Gauss-Legendre integration routines are provided for
fast integration of smooth functions with known polynomial order.  The
n-point Gauss-Legendre rule is exact for polynomials of order 2n - 1 or
less.  For example, these rules are useful when integrating basis
functions to form mass matrices for the Galerkin method.  Unlike other
numerical integration routines within the library, these routines do not
accept absolute or relative error bounds.

 -- Function: gsl_integration_glfixed_table *
          gsl_integration_glfixed_table_alloc (size_t n)

     This function determines the Gauss-Legendre abscissae and weights
     necessary for an n-point fixed order integration scheme.  If
     possible, high precision precomputed coefficients are used.  If
     precomputed weights are not available, lower precision coefficients
     are computed on the fly.

 -- Function: double gsl_integration_glfixed (const gsl_function * f,
          double a, double b, const gsl_integration_glfixed_table * t)

     This function applies the Gauss-Legendre integration rule contained
     in table ‘t’ and returns the result.

 -- Function: int gsl_integration_glfixed_point (double a, double b,
          size_t i, double * xi, double * wi, const
          gsl_integration_glfixed_table * t)

     For ‘i’ in [0, \dots, n - 1], this function obtains the ‘i’-th
     Gauss-Legendre point ‘xi’ and weight ‘wi’ on the interval [‘a’,
     ‘b’].  The points and weights are ordered by increasing point
     value.  A function f may be integrated on [‘a’, ‘b’] by summing wi
     * f(xi) over ‘i’.

 -- Function: void gsl_integration_glfixed_table_free
          (gsl_integration_glfixed_table * t)

     This function frees the memory associated with the table ‘t’.


File: gsl-ref.info,  Node: Fixed point quadratures,  Next: Error codes,  Prev: Gauss-Legendre integration,  Up: Numerical Integration

17.14 Fixed point quadratures
=============================

The routines in this section approximate an integral by the sum

     \int_a^b w(x) f(x) dx = \sum_{i=1}^n w_i f(x_i)

where f(x) is the function to be integrated and w(x) is a weighting
function.  The n weights w_i and nodes x_i are carefully chosen so that
the result is exact when f(x) is a polynomial of degree 2n - 1 or less.
Once the user chooses the order n and weighting function w(x), the
weights w_i and nodes x_i can be precomputed and used to efficiently
evaluate integrals for any number of functions f(x).

This method works best when f(x) is well approximated by a polynomial on
the interval (a,b), and so is not suitable for functions with
singularities.  Since the user specifies ahead of time how many
quadrature nodes will be used, these routines do not accept absolute or
relative error bounds.  The table below lists the weighting functions
currently supported.

Name                 Interval                     Weighting function w(x)                         Constraints
                                                                                                  
--------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                  
Legendre             (a,b)                        1                                               b > a
                                                                                                  
                                                                                                  
Chebyshev Type 1     (a,b)                        1 / \sqrt{(b - x) (x - a)}                      b > a
                                                                                                  
                                                                                                  
Gegenbauer           (a,b)                        ((b - x) (x - a))^{\alpha}                      \alpha > -1, b > a
                                                                                                  
                                                                                                  
Jacobi               (a,b)                        (b - x)^{\alpha} (x - a)^{\beta}                \alpha,\beta > -1, b > a
                                                                                                  
                                                                                                  
Laguerre             (a,\infty)                   (x-a)^{\alpha} \exp{( -b (x - a) )}             \alpha > -1, b > 0
                                                                                                  
                                                                                                  
Hermite              (-\infty,\infty)             |x-a|^{\alpha} \exp{( -b (x-a)^2 )}             \alpha > -1, b > 0
                                                                                                  
                                                                                                  
Exponential          (a,b)                        |x - (a + b)/2|^{\alpha}                        \alpha > -1, b > a
                                                                                                  
                                                                                                  
Rational             (a,\infty)                   (x - a)^{\alpha} (x + b)^{\beta}                \alpha > -1, \alpha + \beta + 2n < 0, a + b > 0
                                                                                                  
                                                                                                  
Chebyshev Type 2     (a,b)                        \sqrt{(b - x) (x - a)}                          b > a
                                                                                                  

The fixed point quadrature routines use the following workspace to store
the nodes and weights, as well as additional variables for intermediate
calculations:

 -- Type: gsl_integration_fixed_workspace

     This workspace is used for fixed point quadrature rules and looks
     like this:

          typedef struct
          {
            size_t n;        /* number of nodes/weights */
            double *weights; /* quadrature weights */
            double *x;       /* quadrature nodes */
            double *diag;    /* diagonal of Jacobi matrix */
            double *subdiag; /* subdiagonal of Jacobi matrix */
            const gsl_integration_fixed_type * type;
          } gsl_integration_fixed_workspace;

 -- Function: gsl_integration_fixed_workspace *
          gsl_integration_fixed_alloc (const gsl_integration_fixed_type
          * T, const size_t n, const double a, const double b, const
          double alpha, const double beta)

     This function allocates a workspace for computing integrals with
     interpolating quadratures using ‘n’ quadrature nodes.  The
     parameters ‘a’, ‘b’, *note alpha: 49a, and ‘beta’ specify the
     integration interval and/or weighting function for the various
     quadrature types.  See the *note table: 650. above for constraints
     on these parameters.  The size of the workspace is O(4n).

      -- Type: gsl_integration_fixed_type

          The type of quadrature used is specified by ‘T’ which can be
          set to the following choices:

           -- Variable: gsl_integration_fixed_legendre

               This specifies Legendre quadrature integration.  The
               parameters *note alpha: 49a. and ‘beta’ are ignored for
               this type.

           -- Variable: gsl_integration_fixed_chebyshev

               This specifies Chebyshev type 1 quadrature integration.
               The parameters *note alpha: 49a. and ‘beta’ are ignored
               for this type.

           -- Variable: gsl_integration_fixed_gegenbauer

               This specifies Gegenbauer quadrature integration.  The
               parameter ‘beta’ is ignored for this type.

           -- Variable: gsl_integration_fixed_jacobi

               This specifies Jacobi quadrature integration.

           -- Variable: gsl_integration_fixed_laguerre

               This specifies Laguerre quadrature integration.  The
               parameter ‘beta’ is ignored for this type.

           -- Variable: gsl_integration_fixed_hermite

               This specifies Hermite quadrature integration.  The
               parameter ‘beta’ is ignored for this type.

           -- Variable: gsl_integration_fixed_exponential

               This specifies exponential quadrature integration.  The
               parameter ‘beta’ is ignored for this type.

           -- Variable: gsl_integration_fixed_rational

               This specifies rational quadrature integration.

           -- Variable: gsl_integration_fixed_chebyshev2

               This specifies Chebyshev type 2 quadrature integration.
               The parameters *note alpha: 49a. and ‘beta’ are ignored
               for this type.

 -- Function: void gsl_integration_fixed_free
          (gsl_integration_fixed_workspace * w)

     This function frees the memory assocated with the workspace ‘w’

 -- Function: size_t gsl_integration_fixed_n (const
          gsl_integration_fixed_workspace * w)

     This function returns the number of quadrature nodes and weights.

 -- Function: double * gsl_integration_fixed_nodes (const
          gsl_integration_fixed_workspace * w)

     This function returns a pointer to an array of size ‘n’ containing
     the quadrature nodes x_i.

 -- Function: double * gsl_integration_fixed_weights (const
          gsl_integration_fixed_workspace * w)

     This function returns a pointer to an array of size ‘n’ containing
     the quadrature weights w_i.

 -- Function: int gsl_integration_fixed (const gsl_function * func,
          double * result, const gsl_integration_fixed_workspace * w)

     This function integrates the function f(x) provided in ‘func’ using
     previously computed fixed quadrature rules.  The integral is
     approximated as

          \sum_{i=1}^n w_i f(x_i)

     where w_i are the quadrature weights and x_i are the quadrature
     nodes computed previously by *note gsl_integration_fixed_alloc():
     652.  The sum is stored in ‘result’ on output.


File: gsl-ref.info,  Node: Error codes,  Next: Examples<11>,  Prev: Fixed point quadratures,  Up: Numerical Integration

17.15 Error codes
=================

In addition to the standard error codes for invalid arguments the
functions can return the following values,

‘GSL_EMAXITER’            the maximum number of subdivisions was exceeded.
                          
                          
‘GSL_EROUND’              cannot reach tolerance because of roundoff error, or roundoff error was detected in the extrapolation table.
                          
                          
‘GSL_ESING’               a non-integrable singularity or other bad integrand behavior was found in the integration interval.
                          
                          
‘GSL_EDIVERGE’            the integral is divergent, or too slowly convergent to be integrated numerically.
                          
                          
*note GSL_EDOM: 28.       error in the values of the input arguments
                          


File: gsl-ref.info,  Node: Examples<11>,  Next: References and Further Reading<12>,  Prev: Error codes,  Up: Numerical Integration

17.16 Examples
==============

* Menu:

* Adaptive integration example::
* Fixed-point quadrature example::


File: gsl-ref.info,  Node: Adaptive integration example,  Next: Fixed-point quadrature example,  Up: Examples<11>

17.16.1 Adaptive integration example
------------------------------------

The integrator ‘QAGS’ will handle a large class of definite integrals.
For example, consider the following integral, which has an
algebraic-logarithmic singularity at the origin,

     \int_0^1 x^{-1/2} \log(x) dx = -4

The program below computes this integral to a relative accuracy bound of
‘1e-7’.

     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_integration.h>

     double f (double x, void * params) {
       double alpha = *(double *) params;
       double f = log(alpha*x) / sqrt(x);
       return f;
     }

     int
     main (void)
     {
       gsl_integration_workspace * w
         = gsl_integration_workspace_alloc (1000);

       double result, error;
       double expected = -4.0;
       double alpha = 1.0;

       gsl_function F;
       F.function = &f;
       F.params = &alpha;

       gsl_integration_qags (&F, 0, 1, 0, 1e-7, 1000,
                             w, &result, &error);

       printf ("result          = % .18f\n", result);
       printf ("exact result    = % .18f\n", expected);
       printf ("estimated error = % .18f\n", error);
       printf ("actual error    = % .18f\n", result - expected);
       printf ("intervals       = %zu\n", w->size);

       gsl_integration_workspace_free (w);

       return 0;
     }

The results below show that the desired accuracy is achieved after 8
subdivisions.

     result          = -4.000000000000085265
     exact result    = -4.000000000000000000
     estimated error =  0.000000000000135447
     actual error    = -0.000000000000085265
     intervals       = 8

In fact, the extrapolation procedure used by ‘QAGS’ produces an accuracy
of almost twice as many digits.  The error estimate returned by the
extrapolation procedure is larger than the actual error, giving a margin
of safety of one order of magnitude.


File: gsl-ref.info,  Node: Fixed-point quadrature example,  Prev: Adaptive integration example,  Up: Examples<11>

17.16.2 Fixed-point quadrature example
--------------------------------------

In this example, we use a fixed-point quadrature rule to integrate the
integral

     \int_{-\infty}^{\infty} e^{-x^2} \left( x^m + 1 \right) dx = \left\{ \begin{array}{cc} \sqrt{\pi} + \Gamma{\left( \frac{m+1}{2} \right)}, & m \textrm{ even} \\ \sqrt{\pi}, & m \textrm{ odd} \end{array} \right.

for integer m.  Consulting our *note table: 650. of fixed point
quadratures, we see that this integral can be evaluated with a Hermite
quadrature rule, setting \alpha = 0, a = 0, b = 1.  Since we are
integrating a polynomial of degree m, we need to choose the number of
nodes n \ge (m+1)/2 to achieve the best results.

First we will try integrating for m = 10, n = 5, which does not satisfy
our criteria above:

     $ ./integration2 10 5

The output is,

     m               = 10
     intervals       = 5
     result          =  47.468529694563351029
     exact result    =  54.115231635459025483
     actual error    = -6.646701940895674454

So, we find a large error.  Now we try integrating for m = 10, n = 6
which does satisfy the criteria above:

     $ ./integration2 10 6

The output is,

     m               = 10
     intervals       = 6
     result          =  54.115231635459096537
     exact result    =  54.115231635459025483
     actual error    =  0.000000000000071054

The program is given below.

     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_integration.h>
     #include <gsl/gsl_sf_gamma.h>

     double
     f(double x, void * params)
     {
       int m = *(int *) params;
       double f = gsl_pow_int(x, m) + 1.0;
       return f;
     }

     int
     main (int argc, char *argv[])
     {
       gsl_integration_fixed_workspace * w;
       const gsl_integration_fixed_type * T = gsl_integration_fixed_hermite;
       int m = 10;
       int n = 6;
       double expected, result;
       gsl_function F;

       if (argc > 1)
         m = atoi(argv[1]);

       if (argc > 2)
         n = atoi(argv[2]);

       w = gsl_integration_fixed_alloc(T, n, 0.0, 1.0, 0.0, 0.0);

       F.function = &f;
       F.params = &m;

       gsl_integration_fixed(&F, &result, w);

       if (m % 2 == 0)
         expected = M_SQRTPI + gsl_sf_gamma(0.5*(1.0 + m));
       else
         expected = M_SQRTPI;

       printf ("m               = %d\n", m);
       printf ("intervals       = %zu\n", gsl_integration_fixed_n(w));
       printf ("result          = % .18f\n", result);
       printf ("exact result    = % .18f\n", expected);
       printf ("actual error    = % .18f\n", result - expected);

       gsl_integration_fixed_free (w);

       return 0;
     }


File: gsl-ref.info,  Node: References and Further Reading<12>,  Prev: Examples<11>,  Up: Numerical Integration

17.17 References and Further Reading
====================================

The following book is the definitive reference for QUADPACK, and was
written by the original authors.  It provides descriptions of the
algorithms, program listings, test programs and examples.  It also
includes useful advice on numerical integration and many references to
the numerical integration literature used in developing QUADPACK.

   * R. Piessens, E. de Doncker-Kapenga, C.W. Ueberhuber, D.K. Kahaner.
     QUADPACK A subroutine package for automatic integration Springer
     Verlag, 1983.

The CQUAD integration algorithm is described in the following paper:

   * P. Gonnet, “Increasing the Reliability of Adaptive Quadrature Using
     Explicit Interpolants”, ACM Transactions on Mathematical Software,
     Volume 37 (2010), Issue 3, Article 26.

The fixed-point quadrature routines are based on IQPACK, described in
the following papers:

   * S. Elhay, J. Kautsky, Algorithm 655: IQPACK, FORTRAN Subroutines
     for the Weights of Interpolatory Quadrature, ACM Transactions on
     Mathematical Software, Volume 13, Number 4, December 1987, pages
     399-415.

   * J. Kautsky, S. Elhay, Calculation of the Weights of Interpolatory
     Quadratures, Numerische Mathematik, Volume 40, Number 3, October
     1982, pages 407-422.


File: gsl-ref.info,  Node: Random Number Generation,  Next: Quasi-Random Sequences,  Prev: Numerical Integration,  Up: Top

18 Random Number Generation
***************************

The library provides a large collection of random number generators
which can be accessed through a uniform interface.  Environment
variables allow you to select different generators and seeds at runtime,
so that you can easily switch between generators without needing to
recompile your program.  Each instance of a generator keeps track of its
own state, allowing the generators to be used in multi-threaded
programs.  Additional functions are available for transforming uniform
random numbers into samples from continuous or discrete probability
distributions such as the Gaussian, log-normal or Poisson distributions.

These functions are declared in the header file ‘gsl_rng.h’.

* Menu:

* General comments on random numbers::
* The Random Number Generator Interface::
* Random number generator initialization::
* Sampling from a random number generator::
* Auxiliary random number generator functions::
* Random number environment variables::
* Copying random number generator state::
* Reading and writing random number generator state::
* Random number generator algorithms::
* Unix random number generators::
* Other random number generators::
* Performance::
* Examples: Examples<12>.
* References and Further Reading: References and Further Reading<13>.
* Acknowledgements::


File: gsl-ref.info,  Node: General comments on random numbers,  Next: The Random Number Generator Interface,  Up: Random Number Generation

18.1 General comments on random numbers
=======================================

In 1988, Park and Miller wrote a paper entitled “Random number
generators: good ones are hard to find.” [Commun.: ACM, 31, 1192–1201].
Fortunately, some excellent random number generators are available,
though poor ones are still in common use.  You may be happy with the
system-supplied random number generator on your computer, but you should
be aware that as computers get faster, requirements on random number
generators increase.  Nowadays, a simulation that calls a random number
generator millions of times can often finish before you can make it down
the hall to the coffee machine and back.

A very nice review of random number generators was written by Pierre
L’Ecuyer, as Chapter 4 of the book: Handbook on Simulation, Jerry Banks,
ed.  (Wiley, 1997).  The chapter is available in postscript from
L’Ecuyer’s ftp site (see references).  Knuth’s volume on Seminumerical
Algorithms (originally published in 1968) devotes 170 pages to random
number generators, and has recently been updated in its 3rd edition
(1997).  It is brilliant, a classic.  If you don’t own it, you should
stop reading right now, run to the nearest bookstore, and buy it.

A good random number generator will satisfy both theoretical and
statistical properties.  Theoretical properties are often hard to obtain
(they require real math!), but one prefers a random number generator
with a long period, low serial correlation, and a tendency `not' to
“fall mainly on the planes.” Statistical tests are performed with
numerical simulations.  Generally, a random number generator is used to
estimate some quantity for which the theory of probability provides an
exact answer.  Comparison to this exact answer provides a measure of
“randomness”.


File: gsl-ref.info,  Node: The Random Number Generator Interface,  Next: Random number generator initialization,  Prev: General comments on random numbers,  Up: Random Number Generation

18.2 The Random Number Generator Interface
==========================================

It is important to remember that a random number generator is not a
“real” function like sine or cosine.  Unlike real functions, successive
calls to a random number generator yield different return values.  Of
course that is just what you want for a random number generator, but to
achieve this effect, the generator must keep track of some kind of
“state” variable.  Sometimes this state is just an integer (sometimes
just the value of the previously generated random number), but often it
is more complicated than that and may involve a whole array of numbers,
possibly with some indices thrown in.  To use the random number
generators, you do not need to know the details of what comprises the
state, and besides that varies from algorithm to algorithm.

 -- Type: gsl_rng_type
 -- Type: gsl_rng

     The random number generator library uses two special structs, *note
     gsl_rng_type: 66b. which holds static information about each type
     of generator and *note gsl_rng: 66c. which describes an instance of
     a generator created from a given *note gsl_rng_type: 66b.

The functions described in this section are declared in the header file
‘gsl_rng.h’.


File: gsl-ref.info,  Node: Random number generator initialization,  Next: Sampling from a random number generator,  Prev: The Random Number Generator Interface,  Up: Random Number Generation

18.3 Random number generator initialization
===========================================

 -- Function: gsl_rng * gsl_rng_alloc (const gsl_rng_type * T)

     This function returns a pointer to a newly-created instance of a
     random number generator of type ‘T’.  For example, the following
     code creates an instance of the Tausworthe generator:

          gsl_rng * r = gsl_rng_alloc (gsl_rng_taus);

     If there is insufficient memory to create the generator then the
     function returns a null pointer and the error handler is invoked
     with an error code of *note GSL_ENOMEM: 2a.

     The generator is automatically initialized with the default seed,
     *note gsl_rng_default_seed: 66f.  This is zero by default but can
     be changed either directly or by using the environment variable
     *note GSL_RNG_SEED: 670.

     The details of the available generator types are described later in
     this chapter.

 -- Function: void gsl_rng_set (const gsl_rng * r, unsigned long int s)

     This function initializes (or “seeds”) the random number generator.
     If the generator is seeded with the same value of ‘s’ on two
     different runs, the same stream of random numbers will be generated
     by successive calls to the routines below.  If different values of
     s \geq 1 are supplied, then the generated streams of random numbers
     should be completely different.  If the seed ‘s’ is zero then the
     standard seed from the original implementation is used instead.
     For example, the original Fortran source code for the ‘ranlux’
     generator used a seed of 314159265, and so choosing ‘s’ equal to
     zero reproduces this when using *note gsl_rng_ranlux: 672.

     When using multiple seeds with the same generator, choose seed
     values greater than zero to avoid collisions with the default
     setting.

     Note that the most generators only accept 32-bit seeds, with higher
     values being reduced modulo 2^{32}.  For generators with smaller
     ranges the maximum seed value will typically be lower.

 -- Function: void gsl_rng_free (gsl_rng * r)

     This function frees all the memory associated with the generator
     ‘r’.


File: gsl-ref.info,  Node: Sampling from a random number generator,  Next: Auxiliary random number generator functions,  Prev: Random number generator initialization,  Up: Random Number Generation

18.4 Sampling from a random number generator
============================================

The following functions return uniformly distributed random numbers,
either as integers or double precision floating point numbers.  Inline
versions of these functions are used when ‘HAVE_INLINE’ is defined.  To
obtain non-uniform distributions, see *note Random Number Distributions:
675.

 -- Function: unsigned long int gsl_rng_get (const gsl_rng * r)

     This function returns a random integer from the generator ‘r’.  The
     minimum and maximum values depend on the algorithm used, but all
     integers in the range [‘min’, ‘max’] are equally likely.  The
     values of ‘min’ and ‘max’ can be determined using the auxiliary
     functions *note gsl_rng_max(): 677. and *note gsl_rng_min(): 678.

 -- Function: double gsl_rng_uniform (const gsl_rng * r)

     This function returns a double precision floating point number
     uniformly distributed in the range [0,1).  The range includes 0.0
     but excludes 1.0.  The value is typically obtained by dividing the
     result of ‘gsl_rng_get(r)’ by ‘gsl_rng_max(r) + 1.0’ in double
     precision.  Some generators compute this ratio internally so that
     they can provide floating point numbers with more than 32 bits of
     randomness (the maximum number of bits that can be portably
     represented in a single ‘unsigned long int’).

 -- Function: double gsl_rng_uniform_pos (const gsl_rng * r)

     This function returns a positive double precision floating point
     number uniformly distributed in the range (0,1), excluding both 0.0
     and 1.0.  The number is obtained by sampling the generator with the
     algorithm of *note gsl_rng_uniform(): 679. until a non-zero value
     is obtained.  You can use this function if you need to avoid a
     singularity at 0.0.

 -- Function: unsigned long int gsl_rng_uniform_int (const gsl_rng * r,
          unsigned long int n)

     This function returns a random integer from 0 to n-1 inclusive by
     scaling down and/or discarding samples from the generator ‘r’.  All
     integers in the range [0,n-1] are produced with equal probability.
     For generators with a non-zero minimum value an offset is applied
     so that zero is returned with the correct probability.

     Note that this function is designed for sampling from ranges
     smaller than the range of the underlying generator.  The parameter
     ‘n’ must be less than or equal to the range of the generator ‘r’.
     If ‘n’ is larger than the range of the generator then the function
     calls the error handler with an error code of *note GSL_EINVAL: 2b.
     and returns zero.

     In particular, this function is not intended for generating the
     full range of unsigned integer values [0,2^{32}-1].  Instead choose
     a generator with the maximal integer range and zero minimum value,
     such as *note gsl_rng_ranlxd1: 67c, *note gsl_rng_mt19937: 67d. or
     *note gsl_rng_taus: 67e, and sample it directly using *note
     gsl_rng_get(): 676.  The range of each generator can be found using
     the auxiliary functions described in the next section.


File: gsl-ref.info,  Node: Auxiliary random number generator functions,  Next: Random number environment variables,  Prev: Sampling from a random number generator,  Up: Random Number Generation

18.5 Auxiliary random number generator functions
================================================

The following functions provide information about an existing generator.
You should use them in preference to hard-coding the generator
parameters into your own code.

 -- Function: const char * gsl_rng_name (const gsl_rng * r)

     This function returns a pointer to the name of the generator.  For
     example:

          printf ("r is a '%s' generator\n", gsl_rng_name (r));

     would print something like:

          r is a 'taus' generator

 -- Function: unsigned long int gsl_rng_max (const gsl_rng * r)

     This function returns the largest value that *note gsl_rng_get():
     676. can return.

 -- Function: unsigned long int gsl_rng_min (const gsl_rng * r)

     This function returns the smallest value that *note gsl_rng_get():
     676. can return.  Usually this value is zero.  There are some
     generators with algorithms that cannot return zero, and for these
     generators the minimum value is 1.

 -- Function: void * gsl_rng_state (const gsl_rng * r)
 -- Function: size_t gsl_rng_size (const gsl_rng * r)

     These functions return a pointer to the state of generator ‘r’ and
     its size.  You can use this information to access the state
     directly.  For example, the following code will write the state of
     a generator to a stream:

          void * state = gsl_rng_state (r);
          size_t n = gsl_rng_size (r);
          fwrite (state, n, 1, stream);

 -- Function: const gsl_rng_type ** gsl_rng_types_setup (void)

     This function returns a pointer to an array of all the available
     generator types, terminated by a null pointer.  The function should
     be called once at the start of the program, if needed.  The
     following code fragment shows how to iterate over the array of
     generator types to print the names of the available algorithms:

          const gsl_rng_type **t, **t0;

          t0 = gsl_rng_types_setup ();

          printf ("Available generators:\n");

          for (t = t0; *t != 0; t++)
            {
              printf ("%s\n", (*t)->name);
            }


File: gsl-ref.info,  Node: Random number environment variables,  Next: Copying random number generator state,  Prev: Auxiliary random number generator functions,  Up: Random Number Generation

18.6 Random number environment variables
========================================

The library allows you to choose a default generator and seed from the
environment variables *note GSL_RNG_TYPE: 685. and *note GSL_RNG_SEED:
670. and the function *note gsl_rng_env_setup(): 686.  This makes it
easy try out different generators and seeds without having to recompile
your program.

 -- Macro: GSL_RNG_TYPE

     This environment variable specifies the default random number
     generator.  It should be the name of a generator, such as ‘taus’ or
     ‘mt19937’.

 -- Macro: GSL_RNG_SEED

     This environment variable specifies the default seed for the random
     number generator

 -- Variable: gsl_rng_default

     This global library variable specifies the default random number
     generator, and can be initialized from *note GSL_RNG_TYPE: 685.
     using *note gsl_rng_env_setup(): 686.  It is defined as follows:

          extern const gsl_rng_type *gsl_rng_default

 -- Variable: gsl_rng_default_seed

     This global library variable specifies the seed for the default
     random number generator, and can be initialized from *note
     GSL_RNG_SEED: 670. using *note gsl_rng_env_setup(): 686.  It is set
     to zero by default and is defined as follows:

          extern unsigned long int gsl_rng_default_seed

 -- Function: const gsl_rng_type * gsl_rng_env_setup (void)

     This function reads the environment variables *note GSL_RNG_TYPE:
     685. and *note GSL_RNG_SEED: 670. and uses their values to set the
     corresponding library variables *note gsl_rng_default: 687. and
     *note gsl_rng_default_seed: 66f.

     The value of *note GSL_RNG_SEED: 670. is converted to an ‘unsigned
     long int’ using the C library function ‘strtoul()’.

     If you don’t specify a generator for *note GSL_RNG_TYPE: 685. then
     *note gsl_rng_mt19937: 67d. is used as the default.  The initial
     value of *note gsl_rng_default_seed: 66f. is zero.

Here is a short program which shows how to create a global generator
using the environment variables *note GSL_RNG_TYPE: 685. and *note
GSL_RNG_SEED: 670,

     #include <stdio.h>
     #include <gsl/gsl_rng.h>

     gsl_rng * r;  /* global generator */

     int
     main (void)
     {
       const gsl_rng_type * T;

       gsl_rng_env_setup();

       T = gsl_rng_default;
       r = gsl_rng_alloc (T);

       printf ("generator type: %s\n", gsl_rng_name (r));
       printf ("seed = %lu\n", gsl_rng_default_seed);
       printf ("first value = %lu\n", gsl_rng_get (r));

       gsl_rng_free (r);
       return 0;
     }

Running the program without any environment variables uses the initial
defaults, an ‘mt19937’ generator with a seed of 0,

     generator type: mt19937
     seed = 0
     first value = 4293858116

By setting the two variables on the command line we can change the
default generator and the seed:

     $ GSL_RNG_TYPE="taus" GSL_RNG_SEED=123 ./a.out
     GSL_RNG_TYPE=taus
     GSL_RNG_SEED=123
     generator type: taus
     seed = 123
     first value = 2720986350


File: gsl-ref.info,  Node: Copying random number generator state,  Next: Reading and writing random number generator state,  Prev: Random number environment variables,  Up: Random Number Generation

18.7 Copying random number generator state
==========================================

The above methods do not expose the random number state which changes
from call to call.  It is often useful to be able to save and restore
the state.  To permit these practices, a few somewhat more advanced
functions are supplied.  These include:

 -- Function: int gsl_rng_memcpy (gsl_rng * dest, const gsl_rng * src)

     This function copies the random number generator ‘src’ into the
     pre-existing generator ‘dest’, making ‘dest’ into an exact copy of
     ‘src’.  The two generators must be of the same type.

 -- Function: gsl_rng * gsl_rng_clone (const gsl_rng * r)

     This function returns a pointer to a newly created generator which
     is an exact copy of the generator ‘r’.


File: gsl-ref.info,  Node: Reading and writing random number generator state,  Next: Random number generator algorithms,  Prev: Copying random number generator state,  Up: Random Number Generation

18.8 Reading and writing random number generator state
======================================================

The library provides functions for reading and writing the random number
state to a file as binary data.

 -- Function: int gsl_rng_fwrite (FILE * stream, const gsl_rng * r)

     This function writes the random number state of the random number
     generator ‘r’ to the stream ‘stream’ in binary format.  The return
     value is 0 for success and ‘GSL_EFAILED’ if there was a problem
     writing to the file.  Since the data is written in the native
     binary format it may not be portable between different
     architectures.

 -- Function: int gsl_rng_fread (FILE * stream, gsl_rng * r)

     This function reads the random number state into the random number
     generator ‘r’ from the open stream ‘stream’ in binary format.  The
     random number generator ‘r’ must be preinitialized with the correct
     random number generator type since type information is not saved.
     The return value is 0 for success and ‘GSL_EFAILED’ if there was a
     problem reading from the file.  The data is assumed to have been
     written in the native binary format on the same architecture.


File: gsl-ref.info,  Node: Random number generator algorithms,  Next: Unix random number generators,  Prev: Reading and writing random number generator state,  Up: Random Number Generation

18.9 Random number generator algorithms
=======================================

The functions described above make no reference to the actual algorithm
used.  This is deliberate so that you can switch algorithms without
having to change any of your application source code.  The library
provides a large number of generators of different types, including
simulation quality generators, generators provided for compatibility
with other libraries and historical generators from the past.

The following generators are recommended for use in simulation.  They
have extremely long periods, low correlation and pass most statistical
tests.  For the most reliable source of uncorrelated numbers, the
second-generation RANLUX generators have the strongest proof of
randomness.

 -- Variable: gsl_rng_mt19937

     The MT19937 generator of Makoto Matsumoto and Takuji Nishimura is a
     variant of the twisted generalized feedback shift-register
     algorithm, and is known as the “Mersenne Twister” generator.  It
     has a Mersenne prime period of 2^{19937} - 1 (about 10^{6000}) and
     is equi-distributed in 623 dimensions.  It has passed the DIEHARD
     statistical tests.  It uses 624 words of state per generator and is
     comparable in speed to the other generators.  The original
     generator used a default seed of 4357 and choosing ‘s’ equal to
     zero in *note gsl_rng_set(): 671. reproduces this.  Later versions
     switched to 5489 as the default seed, you can choose this
     explicitly via *note gsl_rng_set(): 671. instead if you require it.

     For more information see,

        * Makoto Matsumoto and Takuji Nishimura, “Mersenne Twister: A
          623-dimensionally equidistributed uniform pseudorandom number
          generator”.  ACM Transactions on Modeling and Computer
          Simulation, Vol.: 8, No.: 1 (Jan.  1998), Pages 3–30

     The generator *note gsl_rng_mt19937: 67d. uses the second revision
     of the seeding procedure published by the two authors above in
     2002.  The original seeding procedures could cause spurious
     artifacts for some seed values.  They are still available through
     the alternative generators ‘gsl_rng_mt19937_1999’ and
     ‘gsl_rng_mt19937_1998’.

 -- Variable: gsl_rng_ranlxs0
 -- Variable: gsl_rng_ranlxs1
 -- Variable: gsl_rng_ranlxs2

     The generator ‘ranlxs0’ is a second-generation version of the
     RANLUX algorithm of Luscher, which produces “luxury random
     numbers”.  This generator provides single precision output (24
     bits) at three luxury levels ‘ranlxs0’, ‘ranlxs1’ and ‘ranlxs2’, in
     increasing order of strength.  It uses double-precision floating
     point arithmetic internally and can be significantly faster than
     the integer version of ‘ranlux’, particularly on 64-bit
     architectures.  The period of the generator is about 10^{171}.  The
     algorithm has mathematically proven properties and can provide
     truly decorrelated numbers at a known level of randomness.  The
     higher luxury levels provide increased decorrelation between
     samples as an additional safety margin.

     Note that the range of allowed seeds for this generator is
     [0,2^{31}-1].  Higher seed values are wrapped modulo 2^{31}.

 -- Variable: gsl_rng_ranlxd1
 -- Variable: gsl_rng_ranlxd2

     These generators produce double precision output (48 bits) from the
     RANLXS generator.  The library provides two luxury levels ‘ranlxd1’
     and ‘ranlxd2’, in increasing order of strength.

 -- Variable: gsl_rng_ranlux
 -- Variable: gsl_rng_ranlux389

     The ‘ranlux’ generator is an implementation of the original
     algorithm developed by Luscher.  It uses a
     lagged-fibonacci-with-skipping algorithm to produce “luxury random
     numbers”.  It is a 24-bit generator, originally designed for
     single-precision IEEE floating point numbers.  This implementation
     is based on integer arithmetic, while the second-generation
     versions RANLXS and RANLXD described above provide floating-point
     implementations which will be faster on many platforms.  The period
     of the generator is about 10^{171}.  The algorithm has
     mathematically proven properties and it can provide truly
     decorrelated numbers at a known level of randomness.  The default
     level of decorrelation recommended by Luscher is provided by *note
     gsl_rng_ranlux: 672, while *note gsl_rng_ranlux389: 693. gives the
     highest level of randomness, with all 24 bits decorrelated.  Both
     types of generator use 24 words of state per generator.

     For more information see,

        * M. Luscher, “A portable high-quality random number generator
          for lattice field theory calculations”, Computer Physics
          Communications, 79 (1994) 100–110.

        * F. James, “RANLUX: A Fortran implementation of the
          high-quality pseudo-random number generator of Luscher”,
          Computer Physics Communications, 79 (1994) 111–114

 -- Variable: gsl_rng_cmrg

     This is a combined multiple recursive generator by L’Ecuyer.  Its
     sequence is,

          z_n = (x_n - y_n) \mod m_1

     where the two underlying generators x_n and y_n are,

          x_n = (a_1 x_{n-1} + a_2 x_{n-2} + a_3 x_{n-3}) mod m_1
          y_n = (b_1 y_{n-1} + b_2 y_{n-2} + b_3 y_{n-3}) mod m_2

     with coefficients a_1 = 0, a_2 = 63308, a_3 = -183326, b_1 = 86098,
     b_2 = 0, b_3 = -539608, and moduli m_1 = 2^{31} - 1 = 2147483647
     and m_2 = 2145483479.

     The period of this generator is \hbox{lcm}(m_1^3-1, m_2^3-1), which
     is approximately 2^{185} (about 10^{56}).  It uses 6 words of state
     per generator.  For more information see,

        * P. L’Ecuyer, “Combined Multiple Recursive Random Number
          Generators”, Operations Research, 44, 5 (1996), 816–822.

 -- Variable: gsl_rng_mrg

     This is a fifth-order multiple recursive generator by L’Ecuyer,
     Blouin and Coutre.  Its sequence is,

          x_n = (a_1 x_{n-1} + a_5 x_{n-5}) \mod m

     with a_1 = 107374182, a_2 = a_3 = a_4 = 0, a_5 = 104480 and m =
     2^{31}-1.

     The period of this generator is about 10^{46}.  It uses 5 words of
     state per generator.  More information can be found in the
     following paper,

        * P. L’Ecuyer, F. Blouin, and R. Coutre, “A search for good
          multiple recursive random number generators”, ACM Transactions
          on Modeling and Computer Simulation 3, 87–98 (1993).

 -- Variable: gsl_rng_taus
 -- Variable: gsl_rng_taus2

     This is a maximally equidistributed combined Tausworthe generator
     by L’Ecuyer.  The sequence is,

          x_n = (s1_n ^^ s2_n ^^ s3_n)

     where,

          s1_{n+1} = (((s1_n&4294967294)<<12)^^(((s1_n<<13)^^s1_n)>>19))
          s2_{n+1} = (((s2_n&4294967288)<< 4)^^(((s2_n<< 2)^^s2_n)>>25))
          s3_{n+1} = (((s3_n&4294967280)<<17)^^(((s3_n<< 3)^^s3_n)>>11))

     computed modulo 2^{32}.  In the formulas above \oplus denotes
     `exclusive-or'.  Note that the algorithm relies on the properties
     of 32-bit unsigned integers and has been implemented using a
     bitmask of ‘0xFFFFFFFF’ to make it work on 64 bit machines.

     The period of this generator is 2^{88} (about 10^{26}).  It uses 3
     words of state per generator.  For more information see,

        * P. L’Ecuyer, “Maximally Equidistributed Combined Tausworthe
          Generators”, Mathematics of Computation, 65, 213 (1996),
          203–213.

     The generator *note gsl_rng_taus2: 696. uses the same algorithm as
     *note gsl_rng_taus: 67e. but with an improved seeding procedure
     described in the paper,

        * P. L’Ecuyer, “Tables of Maximally Equidistributed Combined
          LFSR Generators”, Mathematics of Computation, 68, 225 (1999),
          261–269

     The generator *note gsl_rng_taus2: 696. should now be used in
     preference to *note gsl_rng_taus: 67e.

 -- Variable: gsl_rng_gfsr4

     The ‘gfsr4’ generator is like a lagged-fibonacci generator, and
     produces each number as an ‘xor’’d sum of four previous values.

          r_n = r_{n-A} ^^ r_{n-B} ^^ r_{n-C} ^^ r_{n-D}

     Ziff (ref below) notes that “it is now widely known” that two-tap
     registers (such as R250, which is described below) have serious
     flaws, the most obvious one being the three-point correlation that
     comes from the definition of the generator.  Nice mathematical
     properties can be derived for GFSR’s, and numerics bears out the
     claim that 4-tap GFSR’s with appropriately chosen offsets are as
     random as can be measured, using the author’s test.

     This implementation uses the values suggested the example on p392
     of Ziff’s article: A=471, B=1586, C=6988, D=9689.

     If the offsets are appropriately chosen (such as the one ones in
     this implementation), then the sequence is said to be maximal; that
     means that the period is 2^D - 1, where D is the longest lag.  (It
     is one less than 2^D because it is not permitted to have all zeros
     in the ‘ra[]’ array.)  For this implementation with D=9689 that
     works out to about 10^{2917}.

     Note that the implementation of this generator using a 32-bit
     integer amounts to 32 parallel implementations of one-bit
     generators.  One consequence of this is that the period of this
     32-bit generator is the same as for the one-bit generator.
     Moreover, this independence means that all 32-bit patterns are
     equally likely, and in particular that 0 is an allowed random
     value.  (We are grateful to Heiko Bauke for clarifying for us these
     properties of GFSR random number generators.)

     For more information see,

        * Robert M. Ziff, “Four-tap shift-register-sequence
          random-number generators”, Computers in Physics, 12(4),
          Jul/Aug 1998, pp 385–392.


File: gsl-ref.info,  Node: Unix random number generators,  Next: Other random number generators,  Prev: Random number generator algorithms,  Up: Random Number Generation

18.10 Unix random number generators
===================================

The standard Unix random number generators ‘rand’, ‘random’ and ‘rand48’
are provided as part of GSL. Although these generators are widely
available individually often they aren’t all available on the same
platform.  This makes it difficult to write portable code using them and
so we have included the complete set of Unix generators in GSL for
convenience.  Note that these generators don’t produce high-quality
randomness and aren’t suitable for work requiring accurate statistics.
However, if you won’t be measuring statistical quantities and just want
to introduce some variation into your program then these generators are
quite acceptable.

 -- Variable: gsl_rng_rand

     This is the BSD ‘rand’ generator.  Its sequence is

          x_{n+1} = (a x_n + c) \mod m

     with a = 1103515245, c = 12345 and m = 2^{31}.  The seed specifies
     the initial value, x_1.  The period of this generator is 2^{31},
     and it uses 1 word of storage per generator.

 -- Variable: gsl_rng_random_bsd
 -- Variable: gsl_rng_random_libc5
 -- Variable: gsl_rng_random_glibc2

     These generators implement the ‘random’ family of functions, a set
     of linear feedback shift register generators originally used in BSD
     Unix.  There are several versions of ‘random’ in use today: the
     original BSD version (e.g.  on SunOS4), a libc5 version (found on
     older GNU/Linux systems) and a glibc2 version.  Each version uses a
     different seeding procedure, and thus produces different sequences.

     The original BSD routines accepted a variable length buffer for the
     generator state, with longer buffers providing higher-quality
     randomness.  The ‘random’ function implemented algorithms for
     buffer lengths of 8, 32, 64, 128 and 256 bytes, and the algorithm
     with the largest length that would fit into the user-supplied
     buffer was used.  To support these algorithms additional generators
     are available with the following names:

          gsl_rng_random8_bsd
          gsl_rng_random32_bsd
          gsl_rng_random64_bsd
          gsl_rng_random128_bsd
          gsl_rng_random256_bsd

     where the numeric suffix indicates the buffer length.  The original
     BSD ‘random’ function used a 128-byte default buffer and so *note
     gsl_rng_random_bsd: 69a. has been made equivalent to
     ‘gsl_rng_random128_bsd’.  Corresponding versions of the ‘libc5’ and
     ‘glibc2’ generators are also available, with the names
     ‘gsl_rng_random8_libc5’, ‘gsl_rng_random8_glibc2’, etc.

 -- Variable: gsl_rng_rand48

     This is the Unix ‘rand48’ generator.  Its sequence is

          x_{n+1} = (a x_n + c) \mod m

     defined on 48-bit unsigned integers with a = 25214903917, c = 11
     and m = 2^{48}.  The seed specifies the upper 32 bits of the
     initial value, x_1, with the lower 16 bits set to ‘0x330E’.  The
     function *note gsl_rng_get(): 676. returns the upper 32 bits from
     each term of the sequence.  This does not have a direct parallel in
     the original ‘rand48’ functions, but forcing the result to type
     ‘long int’ reproduces the output of ‘mrand48’.  The function *note
     gsl_rng_uniform(): 679. uses the full 48 bits of internal state to
     return the double precision number x_n/m, which is equivalent to
     the function ‘drand48’.  Note that some versions of the GNU C
     Library contained a bug in ‘mrand48’ function which caused it to
     produce different results (only the lower 16-bits of the return
     value were set).


File: gsl-ref.info,  Node: Other random number generators,  Next: Performance,  Prev: Unix random number generators,  Up: Random Number Generation

18.11 Other random number generators
====================================

The generators in this section are provided for compatibility with
existing libraries.  If you are converting an existing program to use
GSL then you can select these generators to check your new
implementation against the original one, using the same random number
generator.  After verifying that your new program reproduces the
original results you can then switch to a higher-quality generator.

Note that most of the generators in this section are based on single
linear congruence relations, which are the least sophisticated type of
generator.  In particular, linear congruences have poor properties when
used with a non-prime modulus, as several of these routines do (e.g.
with a power of two modulus, 2^{31} or 2^{32}).  This leads to
periodicity in the least significant bits of each number, with only the
higher bits having any randomness.  Thus if you want to produce a random
bitstream it is best to avoid using the least significant bits.

 -- Variable: gsl_rng_ranf

     This is the CRAY random number generator ‘RANF’.  Its sequence is

          x_{n+1} = (a x_n) \mod m

     defined on 48-bit unsigned integers with a = 44485709377909 and m =
     2^{48}.  The seed specifies the lower 32 bits of the initial value,
     x_1, with the lowest bit set to prevent the seed taking an even
     value.  The upper 16 bits of x_1 are set to 0.  A consequence of
     this procedure is that the pairs of seeds 2 and 3, 4 and 5, etc.:
     produce the same sequences.

     The generator compatible with the CRAY MATHLIB routine RANF. It
     produces double precision floating point numbers which should be
     identical to those from the original RANF.

     There is a subtlety in the implementation of the seeding.  The
     initial state is reversed through one step, by multiplying by the
     modular inverse of a mod m.  This is done for compatibility with
     the original CRAY implementation.

     Note that you can only seed the generator with integers up to
     2^{32}, while the original CRAY implementation uses non-portable
     wide integers which can cover all 2^{48} states of the generator.

     The function *note gsl_rng_get(): 676. returns the upper 32 bits
     from each term of the sequence.  The function *note
     gsl_rng_uniform(): 679. uses the full 48 bits to return the double
     precision number x_n/m.

     The period of this generator is 2^{46}.

 -- Variable: gsl_rng_ranmar

     This is the RANMAR lagged-fibonacci generator of Marsaglia, Zaman
     and Tsang.  It is a 24-bit generator, originally designed for
     single-precision IEEE floating point numbers.  It was included in
     the CERNLIB high-energy physics library.

 -- Variable: gsl_rng_r250

     This is the shift-register generator of Kirkpatrick and Stoll.  The
     sequence is based on the recurrence

          x_n = x_{n-103} ^^ x_{n-250}

     where \oplus denotes `exclusive-or', defined on 32-bit words.  The
     period of this generator is about 2^{250} and it uses 250 words of
     state per generator.

     For more information see,

        * S. Kirkpatrick and E. Stoll, “A very fast shift-register
          sequence random number generator”, Journal of Computational
          Physics, 40, 517–526 (1981)

 -- Variable: gsl_rng_tt800

     This is an earlier version of the twisted generalized feedback
     shift-register generator, and has been superseded by the
     development of MT19937.  However, it is still an acceptable
     generator in its own right.  It has a period of 2^{800} and uses 33
     words of storage per generator.

     For more information see,

        * Makoto Matsumoto and Yoshiharu Kurita, “Twisted GFSR
          Generators II”, ACM Transactions on Modelling and Computer
          Simulation, Vol.: 4, No.: 3, 1994, pages 254–266.

 -- Variable: gsl_rng_vax

     This is the VAX generator ‘MTH$RANDOM’.  Its sequence is,

          x_{n+1} = (a x_n + c) \mod m

     with a = 69069, c = 1 and m = 2^{32}.  The seed specifies the
     initial value, x_1.  The period of this generator is 2^{32} and it
     uses 1 word of storage per generator.

 -- Variable: gsl_rng_transputer

     This is the random number generator from the INMOS Transputer
     Development system.  Its sequence is,

          x_{n+1} = (a x_n) \mod m

     with a = 1664525 and m = 2^{32}.  The seed specifies the initial
     value, x_1.

 -- Variable: gsl_rng_randu

     This is the IBM ‘RANDU’ generator.  Its sequence is

          x_{n+1} = (a x_n) \mod m

     with a = 65539 and m = 2^{31}.  The seed specifies the initial
     value, x_1.  The period of this generator was only 2^{29}.  It has
     become a textbook example of a poor generator.

 -- Variable: gsl_rng_minstd

     This is Park and Miller’s “minimal standard” MINSTD generator, a
     simple linear congruence which takes care to avoid the major
     pitfalls of such algorithms.  Its sequence is,

          x_{n+1} = (a x_n) \mod m

     with a = 16807 and m = 2^{31} - 1 = 2147483647.  The seed specifies
     the initial value, x_1.  The period of this generator is about
     2^{31}.

     This generator was used in the IMSL Library (subroutine RNUN) and
     in MATLAB (the RAND function) in the past.  It is also sometimes
     known by the acronym “GGL” (I’m not sure what that stands for).

     For more information see,

        * Park and Miller, “Random Number Generators: Good ones are hard
          to find”, Communications of the ACM, October 1988, Volume 31,
          No 10, pages 1192–1201.

 -- Variable: gsl_rng_uni
 -- Variable: gsl_rng_uni32

     This is a reimplementation of the 16-bit SLATEC random number
     generator RUNIF. A generalization of the generator to 32 bits is
     provided by *note gsl_rng_uni32: 6a8.  The original source code is
     available from NETLIB.

 -- Variable: gsl_rng_slatec

     This is the SLATEC random number generator RAND. It is ancient.
     The original source code is available from NETLIB.

 -- Variable: gsl_rng_zuf

     This is the ZUFALL lagged Fibonacci series generator of Peterson.
     Its sequence is,

          t = u_{n-273} + u_{n-607}
          u_n  = t - floor(t)

     The original source code is available from NETLIB. For more
     information see,

        * W. Petersen, “Lagged Fibonacci Random Number Generators for
          the NEC SX-3”, International Journal of High Speed Computing
          (1994).

 -- Variable: gsl_rng_knuthran2

     This is a second-order multiple recursive generator described by
     Knuth in Seminumerical Algorithms, 3rd Ed., page 108.  Its sequence
     is,

          x_n = (a_1 x_{n-1} + a_2 x_{n-2}) \mod m

     with a_1 = 271828183, a_2 = 314159269, and m = 2^{31}-1.

 -- Variable: gsl_rng_knuthran2002
 -- Variable: gsl_rng_knuthran

     This is a second-order multiple recursive generator described by
     Knuth in Seminumerical Algorithms, 3rd Ed., Section 3.6.  Knuth
     provides its C code.  The updated routine *note
     gsl_rng_knuthran2002: 6ac. is from the revised 9th printing and
     corrects some weaknesses in the earlier version, which is
     implemented as *note gsl_rng_knuthran: 6ad.

 -- Variable: gsl_rng_borosh13
 -- Variable: gsl_rng_fishman18
 -- Variable: gsl_rng_fishman20
 -- Variable: gsl_rng_lecuyer21
 -- Variable: gsl_rng_waterman14

     These multiplicative generators are taken from Knuth’s
     Seminumerical Algorithms, 3rd Ed., pages 106–108.  Their sequence
     is,

          x_{n+1} = (a x_n) \mod m

     where the seed specifies the initial value, x_1.  The parameters a
     and m are as follows, Borosh-Niederreiter: a = 1812433253, m =
     2^{32}, Fishman18: a = 62089911, m = 2^{31}-1, Fishman20: a =
     48271, m = 2^{31}-1, L’Ecuyer: a = 40692, m = 2^{31}-249, Waterman:
     a = 1566083941, m = 2^{32}.

 -- Variable: gsl_rng_fishman2x

     This is the L’Ecuyer–Fishman random number generator.  It is taken
     from Knuth’s Seminumerical Algorithms, 3rd Ed., page 108.  Its
     sequence is,

          z_{n+1} = (x_n - y_n) \mod m

     with m = 2^{31}-1.  x_n and y_n are given by the ‘fishman20’ and
     ‘lecuyer21’ algorithms.  The seed specifies the initial value, x_1.

 -- Variable: gsl_rng_coveyou

     This is the Coveyou random number generator.  It is taken from
     Knuth’s Seminumerical Algorithms, 3rd Ed., Section 3.2.2.  Its
     sequence is,

          x_{n+1} = (x_n (x_n + 1)) \mod m

     with m = 2^{32}.  The seed specifies the initial value, x_1.


File: gsl-ref.info,  Node: Performance,  Next: Examples<12>,  Prev: Other random number generators,  Up: Random Number Generation

18.12 Performance
=================

The following table shows the relative performance of a selection the
available random number generators.  The fastest simulation quality
generators are ‘taus’, ‘gfsr4’ and ‘mt19937’.  The generators which
offer the best mathematically-proven quality are those based on the
RANLUX algorithm:

     1754 k ints/sec,    870 k doubles/sec, taus
     1613 k ints/sec,    855 k doubles/sec, gfsr4
     1370 k ints/sec,    769 k doubles/sec, mt19937
      565 k ints/sec,    571 k doubles/sec, ranlxs0
      400 k ints/sec,    405 k doubles/sec, ranlxs1
      490 k ints/sec,    389 k doubles/sec, mrg
      407 k ints/sec,    297 k doubles/sec, ranlux
      243 k ints/sec,    254 k doubles/sec, ranlxd1
      251 k ints/sec,    253 k doubles/sec, ranlxs2
      238 k ints/sec,    215 k doubles/sec, cmrg
      247 k ints/sec,    198 k doubles/sec, ranlux389
      141 k ints/sec,    140 k doubles/sec, ranlxd2


File: gsl-ref.info,  Node: Examples<12>,  Next: References and Further Reading<13>,  Prev: Performance,  Up: Random Number Generation

18.13 Examples
==============

The following program demonstrates the use of a random number generator
to produce uniform random numbers in the range [0.0, 1.0),

     #include <stdio.h>
     #include <gsl/gsl_rng.h>

     int
     main (void)
     {
       const gsl_rng_type * T;
       gsl_rng * r;

       int i, n = 10;

       gsl_rng_env_setup();

       T = gsl_rng_default;
       r = gsl_rng_alloc (T);

       for (i = 0; i < n; i++)
         {
           double u = gsl_rng_uniform (r);
           printf ("%.5f\n", u);
         }

       gsl_rng_free (r);

       return 0;
     }

Here is the output of the program,

     0.99974
     0.16291
     0.28262
     0.94720
     0.23166
     0.48497
     0.95748
     0.74431
     0.54004
     0.73995

The numbers depend on the seed used by the generator.  The default seed
can be changed with the *note GSL_RNG_SEED: 670. environment variable to
produce a different stream of numbers.  The generator itself can be
changed using the environment variable *note GSL_RNG_TYPE: 685.  Here is
the output of the program using a seed value of 123 and the
multiple-recursive generator ‘mrg’:

     $ GSL_RNG_SEED=123 GSL_RNG_TYPE=mrg ./a.out

     0.33050
     0.86631
     0.32982
     0.67620
     0.53391
     0.06457
     0.16847
     0.70229
     0.04371
     0.86374


File: gsl-ref.info,  Node: References and Further Reading<13>,  Next: Acknowledgements,  Prev: Examples<12>,  Up: Random Number Generation

18.14 References and Further Reading
====================================

The subject of random number generation and testing is reviewed
extensively in Knuth’s `Seminumerical Algorithms'.

   * Donald E. Knuth, The Art of Computer Programming: Seminumerical
     Algorithms (Vol 2, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896842.

Further information is available in the review paper written by Pierre
L’Ecuyer,

   * P. L’Ecuyer, “Random Number Generation”, Chapter 4 of the Handbook
     on Simulation, Jerry Banks Ed., Wiley, 1998, 93–137.

   * ‘http://www.iro.umontreal.ca/~lecuyer/papers.html’ in the file
     ‘handsim.ps’.

The source code for the DIEHARD random number generator tests is also
available online,

   * DIEHARD source code, G. Marsaglia,
     ‘http://stat.fsu.edu/pub/diehard/’

A comprehensive set of random number generator tests is available from
NIST,

   * NIST Special Publication 800-22, “A Statistical Test Suite for the
     Validation of Random Number Generators and Pseudo Random Number
     Generators for Cryptographic Applications”.

   * ‘http://csrc.nist.gov/rng/’


File: gsl-ref.info,  Node: Acknowledgements,  Prev: References and Further Reading<13>,  Up: Random Number Generation

18.15 Acknowledgements
======================

Thanks to Makoto Matsumoto, Takuji Nishimura and Yoshiharu Kurita for
making the source code to their generators (MT19937, MM&TN; TT800,
MM&YK) available under the GNU General Public License.  Thanks to Martin
Luscher for providing notes and source code for the RANLXS and RANLXD
generators.


File: gsl-ref.info,  Node: Quasi-Random Sequences,  Next: Random Number Distributions,  Prev: Random Number Generation,  Up: Top

19 Quasi-Random Sequences
*************************

This chapter describes functions for generating quasi-random sequences
in arbitrary dimensions.  A quasi-random sequence progressively covers a
d-dimensional space with a set of points that are uniformly distributed.
Quasi-random sequences are also known as low-discrepancy sequences.  The
quasi-random sequence generators use an interface that is similar to the
interface for random number generators, except that seeding is not
required—each generator produces a single sequence.

The functions described in this section are declared in the header file
‘gsl_qrng.h’.

* Menu:

* Quasi-random number generator initialization::
* Sampling from a quasi-random number generator::
* Auxiliary quasi-random number generator functions::
* Saving and restoring quasi-random number generator state::
* Quasi-random number generator algorithms::
* Examples: Examples<13>.
* References::


File: gsl-ref.info,  Node: Quasi-random number generator initialization,  Next: Sampling from a quasi-random number generator,  Up: Quasi-Random Sequences

19.1 Quasi-random number generator initialization
=================================================

 -- Type: gsl_qrng

     This is a workspace for computing quasi-random sequences.

 -- Function: gsl_qrng * gsl_qrng_alloc (const gsl_qrng_type * T,
          unsigned int d)

     This function returns a pointer to a newly-created instance of a
     quasi-random sequence generator of type ‘T’ and dimension ‘d’.  If
     there is insufficient memory to create the generator then the
     function returns a null pointer and the error handler is invoked
     with an error code of *note GSL_ENOMEM: 2a.

 -- Function: void gsl_qrng_free (gsl_qrng * q)

     This function frees all the memory associated with the generator
     ‘q’.

 -- Function: void gsl_qrng_init (gsl_qrng * q)

     This function reinitializes the generator ‘q’ to its starting
     point.  Note that quasi-random sequences do not use a seed and
     always produce the same set of values.


File: gsl-ref.info,  Node: Sampling from a quasi-random number generator,  Next: Auxiliary quasi-random number generator functions,  Prev: Quasi-random number generator initialization,  Up: Quasi-Random Sequences

19.2 Sampling from a quasi-random number generator
==================================================

 -- Function: int gsl_qrng_get (const gsl_qrng * q, double x[])

     This function stores the next point from the sequence generator ‘q’
     in the array ‘x’.  The space available for ‘x’ must match the
     dimension of the generator.  The point ‘x’ will lie in the range 0
     < x_i < 1 for each x_i.  An inline version of this function is used
     when ‘HAVE_INLINE’ is defined.


File: gsl-ref.info,  Node: Auxiliary quasi-random number generator functions,  Next: Saving and restoring quasi-random number generator state,  Prev: Sampling from a quasi-random number generator,  Up: Quasi-Random Sequences

19.3 Auxiliary quasi-random number generator functions
======================================================

 -- Function: const char * gsl_qrng_name (const gsl_qrng * q)

     This function returns a pointer to the name of the generator.

 -- Function: size_t gsl_qrng_size (const gsl_qrng * q)
 -- Function: void * gsl_qrng_state (const gsl_qrng * q)

     These functions return a pointer to the state of generator ‘r’ and
     its size.  You can use this information to access the state
     directly.  For example, the following code will write the state of
     a generator to a stream:

          void * state = gsl_qrng_state (q);
          size_t n = gsl_qrng_size (q);
          fwrite (state, n, 1, stream);


File: gsl-ref.info,  Node: Saving and restoring quasi-random number generator state,  Next: Quasi-random number generator algorithms,  Prev: Auxiliary quasi-random number generator functions,  Up: Quasi-Random Sequences

19.4 Saving and restoring quasi-random number generator state
=============================================================

 -- Function: int gsl_qrng_memcpy (gsl_qrng * dest, const gsl_qrng
          * src)

     This function copies the quasi-random sequence generator ‘src’ into
     the pre-existing generator ‘dest’, making ‘dest’ into an exact copy
     of ‘src’.  The two generators must be of the same type.

 -- Function: gsl_qrng * gsl_qrng_clone (const gsl_qrng * q)

     This function returns a pointer to a newly created generator which
     is an exact copy of the generator ‘q’.


File: gsl-ref.info,  Node: Quasi-random number generator algorithms,  Next: Examples<13>,  Prev: Saving and restoring quasi-random number generator state,  Up: Quasi-Random Sequences

19.5 Quasi-random number generator algorithms
=============================================

The following quasi-random sequence algorithms are available,

 -- Type: gsl_qrng_type

      -- Variable: gsl_qrng_niederreiter_2

          This generator uses the algorithm described in Bratley, Fox,
          Niederreiter, ACM Trans.  Model.  Comp.  Sim.  2, 195 (1992).
          It is valid up to 12 dimensions.

      -- Variable: gsl_qrng_sobol

          This generator uses the Sobol sequence described in Antonov,
          Saleev, USSR Comput.  Maths.  Math.  Phys.  19, 252 (1980).
          It is valid up to 40 dimensions.

      -- Variable: gsl_qrng_halton
      -- Variable: gsl_qrng_reversehalton

          These generators use the Halton and reverse Halton sequences
          described in J.H. Halton, Numerische Mathematik, 2, 84-90
          (1960) and B. Vandewoestyne and R. Cools Computational and
          Applied Mathematics, 189, 1&2, 341-361 (2006).  They are valid
          up to 1229 dimensions.


File: gsl-ref.info,  Node: Examples<13>,  Next: References,  Prev: Quasi-random number generator algorithms,  Up: Quasi-Random Sequences

19.6 Examples
=============

The following program prints the first 1024 points of the 2-dimensional
Sobol sequence.

     #include <stdio.h>
     #include <gsl/gsl_qrng.h>

     int
     main (void)
     {
       int i;
       gsl_qrng * q = gsl_qrng_alloc (gsl_qrng_sobol, 2);

       for (i = 0; i < 1024; i++)
         {
           double v[2];
           gsl_qrng_get (q, v);
           printf ("%.5f %.5f\n", v[0], v[1]);
         }

       gsl_qrng_free (q);
       return 0;
     }

Here is the output from the program:

     $ ./a.out
     0.50000 0.50000
     0.75000 0.25000
     0.25000 0.75000
     0.37500 0.37500
     0.87500 0.87500
     0.62500 0.12500
     0.12500 0.62500
     ....

It can be seen that successive points progressively fill-in the spaces
between previous points.

Fig.  %s shows the distribution in the x-y plane of the first 1024
points from the Sobol sequence,

[image src="qrng.png"]

Figure: Distribution of the first 1024 points from the quasi-random
Sobol sequence


File: gsl-ref.info,  Node: References,  Prev: Examples<13>,  Up: Quasi-Random Sequences

19.7 References
===============

The implementations of the quasi-random sequence routines are based on
the algorithms described in the following paper,

   * P. Bratley and B.L. Fox and H. Niederreiter, “Algorithm 738:
     Programs to Generate Niederreiter’s Low-discrepancy Sequences”, ACM
     Transactions on Mathematical Software, Vol.: 20, No.: 4, December,
     1994, p.: 494–495.


File: gsl-ref.info,  Node: Random Number Distributions,  Next: Statistics,  Prev: Quasi-Random Sequences,  Up: Top

20 Random Number Distributions
******************************

This chapter describes functions for generating random variates and
computing their probability distributions.  Samples from the
distributions described in this chapter can be obtained using any of the
random number generators in the library as an underlying source of
randomness.

In the simplest cases a non-uniform distribution can be obtained
analytically from the uniform distribution of a random number generator
by applying an appropriate transformation.  This method uses one call to
the random number generator.  More complicated distributions are created
by the `acceptance-rejection' method, which compares the desired
distribution against a distribution which is similar and known
analytically.  This usually requires several samples from the generator.

The library also provides cumulative distribution functions and inverse
cumulative distribution functions, sometimes referred to as quantile
functions.  The cumulative distribution functions and their inverses are
computed separately for the upper and lower tails of the distribution,
allowing full accuracy to be retained for small results.

The functions for random variates and probability density functions
described in this section are declared in ‘gsl_randist.h’.  The
corresponding cumulative distribution functions are declared in
‘gsl_cdf.h’.

Note that the discrete random variate functions always return a value of
type ‘unsigned int’, and on most platforms this has a maximum value of

     2^32-1 ~=~ 4.29e9

They should only be called with a safe range of parameters (where there
is a negligible probability of a variate exceeding this limit) to
prevent incorrect results due to overflow.

* Menu:

* Introduction: Introduction<3>.
* The Gaussian Distribution::
* The Gaussian Tail Distribution::
* The Bivariate Gaussian Distribution::
* The Multivariate Gaussian Distribution::
* The Exponential Distribution::
* The Laplace Distribution::
* The Exponential Power Distribution::
* The Cauchy Distribution::
* The Rayleigh Distribution::
* The Rayleigh Tail Distribution::
* The Landau Distribution::
* The Levy alpha-Stable Distributions::
* The Levy skew alpha-Stable Distribution::
* The Gamma Distribution::
* The Flat (Uniform) Distribution: The Flat Uniform Distribution.
* The Lognormal Distribution::
* The Chi-squared Distribution::
* The F-distribution::
* The t-distribution::
* The Beta Distribution::
* The Logistic Distribution::
* The Pareto Distribution::
* Spherical Vector Distributions::
* The Weibull Distribution::
* The Type-1 Gumbel Distribution::
* The Type-2 Gumbel Distribution::
* The Dirichlet Distribution::
* General Discrete Distributions::
* The Poisson Distribution::
* The Bernoulli Distribution::
* The Binomial Distribution::
* The Multinomial Distribution::
* The Negative Binomial Distribution::
* The Pascal Distribution::
* The Geometric Distribution::
* The Hypergeometric Distribution::
* The Logarithmic Distribution::
* The Wishart Distribution::
* Shuffling and Sampling::
* Examples: Examples<14>.
* References and Further Reading: References and Further Reading<14>.


File: gsl-ref.info,  Node: Introduction<3>,  Next: The Gaussian Distribution,  Up: Random Number Distributions

20.1 Introduction
=================

Continuous random number distributions are defined by a probability
density function, p(x), such that the probability of x occurring in the
infinitesimal range x to x + dx is p(x) dx.

The cumulative distribution function for the lower tail P(x) is defined
by the integral,

     P(x) = \int_{-\infty}^{x} dx' p(x')

and gives the probability of a variate taking a value less than x.

The cumulative distribution function for the upper tail Q(x) is defined
by the integral,

     Q(x) = \int_{x}^{+\infty} dx' p(x')

and gives the probability of a variate taking a value greater than x.

The upper and lower cumulative distribution functions are related by
P(x) + Q(x) = 1 and satisfy 0 \le P(x) \le 1, 0 \le Q(x) \le 1.

The inverse cumulative distributions, x = P^{-1}(P) and x = Q^{-1}(Q)
give the values of x which correspond to a specific value of P or Q.
They can be used to find confidence limits from probability values.

For discrete distributions the probability of sampling the integer value
k is given by p(k), where \sum_k p(k) = 1.  The cumulative distribution
for the lower tail P(k) of a discrete distribution is defined as,

     P(k) = \sum_{i \le k} p(i)

where the sum is over the allowed range of the distribution less than or
equal to k.

The cumulative distribution for the upper tail of a discrete
distribution Q(k) is defined as

     Q(k) = \sum_{i > k} p(i)

giving the sum of probabilities for all values greater than k.  These
two definitions satisfy the identity P(k)+Q(k)=1.

If the range of the distribution is 1 to n inclusive then P(n) = 1, Q(n)
= 0 while P(1) = p(1), Q(1) = 1 - p(1).


File: gsl-ref.info,  Node: The Gaussian Distribution,  Next: The Gaussian Tail Distribution,  Prev: Introduction<3>,  Up: Random Number Distributions

20.2 The Gaussian Distribution
==============================

 -- Function: double gsl_ran_gaussian (const gsl_rng * r, double sigma)

     This function returns a Gaussian random variate, with mean zero and
     standard deviation ‘sigma’.  The probability distribution for
     Gaussian random variates is,

          p(x) dx = {1 \over \sqrt{2 \pi \sigma^2}} \exp (-x^2 / 2\sigma^2) dx

     for x in the range -\infty to +\infty.  Use the transformation z =
     \mu + x on the numbers returned by *note gsl_ran_gaussian(): 6d6.
     to obtain a Gaussian distribution with mean \mu.  This function
     uses the Box-Muller algorithm which requires two calls to the
     random number generator ‘r’.

 -- Function: double gsl_ran_gaussian_pdf (double x, double sigma)

     This function computes the probability density p(x) at ‘x’ for a
     Gaussian distribution with standard deviation ‘sigma’, using the
     formula given above.

[image src="rand-gaussian.png"]

 -- Function: double gsl_ran_gaussian_ziggurat (const gsl_rng * r,
          double sigma)
 -- Function: double gsl_ran_gaussian_ratio_method (const gsl_rng * r,
          double sigma)

     This function computes a Gaussian random variate using the
     alternative Marsaglia-Tsang ziggurat and Kinderman-Monahan-Leva
     ratio methods.  The Ziggurat algorithm is the fastest available
     algorithm in most cases.

 -- Function: double gsl_ran_ugaussian (const gsl_rng * r)
 -- Function: double gsl_ran_ugaussian_pdf (double x)
 -- Function: double gsl_ran_ugaussian_ratio_method (const gsl_rng * r)

     These functions compute results for the unit Gaussian distribution.
     They are equivalent to the functions above with a standard
     deviation of one, ‘sigma’ = 1.

 -- Function: double gsl_cdf_gaussian_P (double x, double sigma)
 -- Function: double gsl_cdf_gaussian_Q (double x, double sigma)
 -- Function: double gsl_cdf_gaussian_Pinv (double P, double sigma)
 -- Function: double gsl_cdf_gaussian_Qinv (double Q, double sigma)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the Gaussian distribution with standard
     deviation ‘sigma’.

 -- Function: double gsl_cdf_ugaussian_P (double x)
 -- Function: double gsl_cdf_ugaussian_Q (double x)
 -- Function: double gsl_cdf_ugaussian_Pinv (double P)
 -- Function: double gsl_cdf_ugaussian_Qinv (double Q)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the unit Gaussian distribution.


File: gsl-ref.info,  Node: The Gaussian Tail Distribution,  Next: The Bivariate Gaussian Distribution,  Prev: The Gaussian Distribution,  Up: Random Number Distributions

20.3 The Gaussian Tail Distribution
===================================

 -- Function: double gsl_ran_gaussian_tail (const gsl_rng * r, double a,
          double sigma)

     This function provides random variates from the upper tail of a
     Gaussian distribution with standard deviation ‘sigma’.  The values
     returned are larger than the lower limit ‘a’, which must be
     positive.  The method is based on Marsaglia’s famous
     rectangle-wedge-tail algorithm (Ann.  Math.  Stat.  32, 894–899
     (1961)), with this aspect explained in Knuth, v2, 3rd ed, p139,586
     (exercise 11).

     The probability distribution for Gaussian tail random variates is,

          p(x) dx = {1 \over N(a;\sigma) \sqrt{2 \pi \sigma^2}} \exp (- x^2 / 2\sigma^2) dx

     for x > a where N(a;\sigma) is the normalization constant,

          N(a;\sigma) = (1/2) erfc(a / sqrt(2 sigma^2)).

 -- Function: double gsl_ran_gaussian_tail_pdf (double x, double a,
          double sigma)

     This function computes the probability density p(x) at ‘x’ for a
     Gaussian tail distribution with standard deviation ‘sigma’ and
     lower limit ‘a’, using the formula given above.

[image src="rand-gaussian-tail.png"]

 -- Function: double gsl_ran_ugaussian_tail (const gsl_rng * r,
          double a)
 -- Function: double gsl_ran_ugaussian_tail_pdf (double x, double a)

     These functions compute results for the tail of a unit Gaussian
     distribution.  They are equivalent to the functions above with a
     standard deviation of one, ‘sigma’ = 1.


File: gsl-ref.info,  Node: The Bivariate Gaussian Distribution,  Next: The Multivariate Gaussian Distribution,  Prev: The Gaussian Tail Distribution,  Up: Random Number Distributions

20.4 The Bivariate Gaussian Distribution
========================================

 -- Function: void gsl_ran_bivariate_gaussian (const gsl_rng * r,
          double sigma_x, double sigma_y, double rho, double * x, double
          * y)

     This function generates a pair of correlated Gaussian variates,
     with mean zero, correlation coefficient ‘rho’ and standard
     deviations ‘sigma_x’ and ‘sigma_y’ in the x and y directions.  The
     probability distribution for bivariate Gaussian random variates is,

          p(x,y) dx dy = {1 \over 2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}} \exp (-(x^2/\sigma_x^2 + y^2/\sigma_y^2 - 2 \rho x y/(\sigma_x\sigma_y))/2(1-\rho^2)) dx dy

     for x,y in the range -\infty to +\infty.  The correlation
     coefficient ‘rho’ should lie between 1 and -1.

 -- Function: double gsl_ran_bivariate_gaussian_pdf (double x, double y,
          double sigma_x, double sigma_y, double rho)

     This function computes the probability density p(x,y) at (‘x’, ‘y’)
     for a bivariate Gaussian distribution with standard deviations
     ‘sigma_x’, ‘sigma_y’ and correlation coefficient ‘rho’, using the
     formula given above.

[image src="rand-bivariate-gaussian.png"]


File: gsl-ref.info,  Node: The Multivariate Gaussian Distribution,  Next: The Exponential Distribution,  Prev: The Bivariate Gaussian Distribution,  Up: Random Number Distributions

20.5 The Multivariate Gaussian Distribution
===========================================

 -- Function: int gsl_ran_multivariate_gaussian (const gsl_rng * r,
          const gsl_vector * mu, const gsl_matrix * L, gsl_vector
          * result)

     This function generates a random vector satisfying the
     k-dimensional multivariate Gaussian distribution with mean \mu and
     variance-covariance matrix \Sigma.  On input, the k-vector \mu is
     given in ‘mu’, and the Cholesky factor of the k-by-k matrix \Sigma
     = L L^T is given in the lower triangle of ‘L’, as output from *note
     gsl_linalg_cholesky_decomp(): 536.  The random vector is stored in
     ‘result’ on output.  The probability distribution for multivariate
     Gaussian random variates is

          p(x_1,...,x_k) dx_1 ... dx_k = 1 / ( \sqrt{(2 \pi)^k |\Sigma| ) \exp (-1/2 (x - \mu)^T \Sigma^{-1} (x - \mu)) dx_1 ... dx_k

 -- Function: int gsl_ran_multivariate_gaussian_pdf (const gsl_vector
          * x, const gsl_vector * mu, const gsl_matrix * L, double
          * result, gsl_vector * work)
 -- Function: int gsl_ran_multivariate_gaussian_log_pdf (const
          gsl_vector * x, const gsl_vector * mu, const gsl_matrix * L,
          double * result, gsl_vector * work)

     These functions compute p(x) or \log{p(x)} at the point ‘x’, using
     mean vector ‘mu’ and variance-covariance matrix specified by its
     Cholesky factor ‘L’ using the formula above.  Additional workspace
     of length k is required in ‘work’.

 -- Function: int gsl_ran_multivariate_gaussian_mean (const gsl_matrix
          * X, gsl_vector * mu_hat)

     Given a set of n samples X_j from a k-dimensional multivariate
     Gaussian distribution, this function computes the maximum
     likelihood estimate of the mean of the distribution, given by

          \Hat{\mu} = {1 \over n} \sum_{j=1}^n X_j

     The samples X_1,X_2,\dots,X_n are given in the n-by-k matrix ‘X’,
     and the maximum likelihood estimate of the mean is stored in
     ‘mu_hat’ on output.

 -- Function: int gsl_ran_multivariate_gaussian_vcov (const gsl_matrix
          * X, gsl_matrix * sigma_hat)

     Given a set of n samples X_j from a k-dimensional multivariate
     Gaussian distribution, this function computes the maximum
     likelihood estimate of the variance-covariance matrix of the
     distribution, given by

          \Hat{\Sigma} = (1 / n) \sum_{j=1}^n ( X_j - \Hat{\mu} ) ( X_j - \Hat{\mu} )^T

     The samples X_1,X_2,\dots,X_n are given in the n-by-k matrix ‘X’
     and the maximum likelihood estimate of the variance-covariance
     matrix is stored in ‘sigma_hat’ on output.


File: gsl-ref.info,  Node: The Exponential Distribution,  Next: The Laplace Distribution,  Prev: The Multivariate Gaussian Distribution,  Up: Random Number Distributions

20.6 The Exponential Distribution
=================================

 -- Function: double gsl_ran_exponential (const gsl_rng * r, double mu)

     This function returns a random variate from the exponential
     distribution with mean ‘mu’.  The distribution is,

          p(x) dx = {1 \over \mu} \exp(-x/\mu) dx

     for x \ge 0.

 -- Function: double gsl_ran_exponential_pdf (double x, double mu)

     This function computes the probability density p(x) at ‘x’ for an
     exponential distribution with mean ‘mu’, using the formula given
     above.

[image src="rand-exponential.png"]

 -- Function: double gsl_cdf_exponential_P (double x, double mu)
 -- Function: double gsl_cdf_exponential_Q (double x, double mu)
 -- Function: double gsl_cdf_exponential_Pinv (double P, double mu)
 -- Function: double gsl_cdf_exponential_Qinv (double Q, double mu)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the exponential distribution with mean
     ‘mu’.


File: gsl-ref.info,  Node: The Laplace Distribution,  Next: The Exponential Power Distribution,  Prev: The Exponential Distribution,  Up: Random Number Distributions

20.7 The Laplace Distribution
=============================

 -- Function: double gsl_ran_laplace (const gsl_rng * r, double a)

     This function returns a random variate from the Laplace
     distribution with width ‘a’.  The distribution is,

          p(x) dx = {1 \over 2 a} \exp(-|x/a|) dx

     for -\infty < x < \infty.

 -- Function: double gsl_ran_laplace_pdf (double x, double a)

     This function computes the probability density p(x) at ‘x’ for a
     Laplace distribution with width ‘a’, using the formula given above.

[image src="rand-laplace.png"]

 -- Function: double gsl_cdf_laplace_P (double x, double a)
 -- Function: double gsl_cdf_laplace_Q (double x, double a)
 -- Function: double gsl_cdf_laplace_Pinv (double P, double a)
 -- Function: double gsl_cdf_laplace_Qinv (double Q, double a)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the Laplace distribution with width
     ‘a’.


File: gsl-ref.info,  Node: The Exponential Power Distribution,  Next: The Cauchy Distribution,  Prev: The Laplace Distribution,  Up: Random Number Distributions

20.8 The Exponential Power Distribution
=======================================

 -- Function: double gsl_ran_exppow (const gsl_rng * r, double a,
          double b)

     This function returns a random variate from the exponential power
     distribution with scale parameter ‘a’ and exponent ‘b’.  The
     distribution is,

          p(x) dx = {1 \over 2 a \Gamma(1+1/b)} \exp(-|x/a|^b) dx

     for x \ge 0.  For b = 1 this reduces to the Laplace distribution.
     For b = 2 it has the same form as a Gaussian distribution, but with
     a = \sqrt{2} \sigma.

 -- Function: double gsl_ran_exppow_pdf (double x, double a, double b)

     This function computes the probability density p(x) at ‘x’ for an
     exponential power distribution with scale parameter ‘a’ and
     exponent ‘b’, using the formula given above.

[image src="rand-exppow.png"]

 -- Function: double gsl_cdf_exppow_P (double x, double a, double b)
 -- Function: double gsl_cdf_exppow_Q (double x, double a, double b)

     These functions compute the cumulative distribution functions P(x),
     Q(x) for the exponential power distribution with parameters ‘a’ and
     ‘b’.


File: gsl-ref.info,  Node: The Cauchy Distribution,  Next: The Rayleigh Distribution,  Prev: The Exponential Power Distribution,  Up: Random Number Distributions

20.9 The Cauchy Distribution
============================

 -- Function: double gsl_ran_cauchy (const gsl_rng * r, double a)

     This function returns a random variate from the Cauchy distribution
     with scale parameter ‘a’.  The probability distribution for Cauchy
     random variates is,

          p(x) dx = {1 \over a\pi (1 + (x/a)^2) } dx

     for x in the range -\infty to +\infty.  The Cauchy distribution is
     also known as the Lorentz distribution.

 -- Function: double gsl_ran_cauchy_pdf (double x, double a)

     This function computes the probability density p(x) at ‘x’ for a
     Cauchy distribution with scale parameter ‘a’, using the formula
     given above.

[image src="rand-cauchy.png"]

 -- Function: double gsl_cdf_cauchy_P (double x, double a)
 -- Function: double gsl_cdf_cauchy_Q (double x, double a)
 -- Function: double gsl_cdf_cauchy_Pinv (double P, double a)
 -- Function: double gsl_cdf_cauchy_Qinv (double Q, double a)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the Cauchy distribution with scale
     parameter ‘a’.


File: gsl-ref.info,  Node: The Rayleigh Distribution,  Next: The Rayleigh Tail Distribution,  Prev: The Cauchy Distribution,  Up: Random Number Distributions

20.10 The Rayleigh Distribution
===============================

 -- Function: double gsl_ran_rayleigh (const gsl_rng * r, double sigma)

     This function returns a random variate from the Rayleigh
     distribution with scale parameter ‘sigma’.  The distribution is,

          p(x) dx = {x \over \sigma^2} \exp(- x^2/(2 \sigma^2)) dx

     for x > 0.

 -- Function: double gsl_ran_rayleigh_pdf (double x, double sigma)

     This function computes the probability density p(x) at ‘x’ for a
     Rayleigh distribution with scale parameter ‘sigma’, using the
     formula given above.

[image src="rand-rayleigh.png"]

 -- Function: double gsl_cdf_rayleigh_P (double x, double sigma)
 -- Function: double gsl_cdf_rayleigh_Q (double x, double sigma)
 -- Function: double gsl_cdf_rayleigh_Pinv (double P, double sigma)
 -- Function: double gsl_cdf_rayleigh_Qinv (double Q, double sigma)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the Rayleigh distribution with scale
     parameter ‘sigma’.


File: gsl-ref.info,  Node: The Rayleigh Tail Distribution,  Next: The Landau Distribution,  Prev: The Rayleigh Distribution,  Up: Random Number Distributions

20.11 The Rayleigh Tail Distribution
====================================

 -- Function: double gsl_ran_rayleigh_tail (const gsl_rng * r, double a,
          double sigma)

     This function returns a random variate from the tail of the
     Rayleigh distribution with scale parameter ‘sigma’ and a lower
     limit of ‘a’.  The distribution is,

          p(x) dx = {x \over \sigma^2} \exp ((a^2 - x^2) /(2 \sigma^2)) dx

     for x > a.

 -- Function: double gsl_ran_rayleigh_tail_pdf (double x, double a,
          double sigma)

     This function computes the probability density p(x) at ‘x’ for a
     Rayleigh tail distribution with scale parameter ‘sigma’ and lower
     limit ‘a’, using the formula given above.

[image src="rand-rayleigh-tail.png"]


File: gsl-ref.info,  Node: The Landau Distribution,  Next: The Levy alpha-Stable Distributions,  Prev: The Rayleigh Tail Distribution,  Up: Random Number Distributions

20.12 The Landau Distribution
=============================

 -- Function: double gsl_ran_landau (const gsl_rng * r)

     This function returns a random variate from the Landau
     distribution.  The probability distribution for Landau random
     variates is defined analytically by the complex integral,

          p(x) = (1/(2 \pi i)) \int_{c-i\infty}^{c+i\infty} ds exp(s log(s) + x s)

     For numerical purposes it is more convenient to use the following
     equivalent form of the integral,

          p(x) = (1/\pi) \int_0^\infty dt \exp(-t \log(t) - x t) \sin(\pi t).

 -- Function: double gsl_ran_landau_pdf (double x)

     This function computes the probability density p(x) at ‘x’ for the
     Landau distribution using an approximation to the formula given
     above.

[image src="rand-landau.png"]


File: gsl-ref.info,  Node: The Levy alpha-Stable Distributions,  Next: The Levy skew alpha-Stable Distribution,  Prev: The Landau Distribution,  Up: Random Number Distributions

20.13 The Levy alpha-Stable Distributions
=========================================

 -- Function: double gsl_ran_levy (const gsl_rng * r, double c,
          double alpha)

     This function returns a random variate from the Levy symmetric
     stable distribution with scale ‘c’ and exponent *note alpha: 49a.
     The symmetric stable probability distribution is defined by a
     Fourier transform,

          p(x) = 1 / (2 \pi) \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^alpha)

     There is no explicit solution for the form of p(x) and the library
     does not define a corresponding ‘pdf’ function.  For \alpha = 1 the
     distribution reduces to the Cauchy distribution.  For \alpha = 2 it
     is a Gaussian distribution with \sigma = \sqrt{2} c.  For \alpha <
     1 the tails of the distribution become extremely wide.

     The algorithm only works for 0 < \alpha \le 2.

[image src="rand-levy.png"]


File: gsl-ref.info,  Node: The Levy skew alpha-Stable Distribution,  Next: The Gamma Distribution,  Prev: The Levy alpha-Stable Distributions,  Up: Random Number Distributions

20.14 The Levy skew alpha-Stable Distribution
=============================================

 -- Function: double gsl_ran_levy_skew (const gsl_rng * r, double c,
          double alpha, double beta)

     This function returns a random variate from the Levy skew stable
     distribution with scale ‘c’, exponent *note alpha: 49a. and
     skewness parameter ‘beta’.  The skewness parameter must lie in the
     range [-1,1].  The Levy skew stable probability distribution is
     defined by a Fourier transform,

          p(x) = 1 / (2 \pi) \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^alpha (1-i beta sign(t) tan(pi alpha/2)))

     When \alpha = 1 the term \tan(\pi \alpha/2) is replaced by
     -(2/\pi)\log|t|.  There is no explicit solution for the form of
     p(x) and the library does not define a corresponding ‘pdf’
     function.  For \alpha = 2 the distribution reduces to a Gaussian
     distribution with \sigma = \sqrt{2} c and the skewness parameter
     has no effect.  For \alpha < 1 the tails of the distribution become
     extremely wide.  The symmetric distribution corresponds to \beta =
     0.

     The algorithm only works for 0 < \alpha \le 2.

The Levy alpha-stable distributions have the property that if N
alpha-stable variates are drawn from the distribution p(c, \alpha,
\beta) then the sum Y = X_1 + X_2 + \dots + X_N will also be distributed
as an alpha-stable variate, p(N^{1/\alpha} c, \alpha, \beta).

[image src="rand-levyskew.png"]


File: gsl-ref.info,  Node: The Gamma Distribution,  Next: The Flat Uniform Distribution,  Prev: The Levy skew alpha-Stable Distribution,  Up: Random Number Distributions

20.15 The Gamma Distribution
============================

 -- Function: double gsl_ran_gamma (const gsl_rng * r, double a,
          double b)

     This function returns a random variate from the gamma distribution.
     The distribution function is,

          p(x) dx = {1 \over \Gamma(a) b^a} x^{a-1} e^{-x/b} dx

     for x > 0.

     The gamma distribution with an integer parameter ‘a’ is known as
     the Erlang distribution.

     The variates are computed using the Marsaglia-Tsang fast gamma
     method.  This function for this method was previously called
     ‘gsl_ran_gamma_mt()’ and can still be accessed using this name.

 -- Function: double gsl_ran_gamma_knuth (const gsl_rng * r, double a,
          double b)

     This function returns a gamma variate using the algorithms from
     Knuth (vol 2).

 -- Function: double gsl_ran_gamma_pdf (double x, double a, double b)

     This function computes the probability density p(x) at ‘x’ for a
     gamma distribution with parameters ‘a’ and ‘b’, using the formula
     given above.

[image src="rand-gamma.png"]

 -- Function: double gsl_cdf_gamma_P (double x, double a, double b)
 -- Function: double gsl_cdf_gamma_Q (double x, double a, double b)
 -- Function: double gsl_cdf_gamma_Pinv (double P, double a, double b)
 -- Function: double gsl_cdf_gamma_Qinv (double Q, double a, double b)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the gamma distribution with parameters
     ‘a’ and ‘b’.


File: gsl-ref.info,  Node: The Flat Uniform Distribution,  Next: The Lognormal Distribution,  Prev: The Gamma Distribution,  Up: Random Number Distributions

20.16 The Flat (Uniform) Distribution
=====================================

 -- Function: double gsl_ran_flat (const gsl_rng * r, double a,
          double b)

     This function returns a random variate from the flat (uniform)
     distribution from ‘a’ to ‘b’.  The distribution is,

          p(x) dx = {1 \over (b-a)} dx

     if a \le x < b and 0 otherwise.

 -- Function: double gsl_ran_flat_pdf (double x, double a, double b)

     This function computes the probability density p(x) at ‘x’ for a
     uniform distribution from ‘a’ to ‘b’, using the formula given
     above.

[image src="rand-flat.png"]

 -- Function: double gsl_cdf_flat_P (double x, double a, double b)
 -- Function: double gsl_cdf_flat_Q (double x, double a, double b)
 -- Function: double gsl_cdf_flat_Pinv (double P, double a, double b)
 -- Function: double gsl_cdf_flat_Qinv (double Q, double a, double b)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for a uniform distribution from ‘a’ to ‘b’.


File: gsl-ref.info,  Node: The Lognormal Distribution,  Next: The Chi-squared Distribution,  Prev: The Flat Uniform Distribution,  Up: Random Number Distributions

20.17 The Lognormal Distribution
================================

 -- Function: double gsl_ran_lognormal (const gsl_rng * r, double zeta,
          double sigma)

     This function returns a random variate from the lognormal
     distribution.  The distribution function is,

          p(x) dx = {1 \over x \sqrt{2 \pi \sigma^2}} \exp(-(\ln(x) - \zeta)^2/2 \sigma^2) dx

     for x > 0.

 -- Function: double gsl_ran_lognormal_pdf (double x, double zeta,
          double sigma)

     This function computes the probability density p(x) at ‘x’ for a
     lognormal distribution with parameters ‘zeta’ and ‘sigma’, using
     the formula given above.

[image src="rand-lognormal.png"]

 -- Function: double gsl_cdf_lognormal_P (double x, double zeta,
          double sigma)
 -- Function: double gsl_cdf_lognormal_Q (double x, double zeta,
          double sigma)
 -- Function: double gsl_cdf_lognormal_Pinv (double P, double zeta,
          double sigma)
 -- Function: double gsl_cdf_lognormal_Qinv (double Q, double zeta,
          double sigma)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the lognormal distribution with
     parameters ‘zeta’ and ‘sigma’.


File: gsl-ref.info,  Node: The Chi-squared Distribution,  Next: The F-distribution,  Prev: The Lognormal Distribution,  Up: Random Number Distributions

20.18 The Chi-squared Distribution
==================================

The chi-squared distribution arises in statistics.  If Y_i are n
independent Gaussian random variates with unit variance then the
sum-of-squares,

     X_i = \sum_i Y_i^2

has a chi-squared distribution with n degrees of freedom.

 -- Function: double gsl_ran_chisq (const gsl_rng * r, double nu)

     This function returns a random variate from the chi-squared
     distribution with ‘nu’ degrees of freedom.  The distribution
     function is,

          p(x) dx = {1 \over 2 \Gamma(\nu/2) } (x/2)^{\nu/2 - 1} \exp(-x/2) dx

     for x \ge 0.

 -- Function: double gsl_ran_chisq_pdf (double x, double nu)

     This function computes the probability density p(x) at ‘x’ for a
     chi-squared distribution with ‘nu’ degrees of freedom, using the
     formula given above.

[image src="rand-chisq.png"]

 -- Function: double gsl_cdf_chisq_P (double x, double nu)
 -- Function: double gsl_cdf_chisq_Q (double x, double nu)
 -- Function: double gsl_cdf_chisq_Pinv (double P, double nu)
 -- Function: double gsl_cdf_chisq_Qinv (double Q, double nu)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the chi-squared distribution with ‘nu’
     degrees of freedom.


File: gsl-ref.info,  Node: The F-distribution,  Next: The t-distribution,  Prev: The Chi-squared Distribution,  Up: Random Number Distributions

20.19 The F-distribution
========================

The F-distribution arises in statistics.  If Y_1 and Y_2 are chi-squared
deviates with \nu_1 and \nu_2 degrees of freedom then the ratio,

     X = { (Y_1 / \nu_1) \over (Y_2 / \nu_2) }

has an F-distribution F(x;\nu_1,\nu_2).

 -- Function: double gsl_ran_fdist (const gsl_rng * r, double nu1,
          double nu2)

     This function returns a random variate from the F-distribution with
     degrees of freedom ‘nu1’ and ‘nu2’.  The distribution function is,

          p(x) dx =
             { \Gamma((\nu_1 + \nu_2)/2)
                  \over \Gamma(\nu_1/2) \Gamma(\nu_2/2) }
             \nu_1^{\nu_1/2} \nu_2^{\nu_2/2}
             x^{\nu_1/2 - 1} (\nu_2 + \nu_1 x)^{-\nu_1/2 -\nu_2/2}

     for x \ge 0.

 -- Function: double gsl_ran_fdist_pdf (double x, double nu1,
          double nu2)

     This function computes the probability density p(x) at ‘x’ for an
     F-distribution with ‘nu1’ and ‘nu2’ degrees of freedom, using the
     formula given above.

[image src="rand-fdist.png"]

 -- Function: double gsl_cdf_fdist_P (double x, double nu1, double nu2)
 -- Function: double gsl_cdf_fdist_Q (double x, double nu1, double nu2)
 -- Function: double gsl_cdf_fdist_Pinv (double P, double nu1,
          double nu2)
 -- Function: double gsl_cdf_fdist_Qinv (double Q, double nu1,
          double nu2)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the F-distribution with ‘nu1’ and ‘nu2’
     degrees of freedom.


File: gsl-ref.info,  Node: The t-distribution,  Next: The Beta Distribution,  Prev: The F-distribution,  Up: Random Number Distributions

20.20 The t-distribution
========================

The t-distribution arises in statistics.  If Y_1 has a normal
distribution and Y_2 has a chi-squared distribution with \nu degrees of
freedom then the ratio,

     X = { Y_1 \over \sqrt{Y_2 / \nu} }

has a t-distribution t(x;\nu) with \nu degrees of freedom.

 -- Function: double gsl_ran_tdist (const gsl_rng * r, double nu)

     This function returns a random variate from the t-distribution.
     The distribution function is,

          p(x) dx = {\Gamma((\nu + 1)/2) \over \sqrt{\pi \nu} \Gamma(\nu/2)} (1 + x^2/\nu)^{-(\nu + 1)/2} dx

     for -\infty < x < +\infty.

 -- Function: double gsl_ran_tdist_pdf (double x, double nu)

     This function computes the probability density p(x) at ‘x’ for a
     t-distribution with ‘nu’ degrees of freedom, using the formula
     given above.

[image src="rand-tdist.png"]

 -- Function: double gsl_cdf_tdist_P (double x, double nu)
 -- Function: double gsl_cdf_tdist_Q (double x, double nu)
 -- Function: double gsl_cdf_tdist_Pinv (double P, double nu)
 -- Function: double gsl_cdf_tdist_Qinv (double Q, double nu)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the t-distribution with ‘nu’ degrees of
     freedom.


File: gsl-ref.info,  Node: The Beta Distribution,  Next: The Logistic Distribution,  Prev: The t-distribution,  Up: Random Number Distributions

20.21 The Beta Distribution
===========================

 -- Function: double gsl_ran_beta (const gsl_rng * r, double a,
          double b)

     This function returns a random variate from the beta distribution.
     The distribution function is,

          p(x) dx = {\Gamma(a+b) \over \Gamma(a) \Gamma(b)} x^{a-1} (1-x)^{b-1} dx

     for 0 \le x \le 1.

 -- Function: double gsl_ran_beta_pdf (double x, double a, double b)

     This function computes the probability density p(x) at ‘x’ for a
     beta distribution with parameters ‘a’ and ‘b’, using the formula
     given above.

[image src="rand-beta.png"]

 -- Function: double gsl_cdf_beta_P (double x, double a, double b)
 -- Function: double gsl_cdf_beta_Q (double x, double a, double b)
 -- Function: double gsl_cdf_beta_Pinv (double P, double a, double b)
 -- Function: double gsl_cdf_beta_Qinv (double Q, double a, double b)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the beta distribution with parameters
     ‘a’ and ‘b’.


File: gsl-ref.info,  Node: The Logistic Distribution,  Next: The Pareto Distribution,  Prev: The Beta Distribution,  Up: Random Number Distributions

20.22 The Logistic Distribution
===============================

 -- Function: double gsl_ran_logistic (const gsl_rng * r, double a)

     This function returns a random variate from the logistic
     distribution.  The distribution function is,

          p(x) dx = { \exp(-x/a) \over a (1 + \exp(-x/a))^2 } dx

     for -\infty < x < +\infty.

 -- Function: double gsl_ran_logistic_pdf (double x, double a)

     This function computes the probability density p(x) at ‘x’ for a
     logistic distribution with scale parameter ‘a’, using the formula
     given above.

[image src="rand-logistic.png"]

 -- Function: double gsl_cdf_logistic_P (double x, double a)
 -- Function: double gsl_cdf_logistic_Q (double x, double a)
 -- Function: double gsl_cdf_logistic_Pinv (double P, double a)
 -- Function: double gsl_cdf_logistic_Qinv (double Q, double a)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the logistic distribution with scale
     parameter ‘a’.


File: gsl-ref.info,  Node: The Pareto Distribution,  Next: Spherical Vector Distributions,  Prev: The Logistic Distribution,  Up: Random Number Distributions

20.23 The Pareto Distribution
=============================

 -- Function: double gsl_ran_pareto (const gsl_rng * r, double a,
          double b)

     This function returns a random variate from the Pareto distribution
     of order ‘a’.  The distribution function is,

          p(x) dx = (a/b) / (x/b)^{a+1} dx

     for x \ge b.

 -- Function: double gsl_ran_pareto_pdf (double x, double a, double b)

     This function computes the probability density p(x) at ‘x’ for a
     Pareto distribution with exponent ‘a’ and scale ‘b’, using the
     formula given above.

[image src="rand-pareto.png"]

 -- Function: double gsl_cdf_pareto_P (double x, double a, double b)
 -- Function: double gsl_cdf_pareto_Q (double x, double a, double b)
 -- Function: double gsl_cdf_pareto_Pinv (double P, double a, double b)
 -- Function: double gsl_cdf_pareto_Qinv (double Q, double a, double b)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the Pareto distribution with exponent
     ‘a’ and scale ‘b’.


File: gsl-ref.info,  Node: Spherical Vector Distributions,  Next: The Weibull Distribution,  Prev: The Pareto Distribution,  Up: Random Number Distributions

20.24 Spherical Vector Distributions
====================================

The spherical distributions generate random vectors, located on a
spherical surface.  They can be used as random directions, for example
in the steps of a random walk.

 -- Function: void gsl_ran_dir_2d (const gsl_rng * r, double * x, double
          * y)
 -- Function: void gsl_ran_dir_2d_trig_method (const gsl_rng * r, double
          * x, double * y)

     This function returns a random direction vector v = (‘x’, ‘y’) in
     two dimensions.  The vector is normalized such that |v|^2 = x^2 +
     y^2 = 1.  The obvious way to do this is to take a uniform random
     number between 0 and 2\pi and let ‘x’ and ‘y’ be the sine and
     cosine respectively.  Two trig functions would have been expensive
     in the old days, but with modern hardware implementations, this is
     sometimes the fastest way to go.  This is the case for the Pentium
     (but not the case for the Sun Sparcstation).  One can avoid the
     trig evaluations by choosing ‘x’ and ‘y’ in the interior of a unit
     circle (choose them at random from the interior of the enclosing
     square, and then reject those that are outside the unit circle),
     and then dividing by \sqrt{x^2 + y^2}.  A much cleverer approach,
     attributed to von Neumann (See Knuth, v2, 3rd ed, p140, exercise
     23), requires neither trig nor a square root.  In this approach,
     ‘u’ and ‘v’ are chosen at random from the interior of a unit
     circle, and then x=(u^2-v^2)/(u^2+v^2) and y=2uv/(u^2+v^2).

 -- Function: void gsl_ran_dir_3d (const gsl_rng * r, double * x, double
          * y, double * z)

     This function returns a random direction vector v = (‘x’, ‘y’, ‘z’)
     in three dimensions.  The vector is normalized such that |v|^2 =
     x^2 + y^2 + z^2 = 1.  The method employed is due to Robert E. Knop
     (CACM 13, 326 (1970)), and explained in Knuth, v2, 3rd ed, p136.
     It uses the surprising fact that the distribution projected along
     any axis is actually uniform (this is only true for 3 dimensions).

 -- Function: void gsl_ran_dir_nd (const gsl_rng * r, size_t n, double
          * x)

     This function returns a random direction vector v =
     (x_1,x_2,\ldots,x_n) in ‘n’ dimensions.  The vector is normalized
     such that |v|^2 = x_1^2 + x_2^2 + \cdots + x_n^2 = 1.  The method
     uses the fact that a multivariate Gaussian distribution is
     spherically symmetric.  Each component is generated to have a
     Gaussian distribution, and then the components are normalized.  The
     method is described by Knuth, v2, 3rd ed, p135–136, and attributed
     to G. W. Brown, Modern Mathematics for the Engineer (1956).


File: gsl-ref.info,  Node: The Weibull Distribution,  Next: The Type-1 Gumbel Distribution,  Prev: Spherical Vector Distributions,  Up: Random Number Distributions

20.25 The Weibull Distribution
==============================

 -- Function: double gsl_ran_weibull (const gsl_rng * r, double a,
          double b)

     This function returns a random variate from the Weibull
     distribution.  The distribution function is,

          p(x) dx = {b \over a^b} x^{b-1} \exp(-(x/a)^b) dx

     for x \ge 0.

 -- Function: double gsl_ran_weibull_pdf (double x, double a, double b)

     This function computes the probability density p(x) at ‘x’ for a
     Weibull distribution with scale ‘a’ and exponent ‘b’, using the
     formula given above.

[image src="rand-weibull.png"]

 -- Function: double gsl_cdf_weibull_P (double x, double a, double b)
 -- Function: double gsl_cdf_weibull_Q (double x, double a, double b)
 -- Function: double gsl_cdf_weibull_Pinv (double P, double a, double b)
 -- Function: double gsl_cdf_weibull_Qinv (double Q, double a, double b)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the Weibull distribution with scale ‘a’
     and exponent ‘b’.


File: gsl-ref.info,  Node: The Type-1 Gumbel Distribution,  Next: The Type-2 Gumbel Distribution,  Prev: The Weibull Distribution,  Up: Random Number Distributions

20.26 The Type-1 Gumbel Distribution
====================================

 -- Function: double gsl_ran_gumbel1 (const gsl_rng * r, double a,
          double b)

     This function returns a random variate from the Type-1 Gumbel
     distribution.  The Type-1 Gumbel distribution function is,

          p(x) dx = a b \exp(-(b \exp(-ax) + ax)) dx

     for -\infty < x < \infty.

 -- Function: double gsl_ran_gumbel1_pdf (double x, double a, double b)

     This function computes the probability density p(x) at ‘x’ for a
     Type-1 Gumbel distribution with parameters ‘a’ and ‘b’, using the
     formula given above.

[image src="rand-gumbel1.png"]

 -- Function: double gsl_cdf_gumbel1_P (double x, double a, double b)
 -- Function: double gsl_cdf_gumbel1_Q (double x, double a, double b)
 -- Function: double gsl_cdf_gumbel1_Pinv (double P, double a, double b)
 -- Function: double gsl_cdf_gumbel1_Qinv (double Q, double a, double b)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the Type-1 Gumbel distribution with
     parameters ‘a’ and ‘b’.


File: gsl-ref.info,  Node: The Type-2 Gumbel Distribution,  Next: The Dirichlet Distribution,  Prev: The Type-1 Gumbel Distribution,  Up: Random Number Distributions

20.27 The Type-2 Gumbel Distribution
====================================

 -- Function: double gsl_ran_gumbel2 (const gsl_rng * r, double a,
          double b)

     This function returns a random variate from the Type-2 Gumbel
     distribution.  The Type-2 Gumbel distribution function is,

          p(x) dx = a b x^{-a-1} \exp(-b x^{-a}) dx

     for 0 < x < \infty.

 -- Function: double gsl_ran_gumbel2_pdf (double x, double a, double b)

     This function computes the probability density p(x) at ‘x’ for a
     Type-2 Gumbel distribution with parameters ‘a’ and ‘b’, using the
     formula given above.

[image src="rand-gumbel2.png"]

 -- Function: double gsl_cdf_gumbel2_P (double x, double a, double b)
 -- Function: double gsl_cdf_gumbel2_Q (double x, double a, double b)
 -- Function: double gsl_cdf_gumbel2_Pinv (double P, double a, double b)
 -- Function: double gsl_cdf_gumbel2_Qinv (double Q, double a, double b)

     These functions compute the cumulative distribution functions P(x),
     Q(x) and their inverses for the Type-2 Gumbel distribution with
     parameters ‘a’ and ‘b’.


File: gsl-ref.info,  Node: The Dirichlet Distribution,  Next: General Discrete Distributions,  Prev: The Type-2 Gumbel Distribution,  Up: Random Number Distributions

20.28 The Dirichlet Distribution
================================

 -- Function: void gsl_ran_dirichlet (const gsl_rng * r, size_t K, const
          double alpha[], double theta[])

     This function returns an array of ‘K’ random variates from a
     Dirichlet distribution of order ‘K’-1.  The distribution function
     is

          p(\theta_1, ..., \theta_K) d\theta_1 ... d\theta_K =
            (1/Z) \prod_{i=1}^K \theta_i^{\alpha_i - 1} \delta(1 -\sum_{i=1}^K \theta_i) d\theta_1 ... d\theta_K

     for \theta_i \ge 0 and \alpha_i > 0.  The delta function ensures
     that \sum \theta_i = 1.  The normalization factor Z is

          Z = {\prod_{i=1}^K \Gamma(\alpha_i) \over \Gamma( \sum_{i=1}^K \alpha_i)}

     The random variates are generated by sampling ‘K’ values from gamma
     distributions with parameters a=\alpha_i$, $b=1, and renormalizing.
     See A.M. Law, W.D. Kelton, `Simulation Modeling and Analysis'
     (1991).

 -- Function: double gsl_ran_dirichlet_pdf (size_t K, const
          double alpha[], const double theta[])

     This function computes the probability density p(\theta_1, \ldots ,
     \theta_K) at ‘theta[K]’ for a Dirichlet distribution with
     parameters ‘alpha[K]’, using the formula given above.

 -- Function: double gsl_ran_dirichlet_lnpdf (size_t K, const
          double alpha[], const double theta[])

     This function computes the logarithm of the probability density
     p(\theta_1, \ldots , \theta_K) for a Dirichlet distribution with
     parameters ‘alpha[K]’.


File: gsl-ref.info,  Node: General Discrete Distributions,  Next: The Poisson Distribution,  Prev: The Dirichlet Distribution,  Up: Random Number Distributions

20.29 General Discrete Distributions
====================================

Given K discrete events with different probabilities P[k], produce a
random value k consistent with its probability.

The obvious way to do this is to preprocess the probability list by
generating a cumulative probability array with K + 1 elements:

     C[0] = 0
     C[k+1] = C[k] + P[k]

Note that this construction produces C[K] = 1.  Now choose a uniform
deviate u between 0 and 1, and find the value of k such that C[k] \le u
< C[k+1].  Although this in principle requires of order \log K steps per
random number generation, they are fast steps, and if you use something
like \lfloor uK \rfloor as a starting point, you can often do pretty
well.

But faster methods have been devised.  Again, the idea is to preprocess
the probability list, and save the result in some form of lookup table;
then the individual calls for a random discrete event can go rapidly.
An approach invented by G. Marsaglia (Generating discrete random
variables in a computer, Comm ACM 6, 37–38 (1963)) is very clever, and
readers interested in examples of good algorithm design are directed to
this short and well-written paper.  Unfortunately, for large K,
Marsaglia’s lookup table can be quite large.

A much better approach is due to Alastair J. Walker (An efficient method
for generating discrete random variables with general distributions, ACM
Trans on Mathematical Software 3, 253–256 (1977); see also Knuth, v2,
3rd ed, p120–121,139).  This requires two lookup tables, one floating
point and one integer, but both only of size K. After preprocessing, the
random numbers are generated in O(1) time, even for large K. The
preprocessing suggested by Walker requires O(K^2) effort, but that is
not actually necessary, and the implementation provided here only takes
O(K) effort.  In general, more preprocessing leads to faster generation
of the individual random numbers, but a diminishing return is reached
pretty early.  Knuth points out that the optimal preprocessing is
combinatorially difficult for large K.

This method can be used to speed up some of the discrete random number
generators below, such as the binomial distribution.  To use it for
something like the Poisson Distribution, a modification would have to be
made, since it only takes a finite set of K outcomes.

 -- Type: gsl_ran_discrete_t

     This structure contains the lookup table for the discrete random
     number generator.

 -- Function: gsl_ran_discrete_t * gsl_ran_discrete_preproc (size_t K,
          const double * P)

     This function returns a pointer to a structure that contains the
     lookup table for the discrete random number generator.  The array
     ‘P’ contains the probabilities of the discrete events; these array
     elements must all be positive, but they needn’t add up to one (so
     you can think of them more generally as “weights”)—the preprocessor
     will normalize appropriately.  This return value is used as an
     argument for the *note gsl_ran_discrete(): 77f. function below.

 -- Function: size_t gsl_ran_discrete (const gsl_rng * r, const
          gsl_ran_discrete_t * g)

     After the preprocessor, above, has been called, you use this
     function to get the discrete random numbers.

 -- Function: double gsl_ran_discrete_pdf (size_t k, const
          gsl_ran_discrete_t * g)

     Returns the probability P[k] of observing the variable ‘k’.  Since
     P[k] is not stored as part of the lookup table, it must be
     recomputed; this computation takes O(K), so if ‘K’ is large and you
     care about the original array P[k] used to create the lookup table,
     then you should just keep this original array P[k] around.

 -- Function: void gsl_ran_discrete_free (gsl_ran_discrete_t * g)

     De-allocates the lookup table pointed to by ‘g’.


File: gsl-ref.info,  Node: The Poisson Distribution,  Next: The Bernoulli Distribution,  Prev: General Discrete Distributions,  Up: Random Number Distributions

20.30 The Poisson Distribution
==============================

 -- Function: unsigned int gsl_ran_poisson (const gsl_rng * r,
          double mu)

     This function returns a random integer from the Poisson
     distribution with mean ‘mu’.  The probability distribution for
     Poisson variates is,

          p(k) = {\mu^k \over k!} \exp(-\mu)

     for k \ge 0.

 -- Function: double gsl_ran_poisson_pdf (unsigned int k, double mu)

     This function computes the probability p(k) of obtaining ‘k’ from a
     Poisson distribution with mean ‘mu’, using the formula given above.

[image src="rand-poisson.png"]

 -- Function: double gsl_cdf_poisson_P (unsigned int k, double mu)
 -- Function: double gsl_cdf_poisson_Q (unsigned int k, double mu)

     These functions compute the cumulative distribution functions P(k),
     Q(k) for the Poisson distribution with parameter ‘mu’.


File: gsl-ref.info,  Node: The Bernoulli Distribution,  Next: The Binomial Distribution,  Prev: The Poisson Distribution,  Up: Random Number Distributions

20.31 The Bernoulli Distribution
================================

 -- Function: unsigned int gsl_ran_bernoulli (const gsl_rng * r,
          double p)

     This function returns either 0 or 1, the result of a Bernoulli
     trial with probability ‘p’.  The probability distribution for a
     Bernoulli trial is,

          p(0) = 1 - p
          p(1) = p

 -- Function: double gsl_ran_bernoulli_pdf (unsigned int k, double p)

     This function computes the probability p(k) of obtaining ‘k’ from a
     Bernoulli distribution with probability parameter ‘p’, using the
     formula given above.

[image src="rand-bernoulli.png"]


File: gsl-ref.info,  Node: The Binomial Distribution,  Next: The Multinomial Distribution,  Prev: The Bernoulli Distribution,  Up: Random Number Distributions

20.32 The Binomial Distribution
===============================

 -- Function: unsigned int gsl_ran_binomial (const gsl_rng * r,
          double p, unsigned int n)

     This function returns a random integer from the binomial
     distribution, the number of successes in ‘n’ independent trials
     with probability ‘p’.  The probability distribution for binomial
     variates is,

          p(k) = {n! \over k! (n-k)!} p^k (1-p)^{n-k}

     for 0 \le k \le n.

 -- Function: double gsl_ran_binomial_pdf (unsigned int k, double p,
          unsigned int n)

     This function computes the probability p(k) of obtaining ‘k’ from a
     binomial distribution with parameters ‘p’ and ‘n’, using the
     formula given above.

[image src="rand-binomial.png"]

 -- Function: double gsl_cdf_binomial_P (unsigned int k, double p,
          unsigned int n)
 -- Function: double gsl_cdf_binomial_Q (unsigned int k, double p,
          unsigned int n)

     These functions compute the cumulative distribution functions P(k),
     Q(k) for the binomial distribution with parameters ‘p’ and ‘n’.


File: gsl-ref.info,  Node: The Multinomial Distribution,  Next: The Negative Binomial Distribution,  Prev: The Binomial Distribution,  Up: Random Number Distributions

20.33 The Multinomial Distribution
==================================

 -- Function: void gsl_ran_multinomial (const gsl_rng * r, size_t K,
          unsigned int N, const double p[], unsigned int n[])

     This function computes a random sample ‘n’ from the multinomial
     distribution formed by ‘N’ trials from an underlying distribution
     ‘p[K]’.  The distribution function for ‘n’ is,

          P(n_1, n_2, ..., n_K) =
            (N!/(n_1! n_2! ... n_K!)) p_1^n_1 p_2^n_2 ... p_K^n_K

     where (n_1, n_2, \ldots, n_K) are nonnegative integers with
     \sum_{k=1}^{K} n_k = N, and (p_1, p_2, \ldots, p_K) is a
     probability distribution with \sum p_i = 1.  If the array ‘p[K]’ is
     not normalized then its entries will be treated as weights and
     normalized appropriately.  The arrays ‘n’ and ‘p’ must both be of
     length ‘K’.

     Random variates are generated using the conditional binomial method
     (see C.S. Davis, `The computer generation of multinomial random
     variates', Comp.  Stat.  Data Anal.  16 (1993) 205–217 for
     details).

 -- Function: double gsl_ran_multinomial_pdf (size_t K, const
          double p[], const unsigned int n[])

     This function computes the probability P(n_1, n_2, \ldots, n_K) of
     sampling ‘n[K]’ from a multinomial distribution with parameters
     ‘p[K]’, using the formula given above.

 -- Function: double gsl_ran_multinomial_lnpdf (size_t K, const
          double p[], const unsigned int n[])

     This function returns the logarithm of the probability for the
     multinomial distribution P(n_1, n_2, \ldots, n_K) with parameters
     ‘p[K]’.


File: gsl-ref.info,  Node: The Negative Binomial Distribution,  Next: The Pascal Distribution,  Prev: The Multinomial Distribution,  Up: Random Number Distributions

20.34 The Negative Binomial Distribution
========================================

 -- Function: unsigned int gsl_ran_negative_binomial (const gsl_rng * r,
          double p, double n)

     This function returns a random integer from the negative binomial
     distribution, the number of failures occurring before ‘n’ successes
     in independent trials with probability ‘p’ of success.  The
     probability distribution for negative binomial variates is,

          p(k) = {\Gamma(n + k) \over \Gamma(k+1) \Gamma(n) } p^n (1-p)^k

     Note that n is not required to be an integer.

 -- Function: double gsl_ran_negative_binomial_pdf (unsigned int k,
          double p, double  n)

     This function computes the probability p(k) of obtaining ‘k’ from a
     negative binomial distribution with parameters ‘p’ and ‘n’, using
     the formula given above.

[image src="rand-nbinomial.png"]

 -- Function: double gsl_cdf_negative_binomial_P (unsigned int k,
          double p, double n)
 -- Function: double gsl_cdf_negative_binomial_Q (unsigned int k,
          double p, double n)

     These functions compute the cumulative distribution functions P(k),
     Q(k) for the negative binomial distribution with parameters ‘p’ and
     ‘n’.


File: gsl-ref.info,  Node: The Pascal Distribution,  Next: The Geometric Distribution,  Prev: The Negative Binomial Distribution,  Up: Random Number Distributions

20.35 The Pascal Distribution
=============================

 -- Function: unsigned int gsl_ran_pascal (const gsl_rng * r, double p,
          unsigned int n)

     This function returns a random integer from the Pascal
     distribution.  The Pascal distribution is simply a negative
     binomial distribution with an integer value of n.

          p(k) = {(n + k - 1)! \over k! (n - 1)! } p^n (1-p)^k

     for k \ge 0.

 -- Function: double gsl_ran_pascal_pdf (unsigned int k, double p,
          unsigned int n)

     This function computes the probability p(k) of obtaining ‘k’ from a
     Pascal distribution with parameters ‘p’ and ‘n’, using the formula
     given above.

[image src="rand-pascal.png"]

 -- Function: double gsl_cdf_pascal_P (unsigned int k, double p,
          unsigned int n)
 -- Function: double gsl_cdf_pascal_Q (unsigned int k, double p,
          unsigned int n)

     These functions compute the cumulative distribution functions P(k),
     Q(k) for the Pascal distribution with parameters ‘p’ and ‘n’.


File: gsl-ref.info,  Node: The Geometric Distribution,  Next: The Hypergeometric Distribution,  Prev: The Pascal Distribution,  Up: Random Number Distributions

20.36 The Geometric Distribution
================================

 -- Function: unsigned int gsl_ran_geometric (const gsl_rng * r,
          double p)

     This function returns a random integer from the geometric
     distribution, the number of independent trials with probability ‘p’
     until the first success.  The probability distribution for
     geometric variates is,

          p(k) = p (1-p)^{k-1}

     for k \ge 1.  Note that the distribution begins with k = 1 with
     this definition.  There is another convention in which the exponent
     k - 1 is replaced by k.

 -- Function: double gsl_ran_geometric_pdf (unsigned int k, double p)

     This function computes the probability p(k) of obtaining ‘k’ from a
     geometric distribution with probability parameter ‘p’, using the
     formula given above.

[image src="rand-geometric.png"]

 -- Function: double gsl_cdf_geometric_P (unsigned int k, double p)
 -- Function: double gsl_cdf_geometric_Q (unsigned int k, double p)

     These functions compute the cumulative distribution functions P(k),
     Q(k) for the geometric distribution with parameter ‘p’.


File: gsl-ref.info,  Node: The Hypergeometric Distribution,  Next: The Logarithmic Distribution,  Prev: The Geometric Distribution,  Up: Random Number Distributions

20.37 The Hypergeometric Distribution
=====================================

 -- Function: unsigned int gsl_ran_hypergeometric (const gsl_rng * r,
          unsigned int n1, unsigned int n2, unsigned int t)

     This function returns a random integer from the hypergeometric
     distribution.  The probability distribution for hypergeometric
     random variates is,

          p(k) = C(n_1, k) C(n_2, t - k) / C(n_1 + n_2, t)

     where C(a,b) = a!/(b!(a-b)!) and t \leq n_1 + n_2.  The domain of k
     is \max(0, t - n_2), \ldots, \min(t, n_1)

     If a population contains n_1 elements of “type 1” and n_2 elements
     of “type 2” then the hypergeometric distribution gives the
     probability of obtaining k elements of “type 1” in t samples from
     the population without replacement.

 -- Function: double gsl_ran_hypergeometric_pdf (unsigned int k,
          unsigned int n1, unsigned int n2, unsigned int t)

     This function computes the probability p(k) of obtaining ‘k’ from a
     hypergeometric distribution with parameters ‘n1’, ‘n2’, ‘t’, using
     the formula given above.

[image src="rand-hypergeometric.png"]

 -- Function: double gsl_cdf_hypergeometric_P (unsigned int k, unsigned
          int n1, unsigned int n2, unsigned int t)
 -- Function: double gsl_cdf_hypergeometric_Q (unsigned int k, unsigned
          int n1, unsigned int n2, unsigned int t)

     These functions compute the cumulative distribution functions P(k),
     Q(k) for the hypergeometric distribution with parameters ‘n1’, ‘n2’
     and ‘t’.


File: gsl-ref.info,  Node: The Logarithmic Distribution,  Next: The Wishart Distribution,  Prev: The Hypergeometric Distribution,  Up: Random Number Distributions

20.38 The Logarithmic Distribution
==================================

 -- Function: unsigned int gsl_ran_logarithmic (const gsl_rng * r,
          double p)

     This function returns a random integer from the logarithmic
     distribution.  The probability distribution for logarithmic random
     variates is,

          p(k) = {-1 \over \log(1-p)} {(p^k \over k)}

     for k \ge 1.

 -- Function: double gsl_ran_logarithmic_pdf (unsigned int k, double p)

     This function computes the probability p(k) of obtaining ‘k’ from a
     logarithmic distribution with probability parameter ‘p’, using the
     formula given above.

[image src="rand-logarithmic.png"]


File: gsl-ref.info,  Node: The Wishart Distribution,  Next: Shuffling and Sampling,  Prev: The Logarithmic Distribution,  Up: Random Number Distributions

20.39 The Wishart Distribution
==============================

 -- Function: int gsl_ran_wishart (const gsl_rng * r, const double n,
          const gsl_matrix * L, gsl_matrix * result, gsl_matrix * work)

     This function computes a random symmetric p-by-p matrix from the
     Wishart distribution.  The probability distribution for Wishart
     random variates is,

          p(X) = \frac{|X|^{(n-p-1)/2} e^{-tr( V^{-1} X)/2}}{2^{(np)/2} |V|^{n/2} \Gamma_p(n/2)}

     Here, n > p - 1 is the number of degrees of freedom, V is a
     symmetric positive definite p-by-p scale matrix, whose Cholesky
     factor is specified by ‘L’, and ‘work’ is p-by-p workspace.  The
     p-by-p Wishart distributed matrix X is stored in ‘result’ on
     output.

 -- Function: int gsl_ran_wishart_pdf (const gsl_matrix * X, const
          gsl_matrix * L_X, const double n, const gsl_matrix * L, double
          * result, gsl_matrix * work)
 -- Function: int gsl_ran_wishart_log_pdf (const gsl_matrix * X, const
          gsl_matrix * L_X, const double n, const gsl_matrix * L, double
          * result, gsl_matrix * work)

     These functions compute p(X) or \log{p(X)} for the p-by-p matrix
     ‘X’, whose Cholesky factor is specified in ‘L_X’.  The degrees of
     freedom is given by ‘n’, the Cholesky factor of the scale matrix V
     is specified in ‘L’, and ‘work’ is p-by-p workspace.  The probably
     density value is returned in ‘result’.


File: gsl-ref.info,  Node: Shuffling and Sampling,  Next: Examples<14>,  Prev: The Wishart Distribution,  Up: Random Number Distributions

20.40 Shuffling and Sampling
============================

The following functions allow the shuffling and sampling of a set of
objects.  The algorithms rely on a random number generator as a source
of randomness and a poor quality generator can lead to correlations in
the output.  In particular it is important to avoid generators with a
short period.  For more information see Knuth, v2, 3rd ed, Section
3.4.2, “Random Sampling and Shuffling”.

 -- Function: void gsl_ran_shuffle (const gsl_rng * r, void * base,
          size_t n, size_t size)

     This function randomly shuffles the order of ‘n’ objects, each of
     size ‘size’, stored in the array ‘base[0..n-1]’.  The output of the
     random number generator ‘r’ is used to produce the permutation.
     The algorithm generates all possible n! permutations with equal
     probability, assuming a perfect source of random numbers.

     The following code shows how to shuffle the numbers from 0 to 51:

          int a[52];

          for (i = 0; i < 52; i++)
            {
              a[i] = i;
            }

          gsl_ran_shuffle (r, a, 52, sizeof (int));

 -- Function: int gsl_ran_choose (const gsl_rng * r, void * dest,
          size_t k, void * src, size_t n, size_t size)

     This function fills the array ‘dest[k]’ with ‘k’ objects taken
     randomly from the ‘n’ elements of the array ‘src[0..n-1]’.  The
     objects are each of size ‘size’.  The output of the random number
     generator ‘r’ is used to make the selection.  The algorithm ensures
     all possible samples are equally likely, assuming a perfect source
     of randomness.

     The objects are sampled `without' replacement, thus each object can
     only appear once in ‘dest’.  It is required that ‘k’ be less than
     or equal to ‘n’.  The objects in ‘dest’ will be in the same
     relative order as those in ‘src’.  You will need to call
     ‘gsl_ran_shuffle(r, dest, n, size)’ if you want to randomize the
     order.

     The following code shows how to select a random sample of three
     unique numbers from the set 0 to 99:

          double a[3], b[100];

          for (i = 0; i < 100; i++)
            {
              b[i] = (double) i;
            }

          gsl_ran_choose (r, a, 3, b, 100, sizeof (double));

 -- Function: void gsl_ran_sample (const gsl_rng * r, void * dest,
          size_t k, void * src, size_t n, size_t size)

     This function is like *note gsl_ran_choose(): 7b0. but samples ‘k’
     items from the original array of ‘n’ items ‘src’ with replacement,
     so the same object can appear more than once in the output sequence
     ‘dest’.  There is no requirement that ‘k’ be less than ‘n’ in this
     case.


File: gsl-ref.info,  Node: Examples<14>,  Next: References and Further Reading<14>,  Prev: Shuffling and Sampling,  Up: Random Number Distributions

20.41 Examples
==============

The following program demonstrates the use of a random number generator
to produce variates from a distribution.  It prints 10 samples from the
Poisson distribution with a mean of 3.

     #include <stdio.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>

     int
     main (void)
     {
       const gsl_rng_type * T;
       gsl_rng * r;

       int i, n = 10;
       double mu = 3.0;

       /* create a generator chosen by the
          environment variable GSL_RNG_TYPE */

       gsl_rng_env_setup();

       T = gsl_rng_default;
       r = gsl_rng_alloc (T);

       /* print n random variates chosen from
          the poisson distribution with mean
          parameter mu */

       for (i = 0; i < n; i++)
         {
           unsigned int k = gsl_ran_poisson (r, mu);
           printf (" %u", k);
         }

       printf ("\n");
       gsl_rng_free (r);
       return 0;
     }

If the library and header files are installed under ‘/usr/local’ (the
default location) then the program can be compiled with these options:

     $ gcc -Wall demo.c -lgsl -lgslcblas -lm

Here is the output of the program,

      2 5 5 2 1 0 3 4 1 1

The variates depend on the seed used by the generator.  The seed for the
default generator type *note gsl_rng_default: 687. can be changed with
the *note GSL_RNG_SEED: 670. environment variable to produce a different
stream of variates:

     $ GSL_RNG_SEED=123 ./a.out

giving output

      4 5 6 3 3 1 4 2 5 5

The following program generates a random walk in two dimensions.

     #include <stdio.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>

     int
     main (void)
     {
       int i;
       double x = 0, y = 0, dx, dy;

       const gsl_rng_type * T;
       gsl_rng * r;

       gsl_rng_env_setup();
       T = gsl_rng_default;
       r = gsl_rng_alloc (T);

       printf ("%g %g\n", x, y);

       for (i = 0; i < 10; i++)
         {
           gsl_ran_dir_2d (r, &dx, &dy);
           x += dx; y += dy;
           printf ("%g %g\n", x, y);
         }

       gsl_rng_free (r);
       return 0;
     }

Fig.  %s shows the output from the program.

[image src="random-walk.png"]

Figure: Four 10-step random walks from the origin.

The following program computes the upper and lower cumulative
distribution functions for the standard normal distribution at x = 2.

     #include <stdio.h>
     #include <gsl/gsl_cdf.h>

     int
     main (void)
     {
       double P, Q;
       double x = 2.0;

       P = gsl_cdf_ugaussian_P (x);
       printf ("prob(x < %f) = %f\n", x, P);

       Q = gsl_cdf_ugaussian_Q (x);
       printf ("prob(x > %f) = %f\n", x, Q);

       x = gsl_cdf_ugaussian_Pinv (P);
       printf ("Pinv(%f) = %f\n", P, x);

       x = gsl_cdf_ugaussian_Qinv (Q);
       printf ("Qinv(%f) = %f\n", Q, x);

       return 0;
     }

Here is the output of the program,

     prob(x < 2.000000) = 0.977250
     prob(x > 2.000000) = 0.022750
     Pinv(0.977250) = 2.000000
     Qinv(0.022750) = 2.000000


File: gsl-ref.info,  Node: References and Further Reading<14>,  Prev: Examples<14>,  Up: Random Number Distributions

20.42 References and Further Reading
====================================

For an encyclopaedic coverage of the subject readers are advised to
consult the book “Non-Uniform Random Variate Generation” by Luc Devroye.
It covers every imaginable distribution and provides hundreds of
algorithms.

   * Luc Devroye, “Non-Uniform Random Variate Generation”,
     Springer-Verlag, ISBN 0-387-96305-7.  Available online at
     ‘http://cg.scs.carleton.ca/~luc/rnbookindex.html’.

The subject of random variate generation is also reviewed by Knuth, who
describes algorithms for all the major distributions.

   * Donald E. Knuth, “The Art of Computer Programming: Seminumerical
     Algorithms” (Vol 2, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896842.

The Particle Data Group provides a short review of techniques for
generating distributions of random numbers in the “Monte Carlo” section
of its Annual Review of Particle Physics.

   * Review of Particle Properties, R.M. Barnett et al., Physical Review
     D54, 1 (1996) ‘http://pdg.lbl.gov/’.

The Review of Particle Physics is available online in postscript and pdf
format.

An overview of methods used to compute cumulative distribution functions
can be found in `Statistical Computing' by W.J. Kennedy and J.E. Gentle.
Another general reference is `Elements of Statistical Computing' by R.A.
Thisted.

   * William E. Kennedy and James E. Gentle, Statistical Computing
     (1980), Marcel Dekker, ISBN 0-8247-6898-1.

   * Ronald A. Thisted, Elements of Statistical Computing (1988),
     Chapman & Hall, ISBN 0-412-01371-1.

The cumulative distribution functions for the Gaussian distribution are
based on the following papers,

   * Rational Chebyshev Approximations Using Linear Equations, W.J.
     Cody, W. Fraser, J.F. Hart.  Numerische Mathematik 12, 242–251
     (1968).

   * Rational Chebyshev Approximations for the Error Function, W.J.
     Cody.  Mathematics of Computation 23, n107, 631–637 (July 1969).


File: gsl-ref.info,  Node: Statistics,  Next: Running Statistics,  Prev: Random Number Distributions,  Up: Top

21 Statistics
*************

This chapter describes the statistical functions in the library.  The
basic statistical functions include routines to compute the mean,
variance and standard deviation.  More advanced functions allow you to
calculate absolute deviations, skewness, and kurtosis as well as the
median and arbitrary percentiles.  The algorithms use recurrence
relations to compute average quantities in a stable way, without large
intermediate values that might overflow.

The functions are available in versions for datasets in the standard
floating-point and integer types.  The versions for double precision
floating-point data have the prefix ‘gsl_stats’ and are declared in the
header file ‘gsl_statistics_double.h’.  The versions for integer data
have the prefix ‘gsl_stats_int’ and are declared in the header file
‘gsl_statistics_int.h’.  All the functions operate on C arrays with a
‘stride’ parameter specifying the spacing between elements.

* Menu:

* Mean, Standard Deviation and Variance: Mean Standard Deviation and Variance.
* Absolute deviation::
* Higher moments (skewness and kurtosis): Higher moments skewness and kurtosis.
* Autocorrelation::
* Covariance::
* Correlation::
* Weighted Samples::
* Maximum and Minimum values::
* Median and Percentiles::
* Order Statistics::
* Robust Location Estimates::
* Robust Scale Estimates::
* Examples: Examples<15>.
* References and Further Reading: References and Further Reading<15>.


File: gsl-ref.info,  Node: Mean Standard Deviation and Variance,  Next: Absolute deviation,  Up: Statistics

21.1 Mean, Standard Deviation and Variance
==========================================

 -- Function: double gsl_stats_mean (const double data[], size_t stride,
          size_t n)

     This function returns the arithmetic mean of ‘data’, a dataset of
     length ‘n’ with stride ‘stride’.  The arithmetic mean, or `sample
     mean', is denoted by \Hat\mu and defined as,

          \Hat\mu = {1 \over N} \sum x_i

     where x_i are the elements of the dataset ‘data’.  For samples
     drawn from a gaussian distribution the variance of \Hat\mu is
     \sigma^2 / N.

 -- Function: double gsl_stats_variance (const double data[],
          size_t stride, size_t n)

     This function returns the estimated, or `sample', variance of
     ‘data’, a dataset of length ‘n’ with stride ‘stride’.  The
     estimated variance is denoted by \Hat\sigma^2 and is defined by,

          \Hat\sigma^2 = (1/(N-1)) \sum (x_i - \Hat\mu)^2

     where x_i are the elements of the dataset ‘data’.  Note that the
     normalization factor of 1/(N-1) results from the derivation of
     \Hat\sigma^2 as an unbiased estimator of the population variance
     \sigma^2.  For samples drawn from a Gaussian distribution the
     variance of \Hat\sigma^2 itself is 2 \sigma^4 / N.

     This function computes the mean via a call to *note
     gsl_stats_mean(): 1d.  If you have already computed the mean then
     you can pass it directly to *note gsl_stats_variance_m(): 7b9.

 -- Function: double gsl_stats_variance_m (const double data[],
          size_t stride, size_t n, double mean)

     This function returns the sample variance of ‘data’ relative to the
     given value of ‘mean’.  The function is computed with \Hat\mu
     replaced by the value of ‘mean’ that you supply,

          \Hat\sigma^2 = (1/(N-1)) \sum (x_i - mean)^2

 -- Function: double gsl_stats_sd (const double data[], size_t stride,
          size_t n)
 -- Function: double gsl_stats_sd_m (const double data[], size_t stride,
          size_t n, double mean)

     The standard deviation is defined as the square root of the
     variance.  These functions return the square root of the
     corresponding variance functions above.

 -- Function: double gsl_stats_tss (const double data[], size_t stride,
          size_t n)
 -- Function: double gsl_stats_tss_m (const double data[],
          size_t stride, size_t n, double mean)

     These functions return the total sum of squares (TSS) of ‘data’
     about the mean.  For *note gsl_stats_tss_m(): 7bd. the
     user-supplied value of ‘mean’ is used, and for *note
     gsl_stats_tss(): 7bc. it is computed using *note gsl_stats_mean():
     1d.

          TSS =  \sum (x_i - mean)^2

 -- Function: double gsl_stats_variance_with_fixed_mean (const
          double data[], size_t stride, size_t n, double mean)

     This function computes an unbiased estimate of the variance of
     ‘data’ when the population mean ‘mean’ of the underlying
     distribution is known `a priori'.  In this case the estimator for
     the variance uses the factor 1/N and the sample mean \Hat\mu is
     replaced by the known population mean \mu,

          \Hat\sigma^2 = (1/N) \sum (x_i - \mu)^2

 -- Function: double gsl_stats_sd_with_fixed_mean (const double data[],
          size_t stride, size_t n, double mean)

     This function calculates the standard deviation of ‘data’ for a
     fixed population mean ‘mean’.  The result is the square root of the
     corresponding variance function.


File: gsl-ref.info,  Node: Absolute deviation,  Next: Higher moments skewness and kurtosis,  Prev: Mean Standard Deviation and Variance,  Up: Statistics

21.2 Absolute deviation
=======================

 -- Function: double gsl_stats_absdev (const double data[],
          size_t stride, size_t n)

     This function computes the absolute deviation from the mean of
     ‘data’, a dataset of length ‘n’ with stride ‘stride’.  The absolute
     deviation from the mean is defined as,

          absdev  = (1/N) \sum |x_i - \Hat\mu|

     where x_i are the elements of the dataset ‘data’.  The absolute
     deviation from the mean provides a more robust measure of the width
     of a distribution than the variance.  This function computes the
     mean of ‘data’ via a call to *note gsl_stats_mean(): 1d.

 -- Function: double gsl_stats_absdev_m (const double data[],
          size_t stride, size_t n, double mean)

     This function computes the absolute deviation of the dataset ‘data’
     relative to the given value of ‘mean’,

          absdev  = (1/N) \sum |x_i - mean|

     This function is useful if you have already computed the mean of
     ‘data’ (and want to avoid recomputing it), or wish to calculate the
     absolute deviation relative to another value (such as zero, or the
     median).


File: gsl-ref.info,  Node: Higher moments skewness and kurtosis,  Next: Autocorrelation,  Prev: Absolute deviation,  Up: Statistics

21.3 Higher moments (skewness and kurtosis)
===========================================

 -- Function: double gsl_stats_skew (const double data[], size_t stride,
          size_t n)

     This function computes the skewness of ‘data’, a dataset of length
     ‘n’ with stride ‘stride’.  The skewness is defined as,

          skew = (1/N) \sum ((x_i - \Hat\mu)/\Hat\sigma)^3

     where x_i are the elements of the dataset ‘data’.  The skewness
     measures the asymmetry of the tails of a distribution.

     The function computes the mean and estimated standard deviation of
     ‘data’ via calls to *note gsl_stats_mean(): 1d. and *note
     gsl_stats_sd(): 7ba.

 -- Function: double gsl_stats_skew_m_sd (const double data[],
          size_t stride, size_t n, double mean, double sd)

     This function computes the skewness of the dataset ‘data’ using the
     given values of the mean ‘mean’ and standard deviation ‘sd’,

          skew = (1/N) \sum ((x_i - mean)/sd)^3

     These functions are useful if you have already computed the mean
     and standard deviation of ‘data’ and want to avoid recomputing
     them.

 -- Function: double gsl_stats_kurtosis (const double data[],
          size_t stride, size_t n)

     This function computes the kurtosis of ‘data’, a dataset of length
     ‘n’ with stride ‘stride’.  The kurtosis is defined as,

          kurtosis = ((1/N) \sum ((x_i - \Hat\mu)/\Hat\sigma)^4)  - 3

     The kurtosis measures how sharply peaked a distribution is,
     relative to its width.  The kurtosis is normalized to zero for a
     Gaussian distribution.

 -- Function: double gsl_stats_kurtosis_m_sd (const double data[],
          size_t stride, size_t n, double mean, double sd)

     This function computes the kurtosis of the dataset ‘data’ using the
     given values of the mean ‘mean’ and standard deviation ‘sd’,

          kurtosis = ((1/N) \sum ((x_i - mean)/sd)^4) - 3

     This function is useful if you have already computed the mean and
     standard deviation of ‘data’ and want to avoid recomputing them.


File: gsl-ref.info,  Node: Autocorrelation,  Next: Covariance,  Prev: Higher moments skewness and kurtosis,  Up: Statistics

21.4 Autocorrelation
====================

 -- Function: double gsl_stats_lag1_autocorrelation (const
          double data[], const size_t stride, const size_t n)

     This function computes the lag-1 autocorrelation of the dataset
     ‘data’.

          a_1 = {\sum_{i = 2}^{n} (x_{i} - \Hat\mu) (x_{i-1} - \Hat\mu)
                 \over
                 \sum_{i = 1}^{n} (x_{i} - \Hat\mu) (x_{i} - \Hat\mu)}

 -- Function: double gsl_stats_lag1_autocorrelation_m (const
          double data[], const size_t stride, const size_t n, const
          double mean)

     This function computes the lag-1 autocorrelation of the dataset
     ‘data’ using the given value of the mean ‘mean’.


File: gsl-ref.info,  Node: Covariance,  Next: Correlation,  Prev: Autocorrelation,  Up: Statistics

21.5 Covariance
===============

 -- Function: double gsl_stats_covariance (const double data1[], const
          size_t stride1, const double data2[], const size_t stride2,
          const size_t n)

     This function computes the covariance of the datasets ‘data1’ and
     ‘data2’ which must both be of the same length ‘n’.

          covar = (1/(n - 1)) \sum_{i = 1}^{n} (x_i - \Hat x) (y_i - \Hat y)

 -- Function: double gsl_stats_covariance_m (const double data1[], const
          size_t stride1, const double data2[], const size_t stride2,
          const size_t n, const double mean1, const double mean2)

     This function computes the covariance of the datasets ‘data1’ and
     ‘data2’ using the given values of the means, ‘mean1’ and ‘mean2’.
     This is useful if you have already computed the means of ‘data1’
     and ‘data2’ and want to avoid recomputing them.


File: gsl-ref.info,  Node: Correlation,  Next: Weighted Samples,  Prev: Covariance,  Up: Statistics

21.6 Correlation
================

 -- Function: double gsl_stats_correlation (const double data1[], const
          size_t stride1, const double data2[], const size_t stride2,
          const size_t n)

     This function efficiently computes the Pearson correlation
     coefficient between the datasets ‘data1’ and ‘data2’ which must
     both be of the same length ‘n’.

          r = cov(x, y) / (\Hat\sigma_x \Hat\sigma_y)
            = {1/(n-1) \sum (x_i - \Hat x) (y_i - \Hat y)
               \over
               \sqrt{1/(n-1) \sum (x_i - \Hat x)^2} \sqrt{1/(n-1) \sum (y_i - \Hat y)^2}
              }

 -- Function: double gsl_stats_spearman (const double data1[], const
          size_t stride1, const double data2[], const size_t stride2,
          const size_t n, double work[])

     This function computes the Spearman rank correlation coefficient
     between the datasets ‘data1’ and ‘data2’ which must both be of the
     same length ‘n’.  Additional workspace of size 2 * ‘n’ is required
     in ‘work’.  The Spearman rank correlation between vectors x and y
     is equivalent to the Pearson correlation between the ranked vectors
     x_R and y_R, where ranks are defined to be the average of the
     positions of an element in the ascending order of the values.


File: gsl-ref.info,  Node: Weighted Samples,  Next: Maximum and Minimum values,  Prev: Correlation,  Up: Statistics

21.7 Weighted Samples
=====================

The functions described in this section allow the computation of
statistics for weighted samples.  The functions accept an array of
samples, x_i, with associated weights, w_i.  Each sample x_i is
considered as having been drawn from a Gaussian distribution with
variance \sigma_i^2.  The sample weight w_i is defined as the reciprocal
of this variance, w_i = 1/\sigma_i^2.  Setting a weight to zero
corresponds to removing a sample from a dataset.

 -- Function: double gsl_stats_wmean (const double w[], size_t wstride,
          const double data[], size_t stride, size_t n)

     This function returns the weighted mean of the dataset ‘data’ with
     stride ‘stride’ and length ‘n’, using the set of weights ‘w’ with
     stride ‘wstride’ and length ‘n’.  The weighted mean is defined as,

          \Hat\mu = (\sum w_i x_i) / (\sum w_i)

 -- Function: double gsl_stats_wvariance (const double w[],
          size_t wstride, const double data[], size_t stride, size_t n)

     This function returns the estimated variance of the dataset ‘data’
     with stride ‘stride’ and length ‘n’, using the set of weights ‘w’
     with stride ‘wstride’ and length ‘n’.  The estimated variance of a
     weighted dataset is calculated as,

          \Hat\sigma^2 = ((\sum w_i)/((\sum w_i)^2 - \sum (w_i^2)))
                          \sum w_i (x_i - \Hat\mu)^2

     Note that this expression reduces to an unweighted variance with
     the familiar 1/(N-1) factor when there are N equal non-zero
     weights.

 -- Function: double gsl_stats_wvariance_m (const double w[],
          size_t wstride, const double data[], size_t stride, size_t n,
          double wmean)

     This function returns the estimated variance of the weighted
     dataset ‘data’ using the given weighted mean ‘wmean’.

 -- Function: double gsl_stats_wsd (const double w[], size_t wstride,
          const double data[], size_t stride, size_t n)

     The standard deviation is defined as the square root of the
     variance.  This function returns the square root of the
     corresponding variance function *note gsl_stats_wvariance(): 7d3.
     above.

 -- Function: double gsl_stats_wsd_m (const double w[], size_t wstride,
          const double data[], size_t stride, size_t n, double wmean)

     This function returns the square root of the corresponding variance
     function *note gsl_stats_wvariance_m(): 7d4. above.

 -- Function: double gsl_stats_wvariance_with_fixed_mean (const
          double w[], size_t wstride, const double data[],
          size_t stride, size_t n, const double mean)

     This function computes an unbiased estimate of the variance of the
     weighted dataset ‘data’ when the population mean ‘mean’ of the
     underlying distribution is known `a priori'.  In this case the
     estimator for the variance replaces the sample mean \Hat\mu by the
     known population mean \mu,

          \Hat\sigma^2 = (\sum w_i (x_i - \mu)^2) / (\sum w_i)

 -- Function: double gsl_stats_wsd_with_fixed_mean (const double w[],
          size_t wstride, const double data[], size_t stride, size_t n,
          const double mean)

     The standard deviation is defined as the square root of the
     variance.  This function returns the square root of the
     corresponding variance function above.

 -- Function: double gsl_stats_wtss (const double w[], const
          size_t wstride, const double data[], size_t stride, size_t n)
 -- Function: double gsl_stats_wtss_m (const double w[], const
          size_t wstride, const double data[], size_t stride, size_t n,
          double wmean)

     These functions return the weighted total sum of squares (TSS) of
     ‘data’ about the weighted mean.  For *note gsl_stats_wtss_m(): 7da.
     the user-supplied value of ‘wmean’ is used, and for *note
     gsl_stats_wtss(): 7d9. it is computed using *note
     gsl_stats_wmean(): 7d2.

          TSS =  \sum w_i (x_i - wmean)^2

 -- Function: double gsl_stats_wabsdev (const double w[],
          size_t wstride, const double data[], size_t stride, size_t n)

     This function computes the weighted absolute deviation from the
     weighted mean of ‘data’.  The absolute deviation from the mean is
     defined as,

          absdev = (\sum w_i |x_i - \Hat\mu|) / (\sum w_i)

 -- Function: double gsl_stats_wabsdev_m (const double w[],
          size_t wstride, const double data[], size_t stride, size_t n,
          double wmean)

     This function computes the absolute deviation of the weighted
     dataset ‘data’ about the given weighted mean ‘wmean’.

 -- Function: double gsl_stats_wskew (const double w[], size_t wstride,
          const double data[], size_t stride, size_t n)

     This function computes the weighted skewness of the dataset ‘data’.

          skew = (\sum w_i ((x_i - \Hat x)/\Hat \sigma)^3) / (\sum w_i)

 -- Function: double gsl_stats_wskew_m_sd (const double w[],
          size_t wstride, const double data[], size_t stride, size_t n,
          double wmean, double wsd)

     This function computes the weighted skewness of the dataset ‘data’
     using the given values of the weighted mean and weighted standard
     deviation, ‘wmean’ and ‘wsd’.

 -- Function: double gsl_stats_wkurtosis (const double w[],
          size_t wstride, const double data[], size_t stride, size_t n)

     This function computes the weighted kurtosis of the dataset ‘data’.

          kurtosis = ((\sum w_i ((x_i - \Hat x)/\Hat \sigma)^4) / (\sum w_i)) - 3

 -- Function: double gsl_stats_wkurtosis_m_sd (const double w[],
          size_t wstride, const double data[], size_t stride, size_t n,
          double wmean, double wsd)

     This function computes the weighted kurtosis of the dataset ‘data’
     using the given values of the weighted mean and weighted standard
     deviation, ‘wmean’ and ‘wsd’.


File: gsl-ref.info,  Node: Maximum and Minimum values,  Next: Median and Percentiles,  Prev: Weighted Samples,  Up: Statistics

21.8 Maximum and Minimum values
===============================

The following functions find the maximum and minimum values of a dataset
(or their indices).  If the data contains ‘NaN’-s then a ‘NaN’ will be
returned, since the maximum or minimum value is undefined.  For
functions which return an index, the location of the first ‘NaN’ in the
array is returned.

 -- Function: double gsl_stats_max (const double data[], size_t stride,
          size_t n)

     This function returns the maximum value in ‘data’, a dataset of
     length ‘n’ with stride ‘stride’.  The maximum value is defined as
     the value of the element x_i which satisfies x_i \ge x_j for all j.

     If you want instead to find the element with the largest absolute
     magnitude you will need to apply ‘fabs()’ or ‘abs()’ to your data
     before calling this function.

 -- Function: double gsl_stats_min (const double data[], size_t stride,
          size_t n)

     This function returns the minimum value in ‘data’, a dataset of
     length ‘n’ with stride ‘stride’.  The minimum value is defined as
     the value of the element x_i which satisfies x_i \le x_j for all j.

     If you want instead to find the element with the smallest absolute
     magnitude you will need to apply ‘fabs()’ or ‘abs()’ to your data
     before calling this function.

 -- Function: void gsl_stats_minmax (double * min, double * max, const
          double data[], size_t stride, size_t n)

     This function finds both the minimum and maximum values ‘min’,
     ‘max’ in ‘data’ in a single pass.

 -- Function: size_t gsl_stats_max_index (const double data[],
          size_t stride, size_t n)

     This function returns the index of the maximum value in ‘data’, a
     dataset of length ‘n’ with stride ‘stride’.  The maximum value is
     defined as the value of the element x_i which satisfies x_i \ge x_j
     for all j.  When there are several equal maximum elements then the
     first one is chosen.

 -- Function: size_t gsl_stats_min_index (const double data[],
          size_t stride, size_t n)

     This function returns the index of the minimum value in ‘data’, a
     dataset of length ‘n’ with stride ‘stride’.  The minimum value is
     defined as the value of the element x_i which satisfies x_i \ge x_j
     for all j.  When there are several equal minimum elements then the
     first one is chosen.

 -- Function: void gsl_stats_minmax_index (size_t * min_index, size_t
          * max_index, const double data[], size_t stride, size_t n)

     This function returns the indexes ‘min_index’, ‘max_index’ of the
     minimum and maximum values in ‘data’ in a single pass.


File: gsl-ref.info,  Node: Median and Percentiles,  Next: Order Statistics,  Prev: Maximum and Minimum values,  Up: Statistics

21.9 Median and Percentiles
===========================

The median and percentile functions described in this section operate on
sorted data in O(1) time.  There is also a routine for computing the
median of an unsorted input array in average O(n) time using the
quickselect algorithm.  For convenience we use `quantiles', measured on
a scale of 0 to 1, instead of percentiles (which use a scale of 0 to
100).

 -- Function: double gsl_stats_median_from_sorted_data (const
          double sorted_data[], const size_t stride, const size_t n)

     This function returns the median value of ‘sorted_data’, a dataset
     of length ‘n’ with stride ‘stride’.  The elements of the array must
     be in ascending numerical order.  There are no checks to see
     whether the data are sorted, so the function *note gsl_sort(): 45d.
     should always be used first.

     When the dataset has an odd number of elements the median is the
     value of element (n-1)/2.  When the dataset has an even number of
     elements the median is the mean of the two nearest middle values,
     elements (n-1)/2 and n/2.  Since the algorithm for computing the
     median involves interpolation this function always returns a
     floating-point number, even for integer data types.

 -- Function: double gsl_stats_median (double data[], const
          size_t stride, const size_t n)

     This function returns the median value of ‘data’, a dataset of
     length ‘n’ with stride ‘stride’.  The median is found using the
     quickselect algorithm.  The input array does not need to be sorted,
     but note that the algorithm rearranges the array and so the input
     is not preserved on output.

 -- Function: double gsl_stats_quantile_from_sorted_data (const
          double sorted_data[], size_t stride, size_t n, double f)

     This function returns a quantile value of ‘sorted_data’, a
     double-precision array of length ‘n’ with stride ‘stride’.  The
     elements of the array must be in ascending numerical order.  The
     quantile is determined by the ‘f’, a fraction between 0 and 1.  For
     example, to compute the value of the 75th percentile ‘f’ should
     have the value 0.75.

     There are no checks to see whether the data are sorted, so the
     function *note gsl_sort(): 45d. should always be used first.

     The quantile is found by interpolation, using the formula

          quantile = (1 - \delta) x_i + \delta x_{i+1}

     where i is ‘floor((n - 1)f)’ and \delta is (n-1)f - i.

     Thus the minimum value of the array (‘data[0*stride]’) is given by
     ‘f’ equal to zero, the maximum value (‘data[(n-1)*stride]’) is
     given by ‘f’ equal to one and the median value is given by ‘f’
     equal to 0.5.  Since the algorithm for computing quantiles involves
     interpolation this function always returns a floating-point number,
     even for integer data types.


File: gsl-ref.info,  Node: Order Statistics,  Next: Robust Location Estimates,  Prev: Median and Percentiles,  Up: Statistics

21.10 Order Statistics
======================

The k-th `order statistic' of a sample is equal to its k-th smallest
value.  The k-th order statistic of a set of n values x = \left\{ x_i
\right\}, 1 \le i \le n is denoted x_{(k)}.  The median of the set x is
equal to x_{\left( \frac{n}{2} \right)} if n is odd, or the average of
x_{\left( \frac{n}{2} \right)} and x_{\left( \frac{n}{2} + 1 \right)} if
n is even.  The k-th smallest element of a length n vector can be found
in average O(n) time using the quickselect algorithm.

 -- Function: gsl_stats_select (double data[], const size_t stride,
          const size_t n, const size_t k)

     This function finds the ‘k’-th smallest element of the input array
     ‘data’ of length ‘n’ and stride ‘stride’ using the quickselect
     method.  The algorithm rearranges the elements of ‘data’ and so the
     input array is not preserved on output.


File: gsl-ref.info,  Node: Robust Location Estimates,  Next: Robust Scale Estimates,  Prev: Order Statistics,  Up: Statistics

21.11 Robust Location Estimates
===============================

A `location estimate' refers to a typical or central value which best
describes a given dataset.  The mean and median are both examples of
location estimators.  However, the mean has a severe sensitivity to data
outliers and can give erroneous values when even a small number of
outliers are present.  The median on the other hand, has a strong
insensitivity to data outliers, but due to its non-smoothness it can
behave unexpectedly in certain situations.  GSL offers the following
alternative location estimators, which are robust to the presence of
outliers.

* Menu:

* Trimmed Mean::
* Gastwirth Estimator::


File: gsl-ref.info,  Node: Trimmed Mean,  Next: Gastwirth Estimator,  Up: Robust Location Estimates

21.11.1 Trimmed Mean
--------------------

The trimmed mean, or `truncated mean', discards a certain number of
smallest and largest samples from the input vector before computing the
mean of the remaining samples.  The amount of trimming is specified by a
factor \alpha \in [0,0.5].  Then the number of samples discarded from
both ends of the input vector is \left\lfloor \alpha n \right\rfloor,
where n is the length of the input.  So to discard 25% of the samples
from each end, one would set \alpha = 0.25.

 -- Function: double gsl_stats_trmean_from_sorted_data (const
          double alpha, const double sorted_data[], const size_t stride,
          const size_t n)

     This function returns the trimmed mean of ‘sorted_data’, a dataset
     of length ‘n’ with stride ‘stride’.  The elements of the array must
     be in ascending numerical order.  There are no checks to see
     whether the data are sorted, so the function *note gsl_sort(): 45d.
     should always be used first.  The trimming factor \alpha is given
     in *note alpha: 49a.  If \alpha \ge 0.5, then the median of the
     input is returned.


File: gsl-ref.info,  Node: Gastwirth Estimator,  Prev: Trimmed Mean,  Up: Robust Location Estimates

21.11.2 Gastwirth Estimator
---------------------------

Gastwirth’s location estimator is a weighted sum of three order
statistics,

     gastwirth = 0.3 * Q_{1/3} + 0.4 * Q_{1/2} + 0.3 * Q_{2/3}

where Q_{\frac{1}{3}} is the one-third quantile, Q_{\frac{1}{2}} is the
one-half quantile (i.e.  median), and Q_{\frac{2}{3}} is the two-thirds
quantile.

 -- Function: double gsl_stats_gastwirth_from_sorted_data (const
          double sorted_data[], const size_t stride, const size_t n)

     This function returns the Gastwirth location estimator of
     ‘sorted_data’, a dataset of length ‘n’ with stride ‘stride’.  The
     elements of the array must be in ascending numerical order.  There
     are no checks to see whether the data are sorted, so the function
     *note gsl_sort(): 45d. should always be used first.


File: gsl-ref.info,  Node: Robust Scale Estimates,  Next: Examples<15>,  Prev: Robust Location Estimates,  Up: Statistics

21.12 Robust Scale Estimates
============================

A `robust scale estimate', also known as a robust measure of scale,
attempts to quantify the statistical dispersion (variability, scatter,
spread) in a set of data which may contain outliers.  In such datasets,
the usual variance or standard deviation scale estimate can be rendered
useless by even a single outlier.

* Menu:

* Median Absolute Deviation (MAD): Median Absolute Deviation MAD.
* S_n Statistic::
* Q_n Statistic::


File: gsl-ref.info,  Node: Median Absolute Deviation MAD,  Next: S_n Statistic,  Up: Robust Scale Estimates

21.12.1 Median Absolute Deviation (MAD)
---------------------------------------

The median absolute deviation (MAD) is defined as

     MAD = 1.4826 median { | x_i - median(x) | }

In words, first the median of all samples is computed.  Then the median
is subtracted from all samples in the input to find the deviation of
each sample from the median.  The median of all absolute deviations is
then the MAD. The factor 1.4826 makes the MAD an unbiased estimator of
the standard deviation for Gaussian data.  The median absolute deviation
has an asymptotic efficiency of 37%.

 -- Function: double gsl_stats_mad0 (const double data[], const
          size_t stride, const size_t n, double work[])

 -- Function: double gsl_stats_mad (const double data[], const
          size_t stride, const size_t n, double work[])

     These functions return the median absolute deviation of ‘data’, a
     dataset of length ‘n’ and stride ‘stride’.  The ‘mad0’ function
     calculates \textrm{median} \left\{ \left| x_i - \textrm{median}
     \left( x \right) \right| \right\} (i.e.  the MAD statistic without
     the bias correction scale factor).  These functions require
     additional workspace of size ‘n’ provided in ‘work’.


File: gsl-ref.info,  Node: S_n Statistic,  Next: Q_n Statistic,  Prev: Median Absolute Deviation MAD,  Up: Robust Scale Estimates

21.12.2 S_n Statistic
---------------------

The S_n statistic developed by Croux and Rousseeuw is defined as

     S_n = 1.1926 * c_n * median_i { median_j ( | x_i - x_j | ) }

For each sample x_i, 1 \le i \le n, the median of the values \left| x_i
- x_j \right| is computed for all x_j, 1 \le j \le n.  This yields n
values, whose median then gives the final S_n.  The factor 1.1926 makes
S_n an unbiased estimate of the standard deviation for Gaussian data.
The factor c_n is a correction factor to correct bias in small sample
sizes.  S_n has an asymptotic efficiency of 58%.

 -- Function: double gsl_stats_Sn0_from_sorted_data (const
          double sorted_data[], const size_t stride, const size_t n,
          double work[])

 -- Function: double gsl_stats_Sn_from_sorted_data (const
          double sorted_data[], const size_t stride, const size_t n,
          double work[])

     These functions return the S_n statistic of ‘sorted_data’, a
     dataset of length ‘n’ with stride ‘stride’.  The elements of the
     array must be in ascending numerical order.  There are no checks to
     see whether the data are sorted, so the function *note gsl_sort():
     45d. should always be used first.  The ‘Sn0’ function calculates
     \textrm{median}_i \left\{ \textrm{median}_j \left( \left| x_i - x_j
     \right| \right) \right\} (i.e.  the S_n statistic without the bias
     correction scale factors).  These functions require additional
     workspace of size ‘n’ provided in ‘work’.


File: gsl-ref.info,  Node: Q_n Statistic,  Prev: S_n Statistic,  Up: Robust Scale Estimates

21.12.3 Q_n Statistic
---------------------

The Q_n statistic developed by Croux and Rousseeuw is defined as

     Q_n = 2.21914 * d_n * { | x_i - x_j |, i < j }_{(k)}

The factor 2.21914 makes Q_n an unbiased estimate of the standard
deviation for Gaussian data.  The factor d_n is a correction factor to
correct bias in small sample sizes.  The order statistic is

     k = ( floor(n/2) + 1 )
         (       2        )

Q_n has an asymptotic efficiency of 82%.

 -- Function: double gsl_stats_Qn0_from_sorted_data (const
          double sorted_data[], const size_t stride, const size_t n,
          double work[], int work_int[])
 -- Function: double gsl_stats_Qn_from_sorted_data (const
          double sorted_data[], const size_t stride, const size_t n,
          double work[], int work_int[])

     These functions return the Q_n statistic of ‘sorted_data’, a
     dataset of length ‘n’ with stride ‘stride’.  The elements of the
     array must be in ascending numerical order.  There are no checks to
     see whether the data are sorted, so the function *note gsl_sort():
     45d. should always be used first.  The ‘Qn0’ function calculates
     \left\{ \left| x_i - x_j \right|, i < j \right\}_{(k)} (i.e.  Q_n
     without the bias correction scale factors).  These functions
     require additional workspace of size ‘3n’ provided in ‘work’ and
     integer workspace of size ‘5n’ provided in ‘work_int’.


File: gsl-ref.info,  Node: Examples<15>,  Next: References and Further Reading<15>,  Prev: Robust Scale Estimates,  Up: Statistics

21.13 Examples
==============

Here is a basic example of how to use the statistical functions:

     #include <stdio.h>
     #include <gsl/gsl_statistics.h>

     int
     main(void)
     {
       double data[5] = {17.2, 18.1, 16.5, 18.3, 12.6};
       double mean, variance, largest, smallest;

       mean     = gsl_stats_mean(data, 1, 5);
       variance = gsl_stats_variance(data, 1, 5);
       largest  = gsl_stats_max(data, 1, 5);
       smallest = gsl_stats_min(data, 1, 5);

       printf ("The dataset is %g, %g, %g, %g, %g\n",
              data[0], data[1], data[2], data[3], data[4]);

       printf ("The sample mean is %g\n", mean);
       printf ("The estimated variance is %g\n", variance);
       printf ("The largest value is %g\n", largest);
       printf ("The smallest value is %g\n", smallest);
       return 0;
     }

The program should produce the following output,

     The dataset is 17.2, 18.1, 16.5, 18.3, 12.6
     The sample mean is 16.54
     The estimated variance is 5.373
     The largest value is 18.3
     The smallest value is 12.6

Here is an example using sorted data,

     #include <stdio.h>
     #include <gsl/gsl_sort.h>
     #include <gsl/gsl_statistics.h>

     int
     main(void)
     {
       double data[5] = {17.2, 18.1, 16.5, 18.3, 12.6};
       double median, upperq, lowerq;

       printf ("Original dataset:  %g, %g, %g, %g, %g\n",
              data[0], data[1], data[2], data[3], data[4]);

       gsl_sort (data, 1, 5);

       printf ("Sorted dataset: %g, %g, %g, %g, %g\n",
              data[0], data[1], data[2], data[3], data[4]);

       median
         = gsl_stats_median_from_sorted_data (data,
                                              1, 5);

       upperq
         = gsl_stats_quantile_from_sorted_data (data,
                                                1, 5,
                                                0.75);
       lowerq
         = gsl_stats_quantile_from_sorted_data (data,
                                                1, 5,
                                                0.25);

       printf ("The median is %g\n", median);
       printf ("The upper quartile is %g\n", upperq);
       printf ("The lower quartile is %g\n", lowerq);
       return 0;
     }

This program should produce the following output,

     Original dataset:  17.2, 18.1, 16.5, 18.3, 12.6
     Sorted dataset: 12.6, 16.5, 17.2, 18.1, 18.3
     The median is 17.2
     The upper quartile is 18.1
     The lower quartile is 16.5


File: gsl-ref.info,  Node: References and Further Reading<15>,  Prev: Examples<15>,  Up: Statistics

21.14 References and Further Reading
====================================

The standard reference for almost any topic in statistics is the
multi-volume `Advanced Theory of Statistics' by Kendall and Stuart.

   * Maurice Kendall, Alan Stuart, and J. Keith Ord.  `The Advanced
     Theory of Statistics' (multiple volumes) reprinted as `Kendall’s
     Advanced Theory of Statistics'.  Wiley, ISBN 047023380X.

Many statistical concepts can be more easily understood by a Bayesian
approach.  The following book by Gelman, Carlin, Stern and Rubin gives a
comprehensive coverage of the subject.

   * Andrew Gelman, John B. Carlin, Hal S. Stern, Donald B. Rubin.
     `Bayesian Data Analysis'.  Chapman & Hall, ISBN 0412039915.

For physicists the Particle Data Group provides useful reviews of
Probability and Statistics in the “Mathematical Tools” section of its
Annual Review of Particle Physics.

   * `Review of Particle Properties', R.M. Barnett et al., Physical
     Review D54, 1 (1996)

The Review of Particle Physics is available online at the website
‘http://pdg.lbl.gov/’.

The following papers describe robust scale estimation,

   * C. Croux and P. J. Rousseeuw, `Time-Efficient algorithms for two
     highly robust estimators of scale', Comp.  Stat., Physica,
     Heidelberg, 1992.

   * P. J. Rousseeuw and C. Croux, `Explicit scale estimators with high
     breakdown point', L1-Statistical Analysis and Related Methods, pp.
     77-92, 1992.


File: gsl-ref.info,  Node: Running Statistics,  Next: Moving Window Statistics,  Prev: Statistics,  Up: Top

22 Running Statistics
*********************

This chapter describes routines for computing running statistics, also
known as online statistics, of data.  These routines are suitable for
handling large datasets for which it may be inconvenient or impractical
to store in memory all at once.  The data can be processed in a single
pass, one point at a time.  Each time a data point is added to the
accumulator, internal parameters are updated in order to compute the
current mean, variance, standard deviation, skewness, and kurtosis.
These statistics are exact, and are updated with numerically stable
single-pass algorithms.  The median and arbitrary quantiles are also
available, however these calculations use algorithms which provide
approximations, and grow more accurate as more data is added to the
accumulator.

The functions described in this chapter are declared in the header file
‘gsl_rstat.h’.

* Menu:

* Initializing the Accumulator::
* Adding Data to the Accumulator::
* Current Statistics::
* Quantiles::
* Examples: Examples<16>.
* References and Further Reading: References and Further Reading<16>.


File: gsl-ref.info,  Node: Initializing the Accumulator,  Next: Adding Data to the Accumulator,  Up: Running Statistics

22.1 Initializing the Accumulator
=================================

 -- Type: gsl_rstat_workspace

     This workspace contains parameters used to calculate various
     statistics and are updated after each data point is added to the
     accumulator.

 -- Function: gsl_rstat_workspace * gsl_rstat_alloc (void)

     This function allocates a workspace for computing running
     statistics.  The size of the workspace is O(1).

 -- Function: void gsl_rstat_free (gsl_rstat_workspace * w)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: int gsl_rstat_reset (gsl_rstat_workspace * w)

     This function resets the workspace ‘w’ to its initial state, so it
     can begin working on a new set of data.


File: gsl-ref.info,  Node: Adding Data to the Accumulator,  Next: Current Statistics,  Prev: Initializing the Accumulator,  Up: Running Statistics

22.2 Adding Data to the Accumulator
===================================

 -- Function: int gsl_rstat_add (const double x, gsl_rstat_workspace
          * w)

     This function adds the data point ‘x’ to the statistical
     accumulator, updating calculations of the mean, variance, standard
     deviation, skewness, kurtosis, and median.

 -- Function: size_t gsl_rstat_n (const gsl_rstat_workspace * w)

     This function returns the number of data so far added to the
     accumulator.


File: gsl-ref.info,  Node: Current Statistics,  Next: Quantiles,  Prev: Adding Data to the Accumulator,  Up: Running Statistics

22.3 Current Statistics
=======================

 -- Function: double gsl_rstat_min (const gsl_rstat_workspace * w)

     This function returns the minimum value added to the accumulator.

 -- Function: double gsl_rstat_max (const gsl_rstat_workspace * w)

     This function returns the maximum value added to the accumulator.

 -- Function: double gsl_rstat_mean (const gsl_rstat_workspace * w)

     This function returns the mean of all data added to the
     accumulator, defined as

          \Hat\mu = (1/N) \sum x_i

 -- Function: double gsl_rstat_variance (const gsl_rstat_workspace * w)

     This function returns the variance of all data added to the
     accumulator, defined as

          \Hat\sigma^2 = (1/(N-1)) \sum (x_i - \Hat\mu)^2

 -- Function: double gsl_rstat_sd (const gsl_rstat_workspace * w)

     This function returns the standard deviation of all data added to
     the accumulator, defined as the square root of the variance given
     above.

 -- Function: double gsl_rstat_sd_mean (const gsl_rstat_workspace * w)

     This function returns the standard deviation of the mean, defined
     as

          sd_mean = \Hat\sigma / \sqrt{N}

 -- Function: double gsl_rstat_rms (const gsl_rstat_workspace * w)

     This function returns the root mean square of all data added to the
     accumulator, defined as

          rms = \sqrt{{1 \over N} \sum x_i^2}

 -- Function: double gsl_rstat_skew (const gsl_rstat_workspace * w)

     This function returns the skewness of all data added to the
     accumulator, defined as

          skew = (1/N) \sum ((x_i - \Hat\mu)/\Hat\sigma)^3

 -- Function: double gsl_rstat_kurtosis (const gsl_rstat_workspace * w)

     This function returns the kurtosis of all data added to the
     accumulator, defined as

          kurtosis = ((1/N) \sum ((x_i - \Hat\mu)/\Hat\sigma)^4)  - 3

 -- Function: double gsl_rstat_median (gsl_rstat_workspace * w)

     This function returns an estimate of the median of the data added
     to the accumulator.


File: gsl-ref.info,  Node: Quantiles,  Next: Examples<16>,  Prev: Current Statistics,  Up: Running Statistics

22.4 Quantiles
==============

The functions in this section estimate quantiles dynamically without
storing the entire dataset, using the algorithm of Jain and Chlamtec,
1985.  Only five points (markers) are stored which represent the minimum
and maximum of the data, as well as current estimates of the p/2-, p-,
and (1+p)/2-quantiles.  Each time a new data point is added, the marker
positions and heights are updated.

 -- Type: gsl_rstat_quantile_workspace

     This workspace contains parameters for estimating quantiles of the
     current dataset

 -- Function: gsl_rstat_quantile_workspace * gsl_rstat_quantile_alloc
          (const double p)

     This function allocates a workspace for the dynamic estimation of
     ‘p’-quantiles, where ‘p’ is between 0 and 1.  The median
     corresponds to p = 0.5.  The size of the workspace is O(1).

 -- Function: void gsl_rstat_quantile_free (gsl_rstat_quantile_workspace
          * w)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: int gsl_rstat_quantile_reset (gsl_rstat_quantile_workspace
          * w)

     This function resets the workspace ‘w’ to its initial state, so it
     can begin working on a new set of data.

 -- Function: int gsl_rstat_quantile_add (const double x,
          gsl_rstat_quantile_workspace * w)

     This function updates the estimate of the p-quantile with the new
     data point ‘x’.

 -- Function: double gsl_rstat_quantile_get
          (gsl_rstat_quantile_workspace * w)

     This function returns the current estimate of the p-quantile.


File: gsl-ref.info,  Node: Examples<16>,  Next: References and Further Reading<16>,  Prev: Quantiles,  Up: Running Statistics

22.5 Examples
=============

Here is a basic example of how to use the statistical functions:

     #include <stdio.h>
     #include <gsl/gsl_rstat.h>

     int
     main(void)
     {
       double data[5] = {17.2, 18.1, 16.5, 18.3, 12.6};
       double mean, variance, largest, smallest, sd,
              rms, sd_mean, median, skew, kurtosis;
       gsl_rstat_workspace *rstat_p = gsl_rstat_alloc();
       size_t i, n;

       /* add data to rstat accumulator */
       for (i = 0; i < 5; ++i)
         gsl_rstat_add(data[i], rstat_p);

       mean     = gsl_rstat_mean(rstat_p);
       variance = gsl_rstat_variance(rstat_p);
       largest  = gsl_rstat_max(rstat_p);
       smallest = gsl_rstat_min(rstat_p);
       median   = gsl_rstat_median(rstat_p);
       sd       = gsl_rstat_sd(rstat_p);
       sd_mean  = gsl_rstat_sd_mean(rstat_p);
       skew     = gsl_rstat_skew(rstat_p);
       rms      = gsl_rstat_rms(rstat_p);
       kurtosis = gsl_rstat_kurtosis(rstat_p);
       n        = gsl_rstat_n(rstat_p);

       printf ("The dataset is %g, %g, %g, %g, %g\n",
              data[0], data[1], data[2], data[3], data[4]);

       printf ("The sample mean is %g\n", mean);
       printf ("The estimated variance is %g\n", variance);
       printf ("The largest value is %g\n", largest);
       printf ("The smallest value is %g\n", smallest);
       printf( "The median is %g\n", median);
       printf( "The standard deviation is %g\n", sd);
       printf( "The root mean square is %g\n", rms);
       printf( "The standard devation of the mean is %g\n", sd_mean);
       printf( "The skew is %g\n", skew);
       printf( "The kurtosis %g\n", kurtosis);
       printf( "There are %zu items in the accumulator\n", n);

       gsl_rstat_reset(rstat_p);
       n = gsl_rstat_n(rstat_p);
       printf( "There are %zu items in the accumulator\n", n);

       gsl_rstat_free(rstat_p);

       return 0;
     }

The program should produce the following output,

     The dataset is 17.2, 18.1, 16.5, 18.3, 12.6
     The sample mean is 16.54
     The estimated variance is 5.373
     The largest value is 18.3
     The smallest value is 12.6
     The median is 16.5
     The standard deviation is 2.31797
     The root mean square is 16.6694
     The standard devation of the mean is 1.03663
     The skew is -0.829058
     The kurtosis -1.2217
     There are 5 items in the accumulator
     There are 0 items in the accumulator

This next program estimates the lower quartile, median and upper
quartile from 10,000 samples of a random Rayleigh distribution, using
the P^2 algorithm of Jain and Chlamtec.  For comparison, the exact
values are also computed from the sorted dataset.

     #include <stdio.h>
     #include <stdlib.h>
     #include <gsl/gsl_rstat.h>
     #include <gsl/gsl_statistics.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_sort.h>

     int
     main(void)
     {
       const size_t N = 10000;
       double *data = malloc(N * sizeof(double));
       gsl_rstat_quantile_workspace *work_25 = gsl_rstat_quantile_alloc(0.25);
       gsl_rstat_quantile_workspace *work_50 = gsl_rstat_quantile_alloc(0.5);
       gsl_rstat_quantile_workspace *work_75 = gsl_rstat_quantile_alloc(0.75);
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       double exact_p25, exact_p50, exact_p75;
       double val_p25, val_p50, val_p75;
       size_t i;

       /* add data to quantile accumulators; also store data for exact
        * comparisons */
       for (i = 0; i < N; ++i)
         {
           data[i] = gsl_ran_rayleigh(r, 1.0);
           gsl_rstat_quantile_add(data[i], work_25);
           gsl_rstat_quantile_add(data[i], work_50);
           gsl_rstat_quantile_add(data[i], work_75);
         }

       /* exact values */
       gsl_sort(data, 1, N);
       exact_p25 = gsl_stats_quantile_from_sorted_data(data, 1, N, 0.25);
       exact_p50 = gsl_stats_quantile_from_sorted_data(data, 1, N, 0.5);
       exact_p75 = gsl_stats_quantile_from_sorted_data(data, 1, N, 0.75);

       /* estimated values */
       val_p25 = gsl_rstat_quantile_get(work_25);
       val_p50 = gsl_rstat_quantile_get(work_50);
       val_p75 = gsl_rstat_quantile_get(work_75);

       printf ("The dataset is %g, %g, %g, %g, %g, ...\n",
              data[0], data[1], data[2], data[3], data[4]);

       printf ("0.25 quartile: exact = %.5f, estimated = %.5f, error = %.6e\n",
               exact_p25, val_p25, (val_p25 - exact_p25) / exact_p25);
       printf ("0.50 quartile: exact = %.5f, estimated = %.5f, error = %.6e\n",
               exact_p50, val_p50, (val_p50 - exact_p50) / exact_p50);
       printf ("0.75 quartile: exact = %.5f, estimated = %.5f, error = %.6e\n",
               exact_p75, val_p75, (val_p75 - exact_p75) / exact_p75);

       gsl_rstat_quantile_free(work_25);
       gsl_rstat_quantile_free(work_50);
       gsl_rstat_quantile_free(work_75);
       gsl_rng_free(r);
       free(data);

       return 0;
     }

The program should produce the following output,

     The dataset is 0.00645272, 0.0074002, 0.0120706, 0.0207256, 0.0227282, ...
     0.25 quartile: exact = 0.75766, estimated = 0.75580, error = -2.450209e-03
     0.50 quartile: exact = 1.17508, estimated = 1.17438, error = -5.995912e-04
     0.75 quartile: exact = 1.65347, estimated = 1.65696, error = 2.110571e-03


File: gsl-ref.info,  Node: References and Further Reading<16>,  Prev: Examples<16>,  Up: Running Statistics

22.6 References and Further Reading
===================================

The algorithm used to dynamically estimate p-quantiles is described in
the paper,

   * R. Jain and I. Chlamtac.  `The P^2 algorithm for dynamic
     calculation of quantiles and histograms without storing
     observations', Communications of the ACM, Volume 28 (October),
     Number 10, 1985, p.  1076-1085.


File: gsl-ref.info,  Node: Moving Window Statistics,  Next: Digital Filtering,  Prev: Running Statistics,  Up: Top

23 Moving Window Statistics
***************************

This chapter describes routines for computing `moving window statistics'
(also called rolling statistics and running statistics), using a window
around a sample which is used to calculate various local statistical
properties of an input data stream.  The window is then slid forward by
one sample to process the next data point and so on.

The functions described in this chapter are declared in the header file
‘gsl_movstat.h’.

* Menu:

* Introduction: Introduction<4>.
* Handling Endpoints::
* Allocation for Moving Window Statistics::
* Moving Mean::
* Moving Variance and Standard Deviation::
* Moving Minimum and Maximum::
* Moving Sum::
* Moving Median::
* Robust Scale Estimation::
* User-defined Moving Statistics::
* Accumulators::
* Examples: Examples<17>.
* References and Further Reading: References and Further Reading<17>.


File: gsl-ref.info,  Node: Introduction<4>,  Next: Handling Endpoints,  Up: Moving Window Statistics

23.1 Introduction
=================

This chapter is concerned with calculating various statistics from
subsets of a given dataset.  The main idea is to compute statistics in
the vicinity of a given data sample by defining a `window' which
includes the sample itself as well as some specified number of samples
before and after the sample in question.  For a sample x_i, we define a
window W_i^{H,J} as

     W_i^{H,J} = {x_{i-H},...,x_i,...,x_{i+J}}

The parameters H and J are non-negative integers specifying the number
of samples to include before and after the sample x_i.  Statistics such
as the mean and standard deviation of the window W_i^{H,J} may be
computed, and then the window is shifted forward by one sample to focus
on x_{i+1}.  The total number of samples in the window is K = H + J + 1.
To define a symmetric window centered on x_i, one would set H = J =
\left\lfloor K / 2 \right\rfloor.


File: gsl-ref.info,  Node: Handling Endpoints,  Next: Allocation for Moving Window Statistics,  Prev: Introduction<4>,  Up: Moving Window Statistics

23.2 Handling Endpoints
=======================

When processing samples near the ends of the input signal, there will
not be enough samples to fill the window W_i^{H,J} defined above.
Therefore the user must specify how to construct the windows near the
end points.  This is done by passing an input argument of type *note
gsl_movstat_end_t: 824.:

 -- Type: gsl_movstat_end_t

     This data type specifies how to construct windows near end points
     and can be selected from the following choices:

      -- Macro: GSL_MOVSTAT_END_PADZERO

          With this option, a full window of length K will be
          constructed by inserting zeros into the window near the signal
          end points.  Effectively, the input signal is modified to

               x~ = {0, ..., 0, x_1, x_2, ..., x_{n-1}, x_n, 0, ..., 0}

          to ensure a well-defined window for all x_i.

      -- Macro: GSL_MOVSTAT_END_PADVALUE

          With this option, a full window of length K will be
          constructed by padding the window with the first and last
          sample in the input signal.  Effectively, the input signal is
          modified to

               x~ = {x_1, ..., x_1, x_1, x_2, ..., x_{n-1}, x_n, x_n, ..., x_n}

      -- Macro: GSL_MOVSTAT_END_TRUNCATE

          With this option, no padding is performed, and the windows are
          simply truncated as the end points are approached.


File: gsl-ref.info,  Node: Allocation for Moving Window Statistics,  Next: Moving Mean,  Prev: Handling Endpoints,  Up: Moving Window Statistics

23.3 Allocation for Moving Window Statistics
============================================

 -- Type: gsl_movstat_workspace

     The moving window statistical routines use a common workspace.

 -- Function: gsl_movstat_workspace * gsl_movstat_alloc (const size_t K)

     This function allocates a workspace for computing symmetric,
     centered moving statistics with a window length of K samples.  In
     this case, H = J = \left\lfloor K/2 \right\rfloor.  The size of the
     workspace is O(7K).

 -- Function: gsl_movstat_workspace * gsl_movstat_alloc2 (const
          size_t H, const size_t J)

     This function allocates a workspace for computing moving statistics
     using a window with H samples prior to the current sample, and J
     samples after the current sample.  The total window size is K = H +
     J + 1.  The size of the workspace is O(7K).

 -- Function: void * gsl_movstat_free (gsl_movstat_workspace * w)

     This function frees the memory associated with ‘w’.


File: gsl-ref.info,  Node: Moving Mean,  Next: Moving Variance and Standard Deviation,  Prev: Allocation for Moving Window Statistics,  Up: Moving Window Statistics

23.4 Moving Mean
================

The moving window mean calculates the mean of the values of each window
W_i^{H,J}.

     \hat{\mu}_i = 1/| W_i^{H,J} | \sum_{x_m \in W_i^{H,J}} x_m

Here, \left| W_i^{H,J} \right| represents the number of elements in the
window W_i^{H,J}.  This will normally be K, unless the *note
GSL_MOVSTAT_END_TRUNCATE: 827. option is selected, in which case it
could be less than K near the signal end points.

 -- Function: int gsl_movstat_mean (const gsl_movstat_end_t endtype,
          const gsl_vector * x, gsl_vector * y, gsl_movstat_workspace
          * w)

     This function computes the moving window mean of the input vector
     ‘x’, storing the output in ‘y’.  The parameter ‘endtype’ specifies
     how windows near the ends of the input should be handled.  It is
     allowed to have ‘x’ = ‘y’ for an in-place moving mean.


File: gsl-ref.info,  Node: Moving Variance and Standard Deviation,  Next: Moving Minimum and Maximum,  Prev: Moving Mean,  Up: Moving Window Statistics

23.5 Moving Variance and Standard Deviation
===========================================

The moving window variance calculates the `sample variance' of the
values of each window W_i^{H,J}, defined by

     \hat{\sigma}_i^2 = 1/(|W_i^{H,J}| - 1) \sum_{x_m \in W_i^{H,J}} ( x_m - \hat{\mu}_i )^2

where \hat{\mu}_i is the mean of W_i^{H,J} defined above.  The standard
deviation \hat{\sigma}_i is the square root of the variance.

 -- Function: int gsl_movstat_variance (const gsl_movstat_end_t endtype,
          const gsl_vector * x, gsl_vector * y, gsl_movstat_workspace
          * w)

     This function computes the moving window variance of the input
     vector ‘x’, storing the output in ‘y’.  The parameter ‘endtype’
     specifies how windows near the ends of the input should be handled.
     It is allowed to have ‘x’ = ‘y’ for an in-place moving variance.

 -- Function: int gsl_movstat_sd (const gsl_movstat_end_t endtype, const
          gsl_vector * x, gsl_vector * y, gsl_movstat_workspace * w)

     This function computes the moving window standard deviation of the
     input vector ‘x’, storing the output in ‘y’.  The parameter
     ‘endtype’ specifies how windows near the ends of the input should
     be handled.  It is allowed to have ‘x’ = ‘y’ for an in-place moving
     standard deviation.


File: gsl-ref.info,  Node: Moving Minimum and Maximum,  Next: Moving Sum,  Prev: Moving Variance and Standard Deviation,  Up: Moving Window Statistics

23.6 Moving Minimum and Maximum
===============================

The moving minimum/maximum calculates the minimum and maximum values of
each window W_i^{H,J}.

     y_i^{min} = \min W_i^{H,J}
     y_i^{max} = \max W_i^{H,J}

 -- Function: int gsl_movstat_min (const gsl_movstat_end_t endtype,
          const gsl_vector * x, gsl_vector * y, gsl_movstat_workspace
          * w)

     This function computes the moving minimum of the input vector ‘x’,
     storing the result in ‘y’.  The parameter ‘endtype’ specifies how
     windows near the ends of the input should be handled.  It is
     allowed to have ‘x’ = ‘y’ for an in-place moving minimum.

 -- Function: int gsl_movstat_max (const gsl_movstat_end_t endtype,
          const gsl_vector * x, gsl_vector * y, gsl_movstat_workspace
          * w)

     This function computes the moving maximum of the input vector ‘x’,
     storing the result in ‘y’.  The parameter ‘endtype’ specifies how
     windows near the ends of the input should be handled.  It is
     allowed to have ‘x’ = ‘y’ for an in-place moving maximum.

 -- Function: int gsl_movstat_minmax (const gsl_movstat_end_t endtype,
          const gsl_vector * x, gsl_vector * y_min, gsl_vector * y_max,
          gsl_movstat_workspace * w)

     This function computes the moving minimum and maximum of the input
     vector ‘x’, storing the window minimums in ‘y_min’ and the window
     maximums in ‘y_max’.  The parameter ‘endtype’ specifies how windows
     near the ends of the input should be handled.


File: gsl-ref.info,  Node: Moving Sum,  Next: Moving Median,  Prev: Moving Minimum and Maximum,  Up: Moving Window Statistics

23.7 Moving Sum
===============

The moving window sum calculates the sum of the values of each window
W_i^{H,J}.

     y_i = \sum_{x_m \in W_i^{H,J}} x_m

 -- Function: int gsl_movstat_sum (const gsl_movstat_end_t endtype,
          const gsl_vector * x, gsl_vector * y, gsl_movstat_workspace
          * w)

     This function computes the moving window sum of the input vector
     ‘x’, storing the output in ‘y’.  The parameter ‘endtype’ specifies
     how windows near the ends of the input should be handled.  It is
     allowed to have ‘x’ = ‘y’ for an in-place moving sum.


File: gsl-ref.info,  Node: Moving Median,  Next: Robust Scale Estimation,  Prev: Moving Sum,  Up: Moving Window Statistics

23.8 Moving Median
==================

The moving median calculates the median of the window W_i^{H,J} for each
sample x_i:

     y_i = median(W_i^{H,J})

 -- Function: int gsl_movstat_median (const gsl_movstat_end_t endtype,
          const gsl_vector * x, gsl_vector * y, gsl_movstat_workspace
          * w)

     This function computes the moving median of the input vector ‘x’,
     storing the output in ‘y’.  The parameter ‘endtype’ specifies how
     windows near the ends of the input should be handled.  It is
     allowed for ‘x’ = ‘y’ for an in-place moving window median.


File: gsl-ref.info,  Node: Robust Scale Estimation,  Next: User-defined Moving Statistics,  Prev: Moving Median,  Up: Moving Window Statistics

23.9 Robust Scale Estimation
============================

A common problem in statistics is to quantify the dispersion (also known
as the variability, scatter, and spread) of a set of data.  Often this
is done by calculating the variance or standard deviation.  However
these statistics are strongly influenced by outliers, and can often
provide erroneous results when even a small number of outliers are
present.

Several useful statistics have emerged to provide robust estimates of
scale which are not as susceptible to data outliers.  A few of these
statistical scale estimators are described below.

* Menu:

* Moving MAD::
* Moving QQR::
* Moving S_n::
* Moving Q_n::


File: gsl-ref.info,  Node: Moving MAD,  Next: Moving QQR,  Up: Robust Scale Estimation

23.9.1 Moving MAD
-----------------

The median absolute deviation (MAD) for the window W_i^{H,J} is defined
to be the median of the absolute deviations from the window’s median:

     MAD_i = 1.4826 * median[ |W_i^{H,J} - median(W_i^{H,J})| ]

The factor of 1.4826 makes the MAD an unbiased estimator of the standard
deviation for Gaussian data.  The MAD has an efficiency of 37%.  See
*note here: 7f5. for more information.

 -- Function: int gsl_movstat_mad0 (const gsl_movstat_end_t endtype,
          const gsl_vector * x, gsl_vector * xmedian, gsl_vector * xmad,
          gsl_movstat_workspace * w)
 -- Function: int gsl_movstat_mad (const gsl_movstat_end_t endtype,
          const gsl_vector * x, gsl_vector * xmedian, gsl_vector * xmad,
          gsl_movstat_workspace * w)

     These functions compute the moving MAD of the input vector ‘x’ and
     store the result in ‘xmad’.  The medians of each window W_i^{H,J}
     are stored in ‘xmedian’ on output.  The inputs ‘x’, ‘xmedian’, and
     ‘xmad’ must all be the same length.  The parameter ‘endtype’
     specifies how windows near the ends of the input should be handled.
     The function ‘mad0’ does not include the scale factor of 1.4826,
     while the function ‘mad’ does include this factor.


File: gsl-ref.info,  Node: Moving QQR,  Next: Moving S_n,  Prev: Moving MAD,  Up: Robust Scale Estimation

23.9.2 Moving QQR
-----------------

The q-quantile range (QQR) is the difference between the (1-q) and q
quantiles of a set of data,

     QQR = Q_{1-q} - Q_q

The case q = 0.25 corresponds to the well-known `interquartile range
(IQR)', which is the difference between the 75th and 25th percentiles of
a set of data.  The QQR is a `trimmed estimator', the main idea being to
discard the largest and smallest values in a data window and compute a
scale estimate from the remaining middle values.  In the case of the
IQR, the largest and smallest 25% of the data are discarded and the
scale is estimated from the remaining (middle) 50%.

 -- Function: int gsl_movstat_qqr (const gsl_movstat_end_t endtype,
          const gsl_vector * x, const double q, gsl_vector * xqqr,
          gsl_movstat_workspace * w)

     This function computes the moving QQR of the input vector ‘x’ and
     stores the q-quantile ranges of each window W_i^{H,J} in ‘xqqr’.
     The quantile parameter ‘q’ must be between 0 and 0.5.  The input q
     = 0.25 corresponds to the IQR. The inputs ‘x’ and ‘xqqr’ must be
     the same length.  The parameter ‘endtype’ specifies how windows
     near the ends of the input should be handled.


File: gsl-ref.info,  Node: Moving S_n,  Next: Moving Q_n,  Prev: Moving QQR,  Up: Robust Scale Estimation

23.9.3 Moving S_n
-----------------

The S_n statistic proposed by Croux and Rousseeuw is based on pairwise
differences between all samples in the window.  It has an efficiency of
58%, significantly higher than the MAD. See *note here: 7f9. for more
information.

 -- Function: int gsl_movstat_Sn (const gsl_movstat_end_t endtype, const
          gsl_vector * x, gsl_vector * xscale, gsl_movstat_workspace
          * w)

     This function computes the moving S_n of the input vector ‘x’ and
     stores the output in ‘xscale’.  The inputs ‘x’ and ‘xscale’ must be
     the same length.  The parameter ‘endtype’ specifies how windows
     near the ends of the input should be handled.  It is allowed for
     ‘x’ = ‘xscale’ for an in-place moving window S_n.


File: gsl-ref.info,  Node: Moving Q_n,  Prev: Moving S_n,  Up: Robust Scale Estimation

23.9.4 Moving Q_n
-----------------

The Q_n statistic proposed by Croux and Rousseeuw is loosely based on
the Hodges-Lehmann location estimator.  It has a relatively high
efficiency of 82%.  See *note here: 7fd. for more information.

 -- Function: int gsl_movstat_Qn (const gsl_movstat_end_t endtype, const
          gsl_vector * x, gsl_vector * xscale, gsl_movstat_workspace
          * w)

     This function computes the moving Q_n of the input vector ‘x’ and
     stores the output in ‘xscale’.  The inputs ‘x’ and ‘xscale’ must be
     the same length.  The parameter ‘endtype’ specifies how windows
     near the ends of the input should be handled.  It is allowed for
     ‘x’ = ‘xscale’ for an in-place moving window Q_n.


File: gsl-ref.info,  Node: User-defined Moving Statistics,  Next: Accumulators,  Prev: Robust Scale Estimation,  Up: Moving Window Statistics

23.10 User-defined Moving Statistics
====================================

GSL offers an interface for users to define their own moving window
statistics functions, without needing to implement the edge-handling and
accumulator machinery.  This can be done by explicitly constructing the
windows W_i^{H,J} for a given input signal (*note gsl_movstat_fill():
845.), or by calculating a user-defined function for each window
automatically.  In order to apply a user-defined function to each
window, users must define a variable of type *note gsl_movstat_function:
846. to pass into *note gsl_movstat_apply(): 847.  This structure is
defined as follows.

 -- Type: gsl_movstat_function

     Structure specifying user-defined moving window statistical
     function:

          typedef struct
          {
            double (* function) (const size_t n, double x[], void * params);
            void * params;
          } gsl_movstat_function;

     This structure contains a pointer to the user-defined function as
     well as possible parameters to pass to the function.

      -- Member: double (* function) (const size_t n, double x[], void
               * params)

          This function returns the user-defined statistic of the array
          ‘x’ of length ‘n’.  User-specified parameters are passed in
          via ‘params’.  It is allowed to modify the array ‘x’.

      -- Member: void * params

          User-specified parameters to be passed into the function.

 -- Function: int gsl_movstat_apply (const gsl_movstat_end_t endtype,
          const gsl_movstat_function * F, const gsl_vector * x,
          gsl_vector * y, gsl_movstat_workspace * w)

     This function applies the user-defined moving window statistic
     specified in ‘F’ to the input vector ‘x’, storing the output in
     ‘y’.  The parameter ‘endtype’ specifies how windows near the ends
     of the input should be handled.  It is allowed for ‘x’ = ‘y’ for an
     in-place moving window calculation.

 -- Function: size_t gsl_movstat_fill (const gsl_movstat_end_t endtype,
          const gsl_vector * x, const size_t idx, const size_t H, const
          size_t J, double * window)

     This function explicitly constructs the sliding window for the
     input vector ‘x’ which is centered on the sample ‘idx’.  On output,
     the array ‘window’ will contain W_{idx}^{H,J}.  The number of
     samples to the left and right of the sample ‘idx’ are specified by
     ‘H’ and ‘J’ respectively.  The parameter ‘endtype’ specifies how
     windows near the ends of the input should be handled.  The function
     returns the size of the window.


File: gsl-ref.info,  Node: Accumulators,  Next: Examples<17>,  Prev: User-defined Moving Statistics,  Up: Moving Window Statistics

23.11 Accumulators
==================

Many of the algorithms of this chapter are based on an accumulator
design, which process the input vector one sample at a time, updating
calculations of the desired statistic for the current window.  Each
accumulator is stored in the following structure:

 -- Type: gsl_movstat_accum

     Structure specifying accumulator for moving window statistics:

          typedef struct
          {
            size_t (* size) (const size_t n);
            int (* init) (const size_t n, void * vstate);
            int (* insert) (const double x, void * vstate);
            int (* delete) (void * vstate);
            int (* get) (void * params, double * result, const void * vstate);
          } gsl_movstat_accum;

     The structure contains function pointers responsible for performing
     different tasks for the accumulator.

      -- Member: size_t (* size) (const size_t n)

          This function returns the size of the workspace (in bytes)
          needed by the accumulator for a moving window of length ‘n’.

      -- Member: int (* init) (const size_t n, void * vstate)

          This function initializes the workspace ‘vstate’ for a moving
          window of length ‘n’.

      -- Member: int (* insert) (const double x, void * vstate)

          This function inserts a single sample ‘x’ into the
          accumulator, updating internal calculations of the desired
          statistic.  If the accumulator is full (i.e.  n samples have
          already been inserted), then the oldest sample is deleted from
          the accumulator.

      -- Member: int (* delete) (void * vstate)

          This function deletes the oldest sample from the accumulator,
          updating internal calculations of the desired statistic.

      -- Member: int (* get) (void * params, double * result, const void
               * vstate)

          This function stores the desired statistic for the current
          window in ‘result’.  The input ‘params’ specifies optional
          parameters for calculating the statistic.

The following accumulators of type *note gsl_movstat_accum: 84b. are
defined by GSL to perform moving window statistics calculations.

 -- Variable: gsl_movstat_accum_min
 -- Variable: gsl_movstat_accum_max
 -- Variable: gsl_movstat_accum_minmax

     These accumulators calculate moving window minimum/maximums
     efficiently, using the algorithm of D. Lemire.

 -- Variable: gsl_movstat_accum_mean
 -- Variable: gsl_movstat_accum_sd
 -- Variable: gsl_movstat_accum_variance

     These accumulators calculate the moving window mean, standard
     deviation, and variance, using the algorithm of B. P. Welford.

 -- Variable: gsl_movstat_median

     This accumulator calculates the moving window median using the
     min/max heap algorithm of Härdle and Steiger.

 -- Variable: gsl_movstat_accum_Sn
 -- Variable: gsl_movstat_accum_Qn

     These accumulators calculate the moving window S_n and Q_n
     statistics developed by Croux and Rousseeuw.

 -- Variable: gsl_movstat_accum_sum

     This accumulator calculates the moving window sum.

 -- Variable: gsl_movstat_accum_qqr

     This accumulator calculates the moving window q-quantile range.


File: gsl-ref.info,  Node: Examples<17>,  Next: References and Further Reading<17>,  Prev: Accumulators,  Up: Moving Window Statistics

23.12 Examples
==============

* Menu:

* Example 1::
* Example 2; Robust Scale: Example 2 Robust Scale.
* Example 3; User-defined Moving Window: Example 3 User-defined Moving Window.


File: gsl-ref.info,  Node: Example 1,  Next: Example 2 Robust Scale,  Up: Examples<17>

23.12.1 Example 1
-----------------

The following example program computes the moving mean, minimum and
maximum of a noisy sinusoid signal of length N = 500 with a symmetric
moving window of size K = 11.

[image src="movstat1.png"]

Figure: Original signal time series (gray) with moving mean (green),
moving minimum (blue), and moving maximum (orange).

The program is given below.

     #include <stdio.h>
     #include <stdlib.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_movstat.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_vector.h>

     int
     main(void)
     {
       const size_t N = 500;                      /* length of time series */
       const size_t K = 11;                       /* window size */
       gsl_movstat_workspace * w = gsl_movstat_alloc(K);
       gsl_vector *x = gsl_vector_alloc(N);
       gsl_vector *xmean = gsl_vector_alloc(N);
       gsl_vector *xmin = gsl_vector_alloc(N);
       gsl_vector *xmax = gsl_vector_alloc(N);
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       size_t i;

       for (i = 0; i < N; ++i)
         {
           double xi = cos(4.0 * M_PI * i / (double) N);
           double ei = gsl_ran_gaussian(r, 0.1);

           gsl_vector_set(x, i, xi + ei);
         }

       /* compute moving statistics */
       gsl_movstat_mean(GSL_MOVSTAT_END_PADVALUE, x, xmean, w);
       gsl_movstat_minmax(GSL_MOVSTAT_END_PADVALUE, x, xmin, xmax, w);

       /* print results */
       for (i = 0; i < N; ++i)
         {
           printf("%zu %f %f %f %f\n",
                  i,
                  gsl_vector_get(x, i),
                  gsl_vector_get(xmean, i),
                  gsl_vector_get(xmin, i),
                  gsl_vector_get(xmax, i));
         }

       gsl_vector_free(x);
       gsl_vector_free(xmean);
       gsl_rng_free(r);
       gsl_movstat_free(w);

       return 0;
     }


File: gsl-ref.info,  Node: Example 2 Robust Scale,  Next: Example 3 User-defined Moving Window,  Prev: Example 1,  Up: Examples<17>

23.12.2 Example 2: Robust Scale
-------------------------------

The following example program analyzes a time series of length N = 1000
composed of Gaussian random variates with zero mean whose standard
deviation changes in a piecewise constant fashion as shown in the table
below.

Sample Range     \sigma
                 
------------------------------------
                 
1-200            1.0
                 
                 
201-450          5.0
                 
                 
451-600          1.0
                 
                 
601-850          3.0
                 
                 
851-1000         5.0
                 

Additionally, about 1% of the samples are perturbed to represent
outliers by adding \pm 15 to the random Gaussian variate.  The program
calculates the moving statistics MAD, IQR, S_n, Q_n, and the standard
deviation using a symmetric moving window of length K = 41.  The results
are shown in Fig.  %s.

[image src="movstat2.png"]

Figure: Top: time series of piecewise constant variance.  Bottom: scale
estimates using a moving window; the true sigma value is in light blue,
MAD in green, IQR in red, S_n in yellow, and Q_n in dark blue.  The
moving standard deviation is shown in gray.

The robust statistics follow the true standard deviation piecewise
changes well, without being influenced by the outliers.  The moving
standard deviation (gray curve) is heavily influenced by the presence of
the outliers.  The program is given below.

     #include <stdio.h>
     #include <stdlib.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_movstat.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_vector.h>

     int
     main(void)
     {
       const size_t N = 1000;                                 /* length of time series */
       const double sigma[] = { 1.0, 5.0, 1.0, 3.0, 5.0 };    /* variances */
       const size_t N_sigma[] = { 200, 450, 600, 850, 1000 }; /* samples where variance changes */
       const size_t K = 41;                                   /* window size */
       gsl_vector *x = gsl_vector_alloc(N);
       gsl_vector *xmedian = gsl_vector_alloc(N);
       gsl_vector *xmad = gsl_vector_alloc(N);
       gsl_vector *xiqr = gsl_vector_alloc(N);
       gsl_vector *xSn = gsl_vector_alloc(N);
       gsl_vector *xQn = gsl_vector_alloc(N);
       gsl_vector *xsd = gsl_vector_alloc(N);
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       gsl_movstat_workspace * w = gsl_movstat_alloc(K);
       size_t idx = 0;
       size_t i;

       for (i = 0; i < N; ++i)
         {
           double gi = gsl_ran_gaussian(r, sigma[idx]);
           double u = gsl_rng_uniform(r);
           double outlier = (u < 0.01) ? 15.0*GSL_SIGN(gi) : 0.0;
           double xi = gi + outlier;

           gsl_vector_set(x, i, xi);

           if (i == N_sigma[idx] - 1)
             ++idx;
         }

       /* compute moving statistics */
       gsl_movstat_mad(GSL_MOVSTAT_END_TRUNCATE, x, xmedian, xmad, w);
       gsl_movstat_qqr(GSL_MOVSTAT_END_TRUNCATE, x, 0.25, xiqr, w);
       gsl_movstat_Sn(GSL_MOVSTAT_END_TRUNCATE, x, xSn, w);
       gsl_movstat_Qn(GSL_MOVSTAT_END_TRUNCATE, x, xQn, w);
       gsl_movstat_sd(GSL_MOVSTAT_END_TRUNCATE, x, xsd, w);

       /* scale IQR by factor to approximate standard deviation */
       gsl_vector_scale(xiqr, 0.7413);

       /* print results */
       idx = 0;
       for (i = 0; i < N; ++i)
         {
           printf("%zu %f %f %f %f %f %f %f\n",
                  i,
                  gsl_vector_get(x, i),
                  sigma[idx],
                  gsl_vector_get(xmad, i),
                  gsl_vector_get(xiqr, i),
                  gsl_vector_get(xSn, i),
                  gsl_vector_get(xQn, i),
                  gsl_vector_get(xsd, i));

           if (i == N_sigma[idx] - 1)
             ++idx;
         }

       gsl_vector_free(x);
       gsl_vector_free(xmedian);
       gsl_vector_free(xmad);
       gsl_vector_free(xiqr);
       gsl_vector_free(xSn);
       gsl_vector_free(xQn);
       gsl_vector_free(xsd);
       gsl_rng_free(r);
       gsl_movstat_free(w);

       return 0;
     }


File: gsl-ref.info,  Node: Example 3 User-defined Moving Window,  Prev: Example 2 Robust Scale,  Up: Examples<17>

23.12.3 Example 3: User-defined Moving Window
---------------------------------------------

This example program illustrates how a user can define their own moving
window function to apply to an input vector.  It constructs a random
noisy time series of length N = 1000 with some outliers added.  Then it
applies a moving window trimmed mean to the time series with trim
parameter \alpha = 0.1.  The length of the moving window is K = 11, so
the smallest and largest sample of each window is discarded prior to
computing the mean.  The results are shown in Fig.  %s.

[image src="movstat3.png"]

Figure: Noisy time series data (black) with moving window trimmed mean
(red)

The program is given below.

     #include <stdio.h>
     #include <stdlib.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_movstat.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_sort.h>
     #include <gsl/gsl_statistics.h>

     double
     func(const size_t n, double x[], void * params)
     {
       const double alpha = *(double *) params;

       gsl_sort(x, 1, n);

       return gsl_stats_trmean_from_sorted_data(alpha, x, 1, n);
     }

     int
     main(void)
     {
       const size_t N = 1000;                /* length of time series */
       const size_t K = 11;                  /* window size */
       double alpha = 0.1;                   /* trimmed mean parameter */
       gsl_vector *x = gsl_vector_alloc(N);  /* input vector */
       gsl_vector *y = gsl_vector_alloc(N);  /* filtered output vector for alpha1 */
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       gsl_movstat_workspace *w = gsl_movstat_alloc(K);
       gsl_movstat_function F;
       size_t i;
       double sum = 0.0;

       /* generate input signal */
       for (i = 0; i < N; ++i)
         {
           double ui = gsl_ran_gaussian(r, 1.0);
           double outlier = (gsl_rng_uniform(r) < 0.01) ? 10.0*GSL_SIGN(ui) : 0.0;
           sum += ui;
           gsl_vector_set(x, i, sum + outlier);
         }

       /* apply moving window function */
       F.function = func;
       F.params = &alpha;
       gsl_movstat_apply(GSL_MOVSTAT_END_PADVALUE, &F, x, y, w);

       /* print results */
       for (i = 0; i < N; ++i)
         {
           double xi = gsl_vector_get(x, i);
           double yi = gsl_vector_get(y, i);

           printf("%f %f\n", xi, yi);
         }

       gsl_vector_free(x);
       gsl_vector_free(y);
       gsl_rng_free(r);
       gsl_movstat_free(w);

       return 0;
     }


File: gsl-ref.info,  Node: References and Further Reading<17>,  Prev: Examples<17>,  Up: Moving Window Statistics

23.13 References and Further Reading
====================================

The following publications are relevant to the algorithms described in
this chapter,

   * W.Hardle and W. Steiger, `Optimal Median Smoothing', Appl.
     Statist., 44 (2), 1995.

   * D. Lemire, `Streaming Maximum-Minimum Filter Using No More than
     Three Comparisons per Element', Nordic Journal of Computing, 13
     (4), 2006 (‘https://arxiv.org/abs/cs/0610046’).

   * B. P. Welford, `Note on a method for calculating corrected sums of
     squares and products', Technometrics, 4 (3), 1962.


File: gsl-ref.info,  Node: Digital Filtering,  Next: Histograms,  Prev: Moving Window Statistics,  Up: Top

24 Digital Filtering
********************

* Menu:

* Introduction: Introduction<5>.
* Handling Endpoints: Handling Endpoints<2>.
* Linear Digital Filters::
* Nonlinear Digital Filters::
* Examples: Examples<18>.
* References and Further Reading: References and Further Reading<18>.


File: gsl-ref.info,  Node: Introduction<5>,  Next: Handling Endpoints<2>,  Up: Digital Filtering

24.1 Introduction
=================

The filters discussed in this chapter are based on the following moving
data window which is centered on i-th sample:

     W_i^H = { x_{i-H}, ..., x_i, ..., x_{i+H} }

Here, H is a non-negative integer called the `window half-length', which
represents the number of samples before and after sample i.  The total
window length is K = 2 H + 1.


File: gsl-ref.info,  Node: Handling Endpoints<2>,  Next: Linear Digital Filters,  Prev: Introduction<5>,  Up: Digital Filtering

24.2 Handling Endpoints
=======================

When processing samples near the ends of the input signal, there will
not be enough samples to fill the window W_i^H defined above.  Therefore
the user must specify how to construct the windows near the end points.
This is done by passing an input argument of type *note
gsl_filter_end_t: 867.:

 -- Type: gsl_filter_end_t

     This data type specifies how to construct windows near end points
     and can be selected from the following choices:

      -- Macro: GSL_FILTER_END_PADZERO

          With this option, a full window of length K will be
          constructed by inserting zeros into the window near the signal
          end points.  Effectively, the input signal is modified to

               x~ = { 0, ..., 0, x_1, x_2, ..., x_{n-1}, x_n, 0, ..., 0 }

          to ensure a well-defined window for all x_i.

      -- Macro: GSL_FILTER_END_PADVALUE

          With this option, a full window of length K will be
          constructed by padding the window with the first and last
          sample in the input signal.  Effectively, the input signal is
          modified to

               x~ = { x_1, ..., x_1, x_1, x_2, ..., x_{n-1}, x_n, x_n, ..., x_n }

      -- Macro: GSL_FILTER_END_TRUNCATE

          With this option, no padding is performed, and the windows are
          simply truncated as the end points are approached.


File: gsl-ref.info,  Node: Linear Digital Filters,  Next: Nonlinear Digital Filters,  Prev: Handling Endpoints<2>,  Up: Digital Filtering

24.3 Linear Digital Filters
===========================

* Menu:

* Gaussian Filter::


File: gsl-ref.info,  Node: Gaussian Filter,  Up: Linear Digital Filters

24.3.1 Gaussian Filter
----------------------

The Gaussian filter convolves the input signal with a Gaussian kernel or
window.  This filter is often used as a smoothing or noise reduction
filter.  The Gaussian kernel is defined by

     G(k) = e^{-1/2 ( \alpha k/((K-1)/2) )^2} = e^{-k^2/2\sigma^2}

for -(K-1)/2 \le k \le (K-1)/2, and K is the size of the kernel.  The
parameter \alpha specifies the number of standard deviations \sigma
desired in the kernel.  So for example setting \alpha = 3 would define a
Gaussian window of length K which spans \pm 3 \sigma.  It is often more
convenient to specify the parameter \alpha rather than the standard
deviation \sigma when constructing the kernel, since a fixed value of
\alpha would correspond to the same shape of Gaussian regardless of the
size K. The appropriate value of the standard deviation depends on K and
is related to \alpha as

     \sigma = (K - 1)/(2 \alpha)

The routines below accept \alpha as an input argument instead of \sigma.

The Gaussian filter offers a convenient way of differentiating and
smoothing an input signal in a single pass.  Using the derivative
property of a convolution,

     d/dt ( G * x ) = dG/dt * x

the input signal x(t) can be smoothed and differentiated at the same
time by convolution with a derivative Gaussian kernel, which can be
readily computed from the analytic expression above.  The same principle
applies to higher order derivatives.

 -- Function: gsl_filter_gaussian_workspace * gsl_filter_gaussian_alloc
          (const size_t K)

     This function initializes a workspace for Gaussian filtering using
     a kernel of size ‘K’.  Here, H = K / 2.  If K is even, it is
     rounded up to the next odd integer to ensure a symmetric window.
     The size of the workspace is O(K).

 -- Function: void gsl_filter_gaussian_free
          (gsl_filter_gaussian_workspace * w)

     This function frees the memory associated with ‘w’.

 -- Function: int gsl_filter_gaussian (const gsl_filter_end_t endtype,
          const double alpha, const size_t order, const gsl_vector * x,
          gsl_vector * y, gsl_filter_gaussian_workspace * w)

     This function applies a Gaussian filter parameterized by *note
     alpha: 49a. to the input vector ‘x’, storing the output in ‘y’.
     The derivative order is specified by ‘order’, with ‘0’
     corresponding to a Gaussian, ‘1’ corresponding to a first
     derivative Gaussian, and so on.  The parameter ‘endtype’ specifies
     how the signal end points are handled.  It is allowed for ‘x’ = ‘y’
     for an in-place filter.

 -- Function: int gsl_filter_gaussian_kernel (const double alpha, const
          size_t order, const int normalize, gsl_vector * kernel)

     This function constructs a Gaussian kernel parameterized by *note
     alpha: 49a. and stores the output in ‘kernel’.  The parameter
     ‘order’ specifies the derivative order, with ‘0’ corresponding to a
     Gaussian, ‘1’ corresponding to a first derivative Gaussian, and so
     on.  If ‘normalize’ is set to ‘1’, then the kernel will be
     normalized to sum to one on output.  If ‘normalize’ is set to ‘0’,
     no normalization is performed.


File: gsl-ref.info,  Node: Nonlinear Digital Filters,  Next: Examples<18>,  Prev: Linear Digital Filters,  Up: Digital Filtering

24.4 Nonlinear Digital Filters
==============================

The nonlinear digital filters described below are based on the window
median, which is given by

     m_i = median { W_i^H } = median { x_{i-H}, ..., x_i, ..., x_{i+H} }

The median is considered robust to local outliers, unlike the mean.
Median filters can preserve sharp edges while at the same removing
signal noise, and are used in a wide range of applications.

* Menu:

* Standard Median Filter::
* Recursive Median Filter::
* Impulse Detection Filter::


File: gsl-ref.info,  Node: Standard Median Filter,  Next: Recursive Median Filter,  Up: Nonlinear Digital Filters

24.4.1 Standard Median Filter
-----------------------------

The `standard median filter' (SMF) simply replaces the sample x_i by the
median m_i of the window W_i^H: This filter has one tuning parameter
given by H. The standard median filter is considered highly resistant to
local outliers and local noise in the data sequence \{x_i\}.

 -- Function: gsl_filter_median_workspace * gsl_filter_median_alloc
          (const size_t K)

     This function initializes a workspace for standard median filtering
     using a symmetric centered moving window of size ‘K’.  Here, H = K
     / 2.  If K is even, it is rounded up to the next odd integer to
     ensure a symmetric window.  The size of the workspace is O(7K).

 -- Function: void gsl_filter_median_free (gsl_filter_median_workspace
          * w)

     This function frees the memory associated with ‘w’.

 -- Function: int gsl_filter_median (const gsl_filter_end_t endtype,
          const gsl_vector * x, gsl_vector * y,
          gsl_filter_median_workspace * w)

     This function applies a standard median filter to the input ‘x’,
     storing the output in ‘y’.  The parameter ‘endtype’ specifies how
     the signal end points are handled.  It is allowed to have ‘x’ = ‘y’
     for an in-place filter.


File: gsl-ref.info,  Node: Recursive Median Filter,  Next: Impulse Detection Filter,  Prev: Standard Median Filter,  Up: Nonlinear Digital Filters

24.4.2 Recursive Median Filter
------------------------------

The `recursive median filter' (RMF) is a modification of the SMF to
include previous filter outputs in the window before computing the
median.  The filter’s response is

     y_i = median ( y_{i-H}, ..., y_{i-1}, x_i, x_{i+1}, ..., x_{i+H} )

Sometimes, the SMF must be applied several times in a row to achieve
adequate smoothing (i.e.  a cascade filter).  The RMF, on the other
hand, converges to a `root sequence' in one pass, and can sometimes
provide a smoother result than several passes of the SMF. A root
sequence is an input which is left unchanged by the filter.  So there is
no need to apply a recursive median filter twice to an input vector.

 -- Function: gsl_filter_rmedian_workspace * gsl_filter_rmedian_alloc
          (const size_t K)

     This function initializes a workspace for recursive median
     filtering using a symmetric centered moving window of size ‘K’.
     Here, H = K / 2.  If K is even, it is rounded up to the next odd
     integer to ensure a symmetric window.  The size of the workspace is
     O(K).

 -- Function: void gsl_filter_rmedian_free (gsl_filter_rmedian_workspace
          * w)

     This function frees the memory associated with ‘w’.

 -- Function: int gsl_filter_rmedian (const gsl_filter_end_t endtype,
          const gsl_vector * x, gsl_vector * y,
          gsl_filter_rmedian_workspace * w)

     This function applies a recursive median filter to the input ‘x’,
     storing the output in ‘y’.  The parameter ‘endtype’ specifies how
     the signal end points are handled.  It is allowed to have ‘x’ = ‘y’
     for an in-place filter.


File: gsl-ref.info,  Node: Impulse Detection Filter,  Prev: Recursive Median Filter,  Up: Nonlinear Digital Filters

24.4.3 Impulse Detection Filter
-------------------------------

Impulsive noise is characterized by short sequences of data points
distinct from those in the surrounding neighborhood.  This section
describes a powerful class of filters, also known as `impulse rejection
filters' and `decision-based filters', designed to detect and remove
such outliers from data.  The filter’s response is given by

     y_i = { x_i, |x_i - m_i| <= t * S_i
           { m_i, |x_i - m_i| > t * S_i

where m_i is the median value of the window W_i^H, S_i is a robust
estimate of the scatter or dispersion for the window W_i^H, and t is a
tuning parameter specifying the number of scale factors needed to
determine that a point is an outlier.  The main idea is that the median
m_i will be unaffected by a small number of outliers in the window, and
so a given sample x_i is tested to determine how far away it is from the
median in terms of the local scale estimate S_i.  Samples which are more
than t scale estimates away from the median are labeled as outliers and
replaced by the window median m_i.  Samples which are less than t scale
estimates from the median are left unchanged by the filter.

Note that when t = 0, the impulse detection filter is equivalent to the
standard median filter.  When t \rightarrow \infty, it becomes the
identity filter.  This means the impulse detection filter can be viewed
as a “less aggressive” version of the standard median filter, becoming
less aggressive as t is increased.  Note that this filter modifies only
samples identified as outliers, while the standard median filter changes
all samples to the local median, regardless of whether they are
outliers.  This fact, plus the additional flexibility offered by the
additional tuning parameter t can make the impulse detection filter a
better choice for some applications.

It is important to have a robust and accurate scale estimate S_i in
order to detect impulse outliers even in the presence of noise.  The
window standard deviation is not typically a good choice, as it can be
significantly perturbed by the presence of even one outlier.  GSL offers
the following choices (specified by a parameter of type *note
gsl_filter_scale_t: 87b.) for computing the scale estimate S_i, all of
which are robust to the presence of impulse outliers.

 -- Type: gsl_filter_scale_t

     This type specifies how the scale estimate S_i of the window W_i^H
     is calculated.

      -- Macro: GSL_FILTER_SCALE_MAD

          This option specifies the median absolute deviation (MAD)
          scale estimate, defined by

               S_i = 1.4826 median { | W_i^H - m_i | }

          This choice of scale estimate is also known as the `Hampel
          filter' in the statistical literature.  See *note here: 7f5.
          for more information.

      -- Macro: GSL_FILTER_SCALE_IQR

          This option specifies the interquartile range (IQR) scale
          estimate, defined as the difference between the 75th and 25th
          percentiles of the window W_i^H,

               S_i = 0.7413 ( Q_{0.75} - Q_{0.25} )

          where Q_p is the p-quantile of the window W_i^H. The idea is
          to throw away the largest and smallest 25% of the window
          samples (where the outliers would be), and estimate a scale
          from the middle 50%.  The factor 0.7413 provides an unbiased
          estimate of the standard deviation for Gaussian data.

      -- Macro: GSL_FILTER_SCALE_SN

          This option specifies the so-called S_n statistic proposed by
          Croux and Rousseeuw.  See *note here: 7f9. for more
          information.

      -- Macro: GSL_FILTER_SCALE_QN

          This option specifies the so-called Q_n statistic proposed by
          Croux and Rousseeuw.  See *note here: 7fd. for more
          information.

     Warning: While the scale estimates defined above are much less
     sensitive to outliers than the standard deviation, they can suffer
     from an effect called `implosion'.  The standard deviation of a
     window W_i^H will be zero if and only if all samples in the window
     are equal.  However, it is possible for the MAD of a window to be
     zero even if all the samples in the window are not equal.  For
     example, if K/2 + 1 or more of the K samples in the window are
     equal to some value x^{*}, then the window median will be equal to
     x^{*}.  Consequently, at least K/2 + 1 of the absolute deviations
     |x_j - x^{*}| will be zero, and so the MAD will be zero.  In such a
     case, the Hampel filter will act like the standard median filter
     regardless of the value of t.  Caution should also be exercised if
     dividing by S_i.

 -- Function: gsl_filter_impulse_workspace * gsl_filter_impulse_alloc
          (const size_t K)

     This function initializes a workspace for impulse detection
     filtering using a symmetric moving window of size ‘K’.  Here, H = K
     / 2.  If K is even, it is rounded up to the next odd integer to
     ensure a symmetric window.  The size of the workspace is O(6K).

 -- Function: void gsl_filter_impulse_free (gsl_filter_impulse_workspace
          * w)

     This function frees the memory associated with ‘w’.

 -- Function: int gsl_filter_impulse (const gsl_filter_end_t endtype,
          const gsl_filter_scale_t scale_type, const double t, const
          gsl_vector * x, gsl_vector * y, gsl_vector * xmedian,
          gsl_vector * xsigma, size_t * noutlier, gsl_vector_int
          * ioutlier, gsl_filter_impulse_workspace * w)

     These functions apply an impulse detection filter to the input
     vector ‘x’, storing the filtered output in ‘y’.  The tuning
     parameter t is provided in ‘t’.  The window medians m_i are stored
     in ‘xmedian’ and the S_i are stored in ‘xsigma’ on output.  The
     number of outliers detected is stored in ‘noutlier’ on output,
     while the locations of flagged outliers are stored in the boolean
     array ‘ioutlier’.  The input ‘ioutlier’ may be ‘NULL’ if not
     desired.  It is allowed to have ‘x’ = ‘y’ for an in-place filter.


File: gsl-ref.info,  Node: Examples<18>,  Next: References and Further Reading<18>,  Prev: Nonlinear Digital Filters,  Up: Digital Filtering

24.5 Examples
=============

* Menu:

* Gaussian Example 1::
* Gaussian Example 2::
* Square Wave Signal Example::
* Impulse Detection Example::


File: gsl-ref.info,  Node: Gaussian Example 1,  Next: Gaussian Example 2,  Up: Examples<18>

24.5.1 Gaussian Example 1
-------------------------

This example program illustrates the Gaussian filter applied to
smoothing a time series of length N = 500 with a kernel size of K = 51.
Three filters are applied with parameters \alpha = 0.5, 3, 10.  The
results are shown in Fig.  %s.

[image src="gaussfilt.png"]

Figure: Top panel: Gaussian kernels (unnormalized) for \alpha = 0.5, 3,
10.  Bottom panel: Time series (gray) with Gaussian filter output for
same \alpha values.

We see that the filter corresponding to \alpha = 0.5 applies the most
smoothing, while \alpha = 10 corresponds to the least amount of
smoothing.  The program is given below.

     #include <stdio.h>
     #include <stdlib.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_filter.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_vector.h>

     int
     main(void)
     {
       const size_t N = 500;                        /* length of time series */
       const size_t K = 51;                         /* window size */
       const double alpha[3] = { 0.5, 3.0, 10.0 };  /* alpha values */
       gsl_vector *x = gsl_vector_alloc(N);         /* input vector */
       gsl_vector *y1 = gsl_vector_alloc(N);        /* filtered output vector for alpha1 */
       gsl_vector *y2 = gsl_vector_alloc(N);        /* filtered output vector for alpha2 */
       gsl_vector *y3 = gsl_vector_alloc(N);        /* filtered output vector for alpha3 */
       gsl_vector *k1 = gsl_vector_alloc(K);        /* Gaussian kernel for alpha1 */
       gsl_vector *k2 = gsl_vector_alloc(K);        /* Gaussian kernel for alpha2 */
       gsl_vector *k3 = gsl_vector_alloc(K);        /* Gaussian kernel for alpha3 */
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       gsl_filter_gaussian_workspace *gauss_p = gsl_filter_gaussian_alloc(K);
       size_t i;
       double sum = 0.0;

       /* generate input signal */
       for (i = 0; i < N; ++i)
         {
           double ui = gsl_ran_gaussian(r, 1.0);
           sum += ui;
           gsl_vector_set(x, i, sum);
         }

       /* compute kernels without normalization */
       gsl_filter_gaussian_kernel(alpha[0], 0, 0, k1);
       gsl_filter_gaussian_kernel(alpha[1], 0, 0, k2);
       gsl_filter_gaussian_kernel(alpha[2], 0, 0, k3);

       /* apply filters */
       gsl_filter_gaussian(GSL_FILTER_END_PADVALUE, alpha[0], 0, x, y1, gauss_p);
       gsl_filter_gaussian(GSL_FILTER_END_PADVALUE, alpha[1], 0, x, y2, gauss_p);
       gsl_filter_gaussian(GSL_FILTER_END_PADVALUE, alpha[2], 0, x, y3, gauss_p);

       /* print kernels */
       for (i = 0; i < K; ++i)
         {
           double k1i = gsl_vector_get(k1, i);
           double k2i = gsl_vector_get(k2, i);
           double k3i = gsl_vector_get(k3, i);

           printf("%e %e %e\n", k1i, k2i, k3i);
         }

       printf("\n\n");

       /* print filter results */
       for (i = 0; i < N; ++i)
         {
           double xi = gsl_vector_get(x, i);
           double y1i = gsl_vector_get(y1, i);
           double y2i = gsl_vector_get(y2, i);
           double y3i = gsl_vector_get(y3, i);

           printf("%.12e %.12e %.12e %.12e\n", xi, y1i, y2i, y3i);
         }

       gsl_vector_free(x);
       gsl_vector_free(y1);
       gsl_vector_free(y2);
       gsl_vector_free(y3);
       gsl_vector_free(k1);
       gsl_vector_free(k2);
       gsl_vector_free(k3);
       gsl_rng_free(r);
       gsl_filter_gaussian_free(gauss_p);

       return 0;
     }


File: gsl-ref.info,  Node: Gaussian Example 2,  Next: Square Wave Signal Example,  Prev: Gaussian Example 1,  Up: Examples<18>

24.5.2 Gaussian Example 2
-------------------------

A common application of the Gaussian filter is to detect edges, or
sudden jumps, in a noisy input signal.  It is used both for 1D edge
detection in time series, as well as 2D edge detection in images.  Here
we will examine a noisy time series of length N = 1000 with a single
edge.  The input signal is defined as

     x(n) = e(n) + { 0,   n <= N/2
                   { 0.5, n >  N/2

where e(n) is Gaussian random noise.  The program smooths the input
signal with order 0,1, and 2 Gaussian filters of length K = 61 with
\alpha = 3.  For comparison, the program also computes finite
differences of the input signal without smoothing.  The results are
shown in Fig.  %s.

[image src="gaussfilt2.png"]

Figure: Top row: original input signal x(n) (black) with Gaussian
smoothed signal in red.  Second row: First finite differences of input
signal.  Third row: Input signal smoothed with a first order Gaussian
filter.  Fourth row: Input signal smoothed with a second order Gaussian
filter.

The finite difference approximation of the first derivative (second row)
shows the common problem with differentiating a noisy signal.  The noise
is amplified and makes it extremely difficult to detect the sharp
gradient at sample 500.  The third row shows the first order Gaussian
smoothed signal with a clear peak at the location of the edge.
Alternatively, one could examine the second order Gaussian smoothed
signal (fourth row) and look for zero crossings to determine the edge
location.

The program is given below.

     #include <stdio.h>
     #include <stdlib.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_filter.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_vector.h>

     int
     main(void)
     {
       const size_t N = 1000;                       /* length of time series */
       const size_t K = 61;                         /* window size */
       const double alpha = 3.0;                    /* Gaussian kernel has +/- 3 standard deviations */
       gsl_vector *x = gsl_vector_alloc(N);         /* input vector */
       gsl_vector *y = gsl_vector_alloc(N);         /* filtered output vector */
       gsl_vector *dy = gsl_vector_alloc(N);        /* first derivative filtered vector */
       gsl_vector *d2y = gsl_vector_alloc(N);       /* second derivative filtered vector */
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       gsl_filter_gaussian_workspace *gauss_p = gsl_filter_gaussian_alloc(K);
       size_t i;

       /* generate input signal */
       for (i = 0; i < N; ++i)
         {
           double xi = (i > N / 2) ? 0.5 : 0.0;
           double ei = gsl_ran_gaussian(r, 0.1);

           gsl_vector_set(x, i, xi + ei);
         }

       /* apply filters */
       gsl_filter_gaussian(GSL_FILTER_END_PADVALUE, alpha, 0, x, y, gauss_p);
       gsl_filter_gaussian(GSL_FILTER_END_PADVALUE, alpha, 1, x, dy, gauss_p);
       gsl_filter_gaussian(GSL_FILTER_END_PADVALUE, alpha, 2, x, d2y, gauss_p);

       /* print results */
       for (i = 0; i < N; ++i)
         {
           double xi = gsl_vector_get(x, i);
           double yi = gsl_vector_get(y, i);
           double dyi = gsl_vector_get(dy, i);
           double d2yi = gsl_vector_get(d2y, i);
           double dxi;

           /* compute finite difference of x vector */
           if (i == 0)
             dxi = gsl_vector_get(x, i + 1) - xi;
           else if (i == N - 1)
             dxi = gsl_vector_get(x, i) - gsl_vector_get(x, i - 1);
           else
             dxi = 0.5 * (gsl_vector_get(x, i + 1) - gsl_vector_get(x, i - 1));

           printf("%.12e %.12e %.12e %.12e %.12e\n",
                  xi,
                  yi,
                  dxi,
                  dyi,
                  d2yi);
         }

       gsl_vector_free(x);
       gsl_vector_free(y);
       gsl_vector_free(dy);
       gsl_vector_free(d2y);
       gsl_rng_free(r);
       gsl_filter_gaussian_free(gauss_p);

       return 0;
     }


File: gsl-ref.info,  Node: Square Wave Signal Example,  Next: Impulse Detection Example,  Prev: Gaussian Example 2,  Up: Examples<18>

24.5.3 Square Wave Signal Example
---------------------------------

The following example program illustrates the median filters on a noisy
square wave signal.  Median filters are well known for preserving sharp
edges in the input signal while reducing noise.  The program constructs
a 5 Hz square wave signal with Gaussian noise added.  Then the signal is
filtered with a standard median filter and recursive median filter using
a symmetric window of length K = 7.  The results are shown in Fig.  %s.

[image src="filt_edge.png"]

Figure: Original time series is in gray.  The standard median filter
output is in green and the recursive median filter output is in red.

Both filters preserve the sharp signal edges while reducing the noise.
The recursive median filter achieves a smoother result than the standard
median filter.  The “blocky” nature of the output is characteristic of
all median filters.  The program is given below.

     #include <stdio.h>
     #include <stdlib.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_filter.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_vector.h>

     int
     main(void)
     {
       const size_t N = 1000;                               /* length of time series */
       const size_t K = 7;                                  /* window size */
       const double f = 5.0;                                /* frequency of square wave in Hz */
       gsl_filter_median_workspace *median_p = gsl_filter_median_alloc(K);
       gsl_filter_rmedian_workspace *rmedian_p = gsl_filter_rmedian_alloc(K);
       gsl_vector *t = gsl_vector_alloc(N);                 /* time */
       gsl_vector *x = gsl_vector_alloc(N);                 /* input vector */
       gsl_vector *y_median = gsl_vector_alloc(N);          /* median filtered output */
       gsl_vector *y_rmedian = gsl_vector_alloc(N);         /* recursive median filtered output */
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       size_t i;

       /* generate input signal */
       for (i = 0; i < N; ++i)
         {
           double ti = (double) i / (N - 1.0);
           double tmp = sin(2.0 * M_PI * f * ti);
           double xi = (tmp >= 0.0) ? 1.0 : -1.0;
           double ei = gsl_ran_gaussian(r, 0.1);

           gsl_vector_set(t, i, ti);
           gsl_vector_set(x, i, xi + ei);
         }

       gsl_filter_median(GSL_FILTER_END_PADVALUE, x, y_median, median_p);
       gsl_filter_rmedian(GSL_FILTER_END_PADVALUE, x, y_rmedian, rmedian_p);

       /* print results */
       for (i = 0; i < N; ++i)
         {
           double ti = gsl_vector_get(t, i);
           double xi = gsl_vector_get(x, i);
           double medi = gsl_vector_get(y_median, i);
           double rmedi = gsl_vector_get(y_rmedian, i);

           printf("%f %f %f %f\n",
                  ti,
                  xi,
                  medi,
                  rmedi);
         }

       gsl_vector_free(t);
       gsl_vector_free(x);
       gsl_vector_free(y_median);
       gsl_vector_free(y_rmedian);
       gsl_rng_free(r);
       gsl_filter_median_free(median_p);

       return 0;
     }


File: gsl-ref.info,  Node: Impulse Detection Example,  Prev: Square Wave Signal Example,  Up: Examples<18>

24.5.4 Impulse Detection Example
--------------------------------

The following example program illustrates the impulse detection filter.
First, it constructs a sinusoid signal of length N = 1000 with Gaussian
noise added.  Then, about 1% of the data are perturbed to represent
large outliers.  An impulse detecting filter is applied with a window
size K = 25 and tuning parameter t = 4, using the Q_n statistic as the
robust measure of scale.  The results are plotted in Fig.  %s.

[image src="impulse.png"]

Figure: Original time series is in blue, filter output is in green,
upper and lower intervals for detecting outliers are in red and yellow
respectively.  Detected outliers are marked with squares.

The program is given below.

     #include <stdio.h>
     #include <stdlib.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_filter.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_vector.h>

     int
     main(void)
     {
       const size_t N = 1000;                               /* length of time series */
       const size_t K = 25;                                 /* window size */
       const double t = 4.0;                                /* number of scale factors for outlier detection */
       gsl_vector *x = gsl_vector_alloc(N);                 /* input vector */
       gsl_vector *y = gsl_vector_alloc(N);                 /* output (filtered) vector */
       gsl_vector *xmedian = gsl_vector_alloc(N);           /* window medians */
       gsl_vector *xsigma = gsl_vector_alloc(N);            /* window scale estimates */
       gsl_vector_int *ioutlier = gsl_vector_int_alloc(N);  /* outlier detected? */
       gsl_filter_impulse_workspace * w = gsl_filter_impulse_alloc(K);
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       size_t noutlier;
       size_t i;

       /* generate input signal */
       for (i = 0; i < N; ++i)
         {
           double xi = 10.0 * sin(2.0 * M_PI * i / (double) N);
           double ei = gsl_ran_gaussian(r, 2.0);
           double u = gsl_rng_uniform(r);
           double outlier = (u < 0.01) ? 15.0*GSL_SIGN(ei) : 0.0;

           gsl_vector_set(x, i, xi + ei + outlier);
         }

       /* apply impulse detection filter */
       gsl_filter_impulse(GSL_FILTER_END_TRUNCATE, GSL_FILTER_SCALE_QN, t, x, y,
                          xmedian, xsigma, &noutlier, ioutlier, w);

       /* print results */
       for (i = 0; i < N; ++i)
         {
           double xi = gsl_vector_get(x, i);
           double yi = gsl_vector_get(y, i);
           double xmedi = gsl_vector_get(xmedian, i);
           double xsigmai = gsl_vector_get(xsigma, i);
           int outlier = gsl_vector_int_get(ioutlier, i);

           printf("%zu %f %f %f %f %d\n",
                  i,
                  xi,
                  yi,
                  xmedi + t * xsigmai,
                  xmedi - t * xsigmai,
                  outlier);
         }

       gsl_vector_free(x);
       gsl_vector_free(y);
       gsl_vector_free(xmedian);
       gsl_vector_free(xsigma);
       gsl_vector_int_free(ioutlier);
       gsl_filter_impulse_free(w);
       gsl_rng_free(r);

       return 0;
     }


File: gsl-ref.info,  Node: References and Further Reading<18>,  Prev: Examples<18>,  Up: Digital Filtering

24.6 References and Further Reading
===================================

The following publications are relevant to the algorithms described in
this chapter,

   * F. J. Harris, `On the use of windows for harmonic analysis with the
     discrete Fourier transform', Proceedings of the IEEE, 66 (1), 1978.

   * S-J. Ko, Y-H. Lee, and A. T. Fam.  `Efficient implementation of
     one-dimensional recursive median filters', IEEE transactions on
     circuits and systems 37.11 (1990): 1447-1450.

   * R. K. Pearson and M. Gabbouj, `Nonlinear Digital Filtering with
     Python: An Introduction'.  CRC Press, 2015.


File: gsl-ref.info,  Node: Histograms,  Next: N-tuples,  Prev: Digital Filtering,  Up: Top

25 Histograms
*************

This chapter describes functions for creating histograms.  Histograms
provide a convenient way of summarizing the distribution of a set of
data.  A histogram consists of a set of `bins' which count the number of
events falling into a given range of a continuous variable x.  In GSL
the bins of a histogram contain floating-point numbers, so they can be
used to record both integer and non-integer distributions.  The bins can
use arbitrary sets of ranges (uniformly spaced bins are the default).
Both one and two-dimensional histograms are supported.

Once a histogram has been created it can also be converted into a
probability distribution function.  The library provides efficient
routines for selecting random samples from probability distributions.
This can be useful for generating simulations based on real data.

The functions are declared in the header files ‘gsl_histogram.h’ and
‘gsl_histogram2d.h’.

* Menu:

* The histogram struct::
* Histogram allocation::
* Copying Histograms::
* Updating and accessing histogram elements::
* Searching histogram ranges::
* Histogram Statistics::
* Histogram Operations::
* Reading and writing histograms::
* Resampling from histograms::
* The histogram probability distribution struct::
* Example programs for histograms::
* Two dimensional histograms::
* The 2D histogram struct::
* 2D Histogram allocation::
* Copying 2D Histograms::
* Updating and accessing 2D histogram elements::
* Searching 2D histogram ranges::
* 2D Histogram Statistics::
* 2D Histogram Operations::
* Reading and writing 2D histograms::
* Resampling from 2D histograms::
* Example programs for 2D histograms::


File: gsl-ref.info,  Node: The histogram struct,  Next: Histogram allocation,  Up: Histograms

25.1 The histogram struct
=========================

A histogram is defined by the following struct,

 -- Type: gsl_histogram

     size_t n                          This is the number of histogram bins
                                       
                                       
     double * range                    The ranges of the bins are stored in an array of ‘n+1’ elements pointed to by
                                       range.
                                       
                                       
     double * bin                      The counts for each bin are stored in an array of ‘n’ elements pointed to by
                                       ‘bin’.  The bins are floating-point numbers, so you can increment them by
                                       non-integer values if necessary.
                                       

     The range for ‘bin[i]’ is given by ‘range[i]’ to ‘range[i+1]’.  For
     n bins there are ‘n+1’ entries in the array ‘range’.  Each bin is
     inclusive at the lower end and exclusive at the upper end.
     Mathematically this means that the bins are defined by the
     following inequality,

          bin[i] corresponds to range[i] <= x < range[i+1]

     Here is a diagram of the correspondence between ranges and bins on
     the number-line for x:

             [ bin[0] )[ bin[1] )[ bin[2] )[ bin[3] )[ bin[4] )
          ---|---------|---------|---------|---------|---------|---  x
           r[0]      r[1]      r[2]      r[3]      r[4]      r[5]

     In this picture the values of the ‘range’ array are denoted by r.
     On the left-hand side of each bin the square bracket ‘[’ denotes an
     inclusive lower bound (r \le x), and the round parentheses ‘)’ on
     the right-hand side denote an exclusive upper bound (x < r).  Thus
     any samples which fall on the upper end of the histogram are
     excluded.  If you want to include this value for the last bin you
     will need to add an extra bin to your histogram.

     The *note gsl_histogram: 890. struct and its associated functions
     are defined in the header file ‘gsl_histogram.h’.


File: gsl-ref.info,  Node: Histogram allocation,  Next: Copying Histograms,  Prev: The histogram struct,  Up: Histograms

25.2 Histogram allocation
=========================

The functions for allocating memory to a histogram follow the style of
‘malloc()’ and ‘free()’.  In addition they also perform their own error
checking.  If there is insufficient memory available to allocate a
histogram then the functions call the error handler (with an error
number of *note GSL_ENOMEM: 2a.) in addition to returning a null
pointer.  Thus if you use the library error handler to abort your
program then it isn’t necessary to check every histogram ‘alloc’.

 -- Function: gsl_histogram * gsl_histogram_alloc (size_t n)

     This function allocates memory for a histogram with ‘n’ bins, and
     returns a pointer to a newly created *note gsl_histogram: 890.
     struct.  If insufficient memory is available a null pointer is
     returned and the error handler is invoked with an error code of
     *note GSL_ENOMEM: 2a.  The bins and ranges are not initialized, and
     should be prepared using one of the range-setting functions below
     in order to make the histogram ready for use.

 -- Function: int gsl_histogram_set_ranges (gsl_histogram * h, const
          double range[], size_t size)

     This function sets the ranges of the existing histogram ‘h’ using
     the array ‘range’ of size ‘size’.  The values of the histogram bins
     are reset to zero.  The ‘range’ array should contain the desired
     bin limits.  The ranges can be arbitrary, subject to the
     restriction that they are monotonically increasing.

     The following example shows how to create a histogram with
     logarithmic bins with ranges [1,10), [10,100) and [100,1000):

          gsl_histogram * h = gsl_histogram_alloc (3);

          /* bin[0] covers the range 1 <= x < 10 */
          /* bin[1] covers the range 10 <= x < 100 */
          /* bin[2] covers the range 100 <= x < 1000 */

          double range[4] = { 1.0, 10.0, 100.0, 1000.0 };

          gsl_histogram_set_ranges (h, range, 4);

     Note that the size of the ‘range’ array should be defined to be one
     element bigger than the number of bins.  The additional element is
     required for the upper value of the final bin.

 -- Function: int gsl_histogram_set_ranges_uniform (gsl_histogram * h,
          double xmin, double xmax)

     This function sets the ranges of the existing histogram ‘h’ to
     cover the range ‘xmin’ to ‘xmax’ uniformly.  The values of the
     histogram bins are reset to zero.  The bin ranges are shown in the
     table below,

          bin[0] corresponds to xmin <= x < xmin + d
          bin[1] corresponds to xmin + d <= x < xmin + 2 d
          ......
          bin[n-1] corresponds to xmin + (n-1)d <= x < xmax

     where d is the bin spacing, d = (xmax-xmin)/n.

 -- Function: void gsl_histogram_free (gsl_histogram * h)

     This function frees the histogram ‘h’ and all of the memory
     associated with it.


File: gsl-ref.info,  Node: Copying Histograms,  Next: Updating and accessing histogram elements,  Prev: Histogram allocation,  Up: Histograms

25.3 Copying Histograms
=======================

 -- Function: int gsl_histogram_memcpy (gsl_histogram * dest, const
          gsl_histogram * src)

     This function copies the histogram ‘src’ into the pre-existing
     histogram ‘dest’, making ‘dest’ into an exact copy of ‘src’.  The
     two histograms must be of the same size.

 -- Function: gsl_histogram * gsl_histogram_clone (const gsl_histogram
          * src)

     This function returns a pointer to a newly created histogram which
     is an exact copy of the histogram ‘src’.


File: gsl-ref.info,  Node: Updating and accessing histogram elements,  Next: Searching histogram ranges,  Prev: Copying Histograms,  Up: Histograms

25.4 Updating and accessing histogram elements
==============================================

There are two ways to access histogram bins, either by specifying an x
coordinate or by using the bin-index directly.  The functions for
accessing the histogram through x coordinates use a binary search to
identify the bin which covers the appropriate range.

 -- Function: int gsl_histogram_increment (gsl_histogram * h, double x)

     This function updates the histogram ‘h’ by adding one (1.0) to the
     bin whose range contains the coordinate ‘x’.

     If ‘x’ lies in the valid range of the histogram then the function
     returns zero to indicate success.  If ‘x’ is less than the lower
     limit of the histogram then the function returns *note GSL_EDOM:
     28, and none of bins are modified.  Similarly, if the value of ‘x’
     is greater than or equal to the upper limit of the histogram then
     the function returns *note GSL_EDOM: 28, and none of the bins are
     modified.  The error handler is not called, however, since it is
     often necessary to compute histograms for a small range of a larger
     dataset, ignoring the values outside the range of interest.

 -- Function: int gsl_histogram_accumulate (gsl_histogram * h, double x,
          double weight)

     This function is similar to *note gsl_histogram_increment(): 89a.
     but increases the value of the appropriate bin in the histogram ‘h’
     by the floating-point number ‘weight’.

 -- Function: double gsl_histogram_get (const gsl_histogram * h,
          size_t i)

     This function returns the contents of the ‘i’-th bin of the
     histogram ‘h’.  If ‘i’ lies outside the valid range of indices for
     the histogram then the error handler is called with an error code
     of *note GSL_EDOM: 28. and the function returns 0.

 -- Function: int gsl_histogram_get_range (const gsl_histogram * h,
          size_t i, double * lower, double * upper)

     This function finds the upper and lower range limits of the ‘i’-th
     bin of the histogram ‘h’.  If the index ‘i’ is valid then the
     corresponding range limits are stored in ‘lower’ and ‘upper’.  The
     lower limit is inclusive (i.e.  events with this coordinate are
     included in the bin) and the upper limit is exclusive (i.e.  events
     with the coordinate of the upper limit are excluded and fall in the
     neighboring higher bin, if it exists).  The function returns 0 to
     indicate success.  If ‘i’ lies outside the valid range of indices
     for the histogram then the error handler is called and the function
     returns an error code of *note GSL_EDOM: 28.

 -- Function: double gsl_histogram_max (const gsl_histogram * h)
 -- Function: double gsl_histogram_min (const gsl_histogram * h)
 -- Function: size_t gsl_histogram_bins (const gsl_histogram * h)

     These functions return the maximum upper and minimum lower range
     limits and the number of bins of the histogram ‘h’.  They provide a
     way of determining these values without accessing the *note
     gsl_histogram: 890. struct directly.

 -- Function: void gsl_histogram_reset (gsl_histogram * h)

     This function resets all the bins in the histogram ‘h’ to zero.


File: gsl-ref.info,  Node: Searching histogram ranges,  Next: Histogram Statistics,  Prev: Updating and accessing histogram elements,  Up: Histograms

25.5 Searching histogram ranges
===============================

The following functions are used by the access and update routines to
locate the bin which corresponds to a given x coordinate.

 -- Function: int gsl_histogram_find (const gsl_histogram * h, double x,
          size_t * i)

     This function finds and sets the index ‘i’ to the bin number which
     covers the coordinate ‘x’ in the histogram ‘h’.  The bin is located
     using a binary search.  The search includes an optimization for
     histograms with uniform range, and will return the correct bin
     immediately in this case.  If ‘x’ is found in the range of the
     histogram then the function sets the index ‘i’ and returns
     ‘GSL_SUCCESS’.  If ‘x’ lies outside the valid range of the
     histogram then the function returns *note GSL_EDOM: 28. and the
     error handler is invoked.


File: gsl-ref.info,  Node: Histogram Statistics,  Next: Histogram Operations,  Prev: Searching histogram ranges,  Up: Histograms

25.6 Histogram Statistics
=========================

 -- Function: double gsl_histogram_max_val (const gsl_histogram * h)

     This function returns the maximum value contained in the histogram
     bins.

 -- Function: size_t gsl_histogram_max_bin (const gsl_histogram * h)

     This function returns the index of the bin containing the maximum
     value.  In the case where several bins contain the same maximum
     value the smallest index is returned.

 -- Function: double gsl_histogram_min_val (const gsl_histogram * h)

     This function returns the minimum value contained in the histogram
     bins.

 -- Function: size_t gsl_histogram_min_bin (const gsl_histogram * h)

     This function returns the index of the bin containing the minimum
     value.  In the case where several bins contain the same maximum
     value the smallest index is returned.

 -- Function: double gsl_histogram_mean (const gsl_histogram * h)

     This function returns the mean of the histogrammed variable, where
     the histogram is regarded as a probability distribution.  Negative
     bin values are ignored for the purposes of this calculation.  The
     accuracy of the result is limited by the bin width.

 -- Function: double gsl_histogram_sigma (const gsl_histogram * h)

     This function returns the standard deviation of the histogrammed
     variable, where the histogram is regarded as a probability
     distribution.  Negative bin values are ignored for the purposes of
     this calculation.  The accuracy of the result is limited by the bin
     width.

 -- Function: double gsl_histogram_sum (const gsl_histogram * h)

     This function returns the sum of all bin values.  Negative bin
     values are included in the sum.


File: gsl-ref.info,  Node: Histogram Operations,  Next: Reading and writing histograms,  Prev: Histogram Statistics,  Up: Histograms

25.7 Histogram Operations
=========================

 -- Function: int gsl_histogram_equal_bins_p (const gsl_histogram * h1,
          const gsl_histogram * h2)

     This function returns 1 if the all of the individual bin ranges of
     the two histograms are identical, and 0 otherwise.

 -- Function: int gsl_histogram_add (gsl_histogram * h1, const
          gsl_histogram * h2)

     This function adds the contents of the bins in histogram ‘h2’ to
     the corresponding bins of histogram ‘h1’, i.e.  h'_1(i) = h_1(i) +
     h_2(i).  The two histograms must have identical bin ranges.

 -- Function: int gsl_histogram_sub (gsl_histogram * h1, const
          gsl_histogram * h2)

     This function subtracts the contents of the bins in histogram ‘h2’
     from the corresponding bins of histogram ‘h1’, i.e.  h'_1(i) =
     h_1(i) - h_2(i).  The two histograms must have identical bin
     ranges.

 -- Function: int gsl_histogram_mul (gsl_histogram * h1, const
          gsl_histogram * h2)

     This function multiplies the contents of the bins of histogram ‘h1’
     by the contents of the corresponding bins in histogram ‘h2’, i.e.
     h'_1(i) = h_1(i) * h_2(i).  The two histograms must have identical
     bin ranges.

 -- Function: int gsl_histogram_div (gsl_histogram * h1, const
          gsl_histogram * h2)

     This function divides the contents of the bins of histogram ‘h1’ by
     the contents of the corresponding bins in histogram ‘h2’, i.e.
     h'_1(i) = h_1(i) / h_2(i).  The two histograms must have identical
     bin ranges.

 -- Function: int gsl_histogram_scale (gsl_histogram * h, double scale)

     This function multiplies the contents of the bins of histogram ‘h’
     by the constant ‘scale’, i.e.

          h'_1(i) = h_1(i) * scale

 -- Function: int gsl_histogram_shift (gsl_histogram * h, double offset)

     This function shifts the contents of the bins of histogram ‘h’ by
     the constant ‘offset’, i.e.

          h'_1(i) = h_1(i) + offset


File: gsl-ref.info,  Node: Reading and writing histograms,  Next: Resampling from histograms,  Prev: Histogram Operations,  Up: Histograms

25.8 Reading and writing histograms
===================================

The library provides functions for reading and writing histograms to a
file as binary data or formatted text.

 -- Function: int gsl_histogram_fwrite (FILE * stream, const
          gsl_histogram * h)

     This function writes the ranges and bins of the histogram ‘h’ to
     the stream ‘stream’ in binary format.  The return value is 0 for
     success and ‘GSL_EFAILED’ if there was a problem writing to the
     file.  Since the data is written in the native binary format it may
     not be portable between different architectures.

 -- Function: int gsl_histogram_fread (FILE * stream, gsl_histogram * h)

     This function reads into the histogram ‘h’ from the open stream
     ‘stream’ in binary format.  The histogram ‘h’ must be preallocated
     with the correct size since the function uses the number of bins in
     ‘h’ to determine how many bytes to read.  The return value is 0 for
     success and ‘GSL_EFAILED’ if there was a problem reading from the
     file.  The data is assumed to have been written in the native
     binary format on the same architecture.

 -- Function: int gsl_histogram_fprintf (FILE * stream, const
          gsl_histogram * h, const char * range_format, const char
          * bin_format)

     This function writes the ranges and bins of the histogram ‘h’
     line-by-line to the stream ‘stream’ using the format specifiers
     ‘range_format’ and ‘bin_format’.  These should be one of the ‘%g’,
     ‘%e’ or ‘%f’ formats for floating point numbers.  The function
     returns 0 for success and ‘GSL_EFAILED’ if there was a problem
     writing to the file.  The histogram output is formatted in three
     columns, and the columns are separated by spaces, like this:

          range[0] range[1] bin[0]
          range[1] range[2] bin[1]
          range[2] range[3] bin[2]
          ....
          range[n-1] range[n] bin[n-1]

     The values of the ranges are formatted using ‘range_format’ and the
     value of the bins are formatted using ‘bin_format’.  Each line
     contains the lower and upper limit of the range of the bins and the
     value of the bin itself.  Since the upper limit of one bin is the
     lower limit of the next there is duplication of these values
     between lines but this allows the histogram to be manipulated with
     line-oriented tools.

 -- Function: int gsl_histogram_fscanf (FILE * stream, gsl_histogram
          * h)

     This function reads formatted data from the stream ‘stream’ into
     the histogram ‘h’.  The data is assumed to be in the three-column
     format used by *note gsl_histogram_fprintf(): 8b7.  The histogram
     ‘h’ must be preallocated with the correct length since the function
     uses the size of ‘h’ to determine how many numbers to read.  The
     function returns 0 for success and ‘GSL_EFAILED’ if there was a
     problem reading from the file.


File: gsl-ref.info,  Node: Resampling from histograms,  Next: The histogram probability distribution struct,  Prev: Reading and writing histograms,  Up: Histograms

25.9 Resampling from histograms
===============================

A histogram made by counting events can be regarded as a measurement of
a probability distribution.  Allowing for statistical error, the height
of each bin represents the probability of an event where the value of x
falls in the range of that bin.  The probability distribution function
has the one-dimensional form p(x)dx where,

     p(x) = n_i / (N w_i)

In this equation n_i is the number of events in the bin which contains
x, w_i is the width of the bin and N is the total number of events.  The
distribution of events within each bin is assumed to be uniform.


File: gsl-ref.info,  Node: The histogram probability distribution struct,  Next: Example programs for histograms,  Prev: Resampling from histograms,  Up: Histograms

25.10 The histogram probability distribution struct
===================================================

The probability distribution function for a histogram consists of a set
of `bins' which measure the probability of an event falling into a given
range of a continuous variable x.  A probability distribution function
is defined by the following struct, which actually stores the cumulative
probability distribution function.  This is the natural quantity for
generating samples via the inverse transform method, because there is a
one-to-one mapping between the cumulative probability distribution and
the range [0,1].  It can be shown that by taking a uniform random number
in this range and finding its corresponding coordinate in the cumulative
probability distribution we obtain samples with the desired probability
distribution.

 -- Type: gsl_histogram_pdf

     ‘size_t n’                           This is the number of bins used to approximate the probability
                                          distribution function.
                                          
                                          
     ‘double * range’                     The ranges of the bins are stored in an array of n + 1 elements pointed
                                          to by ‘range’.
                                          
                                          
     ‘double * sum’                       The cumulative probability for the bins is stored in an array of ‘n’
                                          elements pointed to by ‘sum’.
                                          

The following functions allow you to create a *note gsl_histogram_pdf:
8bb. struct which represents this probability distribution and generate
random samples from it.

 -- Function: gsl_histogram_pdf * gsl_histogram_pdf_alloc (size_t n)

     This function allocates memory for a probability distribution with
     ‘n’ bins and returns a pointer to a newly initialized *note
     gsl_histogram_pdf: 8bb. struct.  If insufficient memory is
     available a null pointer is returned and the error handler is
     invoked with an error code of *note GSL_ENOMEM: 2a.

 -- Function: int gsl_histogram_pdf_init (gsl_histogram_pdf * p, const
          gsl_histogram * h)

     This function initializes the probability distribution ‘p’ with the
     contents of the histogram ‘h’.  If any of the bins of ‘h’ are
     negative then the error handler is invoked with an error code of
     *note GSL_EDOM: 28. because a probability distribution cannot
     contain negative values.

 -- Function: void gsl_histogram_pdf_free (gsl_histogram_pdf * p)

     This function frees the probability distribution function ‘p’ and
     all of the memory associated with it.

 -- Function: double gsl_histogram_pdf_sample (const gsl_histogram_pdf
          * p, double r)

     This function uses ‘r’, a uniform random number between zero and
     one, to compute a single random sample from the probability
     distribution ‘p’.  The algorithm used to compute the sample s is
     given by the following formula,

          s = range[i] + delta * (range[i+1] - range[i])

     where i is the index which satisfies sum[i] \le r < sum[i+1] and
     delta is (r - sum[i])/(sum[i+1] - sum[i]).


File: gsl-ref.info,  Node: Example programs for histograms,  Next: Two dimensional histograms,  Prev: The histogram probability distribution struct,  Up: Histograms

25.11 Example programs for histograms
=====================================

The following program shows how to make a simple histogram of a column
of numerical data supplied on ‘stdin’.  The program takes three
arguments, specifying the upper and lower bounds of the histogram and
the number of bins.  It then reads numbers from ‘stdin’, one line at a
time, and adds them to the histogram.  When there is no more data to
read it prints out the accumulated histogram using *note
gsl_histogram_fprintf(): 8b7.

     #include <stdio.h>
     #include <stdlib.h>
     #include <gsl/gsl_histogram.h>

     int
     main (int argc, char **argv)
     {
       double a, b;
       size_t n;

       if (argc != 4)
         {
           printf ("Usage: gsl-histogram xmin xmax n\n"
                   "Computes a histogram of the data "
                   "on stdin using n bins from xmin "
                   "to xmax\n");
           exit (0);
         }

       a = atof (argv[1]);
       b = atof (argv[2]);
       n = atoi (argv[3]);

       {
         double x;
         gsl_histogram * h = gsl_histogram_alloc (n);
         gsl_histogram_set_ranges_uniform (h, a, b);

         while (fscanf (stdin, "%lg", &x) == 1)
           {
             gsl_histogram_increment (h, x);
           }
         gsl_histogram_fprintf (stdout, h, "%g", "%g");
         gsl_histogram_free (h);
       }
       exit (0);
     }

Here is an example of the program in use.  We generate 10000 random
samples from a Cauchy distribution with a width of 30 and histogram them
over the range -100 to 100, using 200 bins:

     $ gsl-randist 0 10000 cauchy 30
        | gsl-histogram -- -100 100 200 > histogram.dat

Fig.  %s shows the familiar shape of the Cauchy distribution and the
fluctuations caused by the finite sample size.

[image src="histogram.png"]

Figure: Histogram output from example program


File: gsl-ref.info,  Node: Two dimensional histograms,  Next: The 2D histogram struct,  Prev: Example programs for histograms,  Up: Histograms

25.12 Two dimensional histograms
================================

A two dimensional histogram consists of a set of `bins' which count the
number of events falling in a given area of the (x,y) plane.  The
simplest way to use a two dimensional histogram is to record
two-dimensional position information, n(x,y).  Another possibility is to
form a `joint distribution' by recording related variables.  For example
a detector might record both the position of an event (x) and the amount
of energy it deposited E. These could be histogrammed as the joint
distribution n(x,E).


File: gsl-ref.info,  Node: The 2D histogram struct,  Next: 2D Histogram allocation,  Prev: Two dimensional histograms,  Up: Histograms

25.13 The 2D histogram struct
=============================

Two dimensional histograms are defined by the following struct,

 -- Type: gsl_histogram2d

     ‘size_t nx, ny’                 This is the number of histogram bins in the x and y directions.
                                     
                                     
     ‘double * xrange’               The ranges of the bins in the x-direction are stored in an array of ‘nx + 1’
                                     elements pointed to by ‘xrange’.
                                     
                                     
     ‘double * yrange’               The ranges of the bins in the y-direction are stored in an array of ‘ny + 1’
                                     elements pointed to by ‘yrange’.
                                     
                                     
     ‘double * bin’                  The counts for each bin are stored in an array pointed to by ‘bin’.  The bins
                                     are floating-point numbers, so you can increment them by non-integer values if
                                     necessary.  The array ‘bin’ stores the two dimensional array of bins in a
                                     single block of memory according to the mapping ‘bin(i,j)’ = ‘bin[i * ny +
                                     j]’.
                                     

The range for ‘bin(i,j)’ is given by ‘xrange[i]’ to ‘xrange[i+1]’ in the
x-direction and ‘yrange[j]’ to ‘yrange[j+1]’ in the y-direction.  Each
bin is inclusive at the lower end and exclusive at the upper end.
Mathematically this means that the bins are defined by the following
inequality,

     bin(i,j) corresponds to xrange[i] <= x < xrange[i+1]
                         and yrange[j] <= y < yrange[j+1]

Note that any samples which fall on the upper sides of the histogram are
excluded.  If you want to include these values for the side bins you
will need to add an extra row or column to your histogram.

The *note gsl_histogram2d: 8c4. struct and its associated functions are
defined in the header file ‘gsl_histogram2d.h’.


File: gsl-ref.info,  Node: 2D Histogram allocation,  Next: Copying 2D Histograms,  Prev: The 2D histogram struct,  Up: Histograms

25.14 2D Histogram allocation
=============================

The functions for allocating memory to a 2D histogram follow the style
of ‘malloc()’ and ‘free()’.  In addition they also perform their own
error checking.  If there is insufficient memory available to allocate a
histogram then the functions call the error handler (with an error
number of *note GSL_ENOMEM: 2a.) in addition to returning a null
pointer.  Thus if you use the library error handler to abort your
program then it isn’t necessary to check every 2D histogram ‘alloc’.

 -- Function: gsl_histogram2d * gsl_histogram2d_alloc (size_t nx,
          size_t ny)

     This function allocates memory for a two-dimensional histogram with
     ‘nx’ bins in the x direction and ‘ny’ bins in the y direction.  The
     function returns a pointer to a newly created *note
     gsl_histogram2d: 8c4. struct.  If insufficient memory is available
     a null pointer is returned and the error handler is invoked with an
     error code of *note GSL_ENOMEM: 2a.  The bins and ranges must be
     initialized with one of the functions below before the histogram is
     ready for use.

 -- Function: int gsl_histogram2d_set_ranges (gsl_histogram2d * h, const
          double xrange[], size_t xsize, const double yrange[],
          size_t ysize)

     This function sets the ranges of the existing histogram ‘h’ using
     the arrays ‘xrange’ and ‘yrange’ of size ‘xsize’ and ‘ysize’
     respectively.  The values of the histogram bins are reset to zero.

 -- Function: int gsl_histogram2d_set_ranges_uniform (gsl_histogram2d
          * h, double xmin, double xmax, double ymin, double ymax)

     This function sets the ranges of the existing histogram ‘h’ to
     cover the ranges ‘xmin’ to ‘xmax’ and ‘ymin’ to ‘ymax’ uniformly.
     The values of the histogram bins are reset to zero.

 -- Function: void gsl_histogram2d_free (gsl_histogram2d * h)

     This function frees the 2D histogram ‘h’ and all of the memory
     associated with it.


File: gsl-ref.info,  Node: Copying 2D Histograms,  Next: Updating and accessing 2D histogram elements,  Prev: 2D Histogram allocation,  Up: Histograms

25.15 Copying 2D Histograms
===========================

 -- Function: int gsl_histogram2d_memcpy (gsl_histogram2d * dest, const
          gsl_histogram2d * src)

     This function copies the histogram ‘src’ into the pre-existing
     histogram ‘dest’, making ‘dest’ into an exact copy of ‘src’.  The
     two histograms must be of the same size.

 -- Function: gsl_histogram2d * gsl_histogram2d_clone (const
          gsl_histogram2d * src)

     This function returns a pointer to a newly created histogram which
     is an exact copy of the histogram ‘src’.


File: gsl-ref.info,  Node: Updating and accessing 2D histogram elements,  Next: Searching 2D histogram ranges,  Prev: Copying 2D Histograms,  Up: Histograms

25.16 Updating and accessing 2D histogram elements
==================================================

You can access the bins of a two-dimensional histogram either by
specifying a pair of (x,y) coordinates or by using the bin indices (i,j)
directly.  The functions for accessing the histogram through (x,y)
coordinates use binary searches in the x and y directions to identify
the bin which covers the appropriate range.

 -- Function: int gsl_histogram2d_increment (gsl_histogram2d * h,
          double x, double y)

     This function updates the histogram ‘h’ by adding one (1.0) to the
     bin whose x and y ranges contain the coordinates (‘x’, ‘y’).

     If the point (x,y) lies inside the valid ranges of the histogram
     then the function returns zero to indicate success.  If (x,y) lies
     outside the limits of the histogram then the function returns *note
     GSL_EDOM: 28, and none of the bins are modified.  The error handler
     is not called, since it is often necessary to compute histograms
     for a small range of a larger dataset, ignoring any coordinates
     outside the range of interest.

 -- Function: int gsl_histogram2d_accumulate (gsl_histogram2d * h,
          double x, double y, double weight)

     This function is similar to *note gsl_histogram2d_increment(): 8ce.
     but increases the value of the appropriate bin in the histogram ‘h’
     by the floating-point number ‘weight’.

 -- Function: double gsl_histogram2d_get (const gsl_histogram2d * h,
          size_t i, size_t j)

     This function returns the contents of the (‘i’, ‘j’)-th bin of the
     histogram ‘h’.  If (‘i’, ‘j’) lies outside the valid range of
     indices for the histogram then the error handler is called with an
     error code of *note GSL_EDOM: 28. and the function returns 0.

 -- Function: int gsl_histogram2d_get_xrange (const gsl_histogram2d * h,
          size_t i, double * xlower, double * xupper)
 -- Function: int gsl_histogram2d_get_yrange (const gsl_histogram2d * h,
          size_t j, double * ylower, double * yupper)

     These functions find the upper and lower range limits of the ‘i’-th
     and ‘j’-th bins in the x and y directions of the histogram ‘h’.
     The range limits are stored in ‘xlower’ and ‘xupper’ or ‘ylower’
     and ‘yupper’.  The lower limits are inclusive (i.e.  events with
     these coordinates are included in the bin) and the upper limits are
     exclusive (i.e.  events with the value of the upper limit are not
     included and fall in the neighboring higher bin, if it exists).
     The functions return 0 to indicate success.  If ‘i’ or ‘j’ lies
     outside the valid range of indices for the histogram then the error
     handler is called with an error code of *note GSL_EDOM: 28.

 -- Function: double gsl_histogram2d_xmax (const gsl_histogram2d * h)
 -- Function: double gsl_histogram2d_xmin (const gsl_histogram2d * h)
 -- Function: size_t gsl_histogram2d_nx (const gsl_histogram2d * h)
 -- Function: double gsl_histogram2d_ymax (const gsl_histogram2d * h)
 -- Function: double gsl_histogram2d_ymin (const gsl_histogram2d * h)
 -- Function: size_t gsl_histogram2d_ny (const gsl_histogram2d * h)

     These functions return the maximum upper and minimum lower range
     limits and the number of bins for the x and y directions of the
     histogram ‘h’.  They provide a way of determining these values
     without accessing the *note gsl_histogram2d: 8c4. struct directly.

 -- Function: void gsl_histogram2d_reset (gsl_histogram2d * h)

     This function resets all the bins of the histogram ‘h’ to zero.


File: gsl-ref.info,  Node: Searching 2D histogram ranges,  Next: 2D Histogram Statistics,  Prev: Updating and accessing 2D histogram elements,  Up: Histograms

25.17 Searching 2D histogram ranges
===================================

The following functions are used by the access and update routines to
locate the bin which corresponds to a given (x,y) coordinate.

 -- Function: int gsl_histogram2d_find (const gsl_histogram2d * h,
          double x, double y, size_t * i, size_t * j)

     This function finds and sets the indices ‘i’ and ‘j’ to the bin
     which covers the coordinates (‘x’, ‘y’).  The bin is located using
     a binary search.  The search includes an optimization for
     histograms with uniform ranges, and will return the correct bin
     immediately in this case.  If (x,y) is found then the function sets
     the indices (‘i’, ‘j’) and returns ‘GSL_SUCCESS’.  If (x,y) lies
     outside the valid range of the histogram then the function returns
     *note GSL_EDOM: 28. and the error handler is invoked.


File: gsl-ref.info,  Node: 2D Histogram Statistics,  Next: 2D Histogram Operations,  Prev: Searching 2D histogram ranges,  Up: Histograms

25.18 2D Histogram Statistics
=============================

 -- Function: double gsl_histogram2d_max_val (const gsl_histogram2d * h)

     This function returns the maximum value contained in the histogram
     bins.

 -- Function: void gsl_histogram2d_max_bin (const gsl_histogram2d * h,
          size_t * i, size_t * j)

     This function finds the indices of the bin containing the maximum
     value in the histogram ‘h’ and stores the result in (‘i’, ‘j’).  In
     the case where several bins contain the same maximum value the
     first bin found is returned.

 -- Function: double gsl_histogram2d_min_val (const gsl_histogram2d * h)

     This function returns the minimum value contained in the histogram
     bins.

 -- Function: void gsl_histogram2d_min_bin (const gsl_histogram2d * h,
          size_t * i, size_t * j)

     This function finds the indices of the bin containing the minimum
     value in the histogram ‘h’ and stores the result in (‘i’, ‘j’).  In
     the case where several bins contain the same maximum value the
     first bin found is returned.

 -- Function: double gsl_histogram2d_xmean (const gsl_histogram2d * h)

     This function returns the mean of the histogrammed x variable,
     where the histogram is regarded as a probability distribution.
     Negative bin values are ignored for the purposes of this
     calculation.

 -- Function: double gsl_histogram2d_ymean (const gsl_histogram2d * h)

     This function returns the mean of the histogrammed y variable,
     where the histogram is regarded as a probability distribution.
     Negative bin values are ignored for the purposes of this
     calculation.

 -- Function: double gsl_histogram2d_xsigma (const gsl_histogram2d * h)

     This function returns the standard deviation of the histogrammed x
     variable, where the histogram is regarded as a probability
     distribution.  Negative bin values are ignored for the purposes of
     this calculation.

 -- Function: double gsl_histogram2d_ysigma (const gsl_histogram2d * h)

     This function returns the standard deviation of the histogrammed y
     variable, where the histogram is regarded as a probability
     distribution.  Negative bin values are ignored for the purposes of
     this calculation.

 -- Function: double gsl_histogram2d_cov (const gsl_histogram2d * h)

     This function returns the covariance of the histogrammed x and y
     variables, where the histogram is regarded as a probability
     distribution.  Negative bin values are ignored for the purposes of
     this calculation.

 -- Function: double gsl_histogram2d_sum (const gsl_histogram2d * h)

     This function returns the sum of all bin values.  Negative bin
     values are included in the sum.


File: gsl-ref.info,  Node: 2D Histogram Operations,  Next: Reading and writing 2D histograms,  Prev: 2D Histogram Statistics,  Up: Histograms

25.19 2D Histogram Operations
=============================

 -- Function: int gsl_histogram2d_equal_bins_p (const gsl_histogram2d
          * h1, const gsl_histogram2d * h2)

     This function returns 1 if all the individual bin ranges of the two
     histograms are identical, and 0 otherwise.

 -- Function: int gsl_histogram2d_add (gsl_histogram2d * h1, const
          gsl_histogram2d * h2)

     This function adds the contents of the bins in histogram ‘h2’ to
     the corresponding bins of histogram ‘h1’, i.e.  h'_1(i,j) =
     h_1(i,j) + h_2(i,j).  The two histograms must have identical bin
     ranges.

 -- Function: int gsl_histogram2d_sub (gsl_histogram2d * h1, const
          gsl_histogram2d * h2)

     This function subtracts the contents of the bins in histogram ‘h2’
     from the corresponding bins of histogram ‘h1’, i.e.  h'_1(i,j) =
     h_1(i,j) - h_2(i,j).  The two histograms must have identical bin
     ranges.

 -- Function: int gsl_histogram2d_mul (gsl_histogram2d * h1, const
          gsl_histogram2d * h2)

     This function multiplies the contents of the bins of histogram ‘h1’
     by the contents of the corresponding bins in histogram ‘h2’, i.e.
     h'_1(i,j) = h_1(i,j) * h_2(i,j).  The two histograms must have
     identical bin ranges.

 -- Function: int gsl_histogram2d_div (gsl_histogram2d * h1, const
          gsl_histogram2d * h2)

     This function divides the contents of the bins of histogram ‘h1’ by
     the contents of the corresponding bins in histogram ‘h2’, i.e.
     h'_1(i,j) = h_1(i,j) / h_2(i,j).  The two histograms must have
     identical bin ranges.

 -- Function: int gsl_histogram2d_scale (gsl_histogram2d * h,
          double scale)

     This function multiplies the contents of the bins of histogram ‘h’
     by the constant ‘scale’, i.e.

          h'_1(i,j) = h_1(i,j) scale

 -- Function: int gsl_histogram2d_shift (gsl_histogram2d * h,
          double offset)

     This function shifts the contents of the bins of histogram ‘h’ by
     the constant ‘offset’, i.e.

          h'_1(i,j) = h_1(i,j) + offset


File: gsl-ref.info,  Node: Reading and writing 2D histograms,  Next: Resampling from 2D histograms,  Prev: 2D Histogram Operations,  Up: Histograms

25.20 Reading and writing 2D histograms
=======================================

The library provides functions for reading and writing two dimensional
histograms to a file as binary data or formatted text.

 -- Function: int gsl_histogram2d_fwrite (FILE * stream, const
          gsl_histogram2d * h)

     This function writes the ranges and bins of the histogram ‘h’ to
     the stream ‘stream’ in binary format.  The return value is 0 for
     success and ‘GSL_EFAILED’ if there was a problem writing to the
     file.  Since the data is written in the native binary format it may
     not be portable between different architectures.

 -- Function: int gsl_histogram2d_fread (FILE * stream, gsl_histogram2d
          * h)

     This function reads into the histogram ‘h’ from the stream ‘stream’
     in binary format.  The histogram ‘h’ must be preallocated with the
     correct size since the function uses the number of x and y bins in
     ‘h’ to determine how many bytes to read.  The return value is 0 for
     success and ‘GSL_EFAILED’ if there was a problem reading from the
     file.  The data is assumed to have been written in the native
     binary format on the same architecture.

 -- Function: int gsl_histogram2d_fprintf (FILE * stream, const
          gsl_histogram2d * h, const char * range_format, const char
          * bin_format)

     This function writes the ranges and bins of the histogram ‘h’
     line-by-line to the stream ‘stream’ using the format specifiers
     ‘range_format’ and ‘bin_format’.  These should be one of the ‘%g’,
     ‘%e’ or ‘%f’ formats for floating point numbers.  The function
     returns 0 for success and ‘GSL_EFAILED’ if there was a problem
     writing to the file.  The histogram output is formatted in five
     columns, and the columns are separated by spaces, like this:

          xrange[0] xrange[1] yrange[0] yrange[1] bin(0,0)
          xrange[0] xrange[1] yrange[1] yrange[2] bin(0,1)
          xrange[0] xrange[1] yrange[2] yrange[3] bin(0,2)
          ....
          xrange[0] xrange[1] yrange[ny-1] yrange[ny] bin(0,ny-1)

          xrange[1] xrange[2] yrange[0] yrange[1] bin(1,0)
          xrange[1] xrange[2] yrange[1] yrange[2] bin(1,1)
          xrange[1] xrange[2] yrange[1] yrange[2] bin(1,2)
          ....
          xrange[1] xrange[2] yrange[ny-1] yrange[ny] bin(1,ny-1)

          ....

          xrange[nx-1] xrange[nx] yrange[0] yrange[1] bin(nx-1,0)
          xrange[nx-1] xrange[nx] yrange[1] yrange[2] bin(nx-1,1)
          xrange[nx-1] xrange[nx] yrange[1] yrange[2] bin(nx-1,2)
          ....
          xrange[nx-1] xrange[nx] yrange[ny-1] yrange[ny] bin(nx-1,ny-1)

     Each line contains the lower and upper limits of the bin and the
     contents of the bin.  Since the upper limits of the each bin are
     the lower limits of the neighboring bins there is duplication of
     these values but this allows the histogram to be manipulated with
     line-oriented tools.

 -- Function: int gsl_histogram2d_fscanf (FILE * stream, gsl_histogram2d
          * h)

     This function reads formatted data from the stream ‘stream’ into
     the histogram ‘h’.  The data is assumed to be in the five-column
     format used by *note gsl_histogram2d_fprintf(): 8f2.  The histogram
     ‘h’ must be preallocated with the correct lengths since the
     function uses the sizes of ‘h’ to determine how many numbers to
     read.  The function returns 0 for success and ‘GSL_EFAILED’ if
     there was a problem reading from the file.


File: gsl-ref.info,  Node: Resampling from 2D histograms,  Next: Example programs for 2D histograms,  Prev: Reading and writing 2D histograms,  Up: Histograms

25.21 Resampling from 2D histograms
===================================

As in the one-dimensional case, a two-dimensional histogram made by
counting events can be regarded as a measurement of a probability
distribution.  Allowing for statistical error, the height of each bin
represents the probability of an event where (x, y) falls in the range
of that bin.  For a two-dimensional histogram the probability
distribution takes the form p(x,y) dx dy where,

     p(x,y) = n_{ij} / (N A_{ij})

In this equation n_{ij} is the number of events in the bin which
contains (x,y), A_{ij} is the area of the bin and N is the total number
of events.  The distribution of events within each bin is assumed to be
uniform.

 -- Type: gsl_histogram2d_pdf

     ‘size_t nx, ny’                   This is the number of histogram bins used to approximate the probability
                                       distribution function in the x and y directions.
                                       
                                       
     ‘double * xrange’                 The ranges of the bins in the x-direction are stored in an array of ‘nx + 1’
                                       elements pointed to by ‘xrange’.
                                       
                                       
     ‘double * yrange’                 The ranges of the bins in the y-direction are stored in an array of ‘ny + 1’
                                       pointed to by ‘yrange’.
                                       
                                       
     ‘double * sum’                    The cumulative probability for the bins is stored in an array of ‘nx’ * ‘ny’
                                       elements pointed to by ‘sum’.
                                       

The following functions allow you to create a *note gsl_histogram2d_pdf:
8f5. struct which represents a two dimensional probability distribution
and generate random samples from it.

 -- Function: gsl_histogram2d_pdf * gsl_histogram2d_pdf_alloc
          (size_t nx, size_t ny)

     This function allocates memory for a two-dimensional probability
     distribution of size ‘nx’-by-‘ny’ and returns a pointer to a newly
     initialized *note gsl_histogram2d_pdf: 8f5. struct.  If
     insufficient memory is available a null pointer is returned and the
     error handler is invoked with an error code of *note GSL_ENOMEM:
     2a.

 -- Function: int gsl_histogram2d_pdf_init (gsl_histogram2d_pdf * p,
          const gsl_histogram2d * h)

     This function initializes the two-dimensional probability
     distribution calculated ‘p’ from the histogram ‘h’.  If any of the
     bins of ‘h’ are negative then the error handler is invoked with an
     error code of *note GSL_EDOM: 28. because a probability
     distribution cannot contain negative values.

 -- Function: void gsl_histogram2d_pdf_free (gsl_histogram2d_pdf * p)

     This function frees the two-dimensional probability distribution
     function ‘p’ and all of the memory associated with it.

 -- Function: int gsl_histogram2d_pdf_sample (const gsl_histogram2d_pdf
          * p, double r1, double r2, double * x, double * y)

     This function uses two uniform random numbers between zero and one,
     ‘r1’ and ‘r2’, to compute a single random sample from the
     two-dimensional probability distribution ‘p’.


File: gsl-ref.info,  Node: Example programs for 2D histograms,  Prev: Resampling from 2D histograms,  Up: Histograms

25.22 Example programs for 2D histograms
========================================

This program demonstrates two features of two-dimensional histograms.
First a 10-by-10 two-dimensional histogram is created with x and y
running from 0 to 1.  Then a few sample points are added to the
histogram, at (0.3,0.3) with a height of 1, at (0.8,0.1) with a height
of 5 and at (0.7,0.9) with a height of 0.5.  This histogram with three
events is used to generate a random sample of 1000 simulated events,
which are printed out.

     #include <stdio.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_histogram2d.h>

     int
     main (void)
     {
       const gsl_rng_type * T;
       gsl_rng * r;

       gsl_histogram2d * h = gsl_histogram2d_alloc (10, 10);

       gsl_histogram2d_set_ranges_uniform (h,
                                           0.0, 1.0,
                                           0.0, 1.0);

       gsl_histogram2d_accumulate (h, 0.3, 0.3, 1);
       gsl_histogram2d_accumulate (h, 0.8, 0.1, 5);
       gsl_histogram2d_accumulate (h, 0.7, 0.9, 0.5);

       gsl_rng_env_setup ();

       T = gsl_rng_default;
       r = gsl_rng_alloc (T);

       {
         int i;
         gsl_histogram2d_pdf * p
           = gsl_histogram2d_pdf_alloc (h->nx, h->ny);

         gsl_histogram2d_pdf_init (p, h);

         for (i = 0; i < 1000; i++) {
           double x, y;
           double u = gsl_rng_uniform (r);
           double v = gsl_rng_uniform (r);

           gsl_histogram2d_pdf_sample (p, u, v, &x, &y);

           printf ("%g %g\n", x, y);
         }

         gsl_histogram2d_pdf_free (p);
       }

       gsl_histogram2d_free (h);
       gsl_rng_free (r);

       return 0;
     }

The following plot shows the distribution of the simulated events.
Using a higher resolution grid we can see the original underlying
histogram and also the statistical fluctuations caused by the events
being uniformly distributed over the area of the original bins.

[image src="histogram2d.png"]

Figure: Distribution of simulated events from example program


File: gsl-ref.info,  Node: N-tuples,  Next: Monte Carlo Integration,  Prev: Histograms,  Up: Top

26 N-tuples
***********

This chapter describes functions for creating and manipulating
`ntuples', sets of values associated with events.  The ntuples are
stored in files.  Their values can be extracted in any combination and
`booked' in a histogram using a selection function.

The values to be stored are held in a user-defined data structure, and
an ntuple is created associating this data structure with a file.  The
values are then written to the file (normally inside a loop) using the
ntuple functions described below.

A histogram can be created from ntuple data by providing a selection
function and a value function.  The selection function specifies whether
an event should be included in the subset to be analyzed or not.  The
value function computes the entry to be added to the histogram for each
event.

All the ntuple functions are defined in the header file ‘gsl_ntuple.h’.

* Menu:

* The ntuple struct::
* Creating ntuples::
* Opening an existing ntuple file::
* Writing ntuples::
* Reading ntuples::
* Closing an ntuple file::
* Histogramming ntuple values::
* Examples: Examples<19>.
* References and Further Reading: References and Further Reading<19>.


File: gsl-ref.info,  Node: The ntuple struct,  Next: Creating ntuples,  Up: N-tuples

26.1 The ntuple struct
======================

 -- Type: gsl_ntuple

     Ntuples are manipulated using the *note gsl_ntuple: 8fe. struct.
     This struct contains information on the file where the ntuple data
     is stored, a pointer to the current ntuple data row and the size of
     the user-defined ntuple data struct:

          typedef struct
            {
              FILE * file;
              void * ntuple_data;
              size_t size;
            } gsl_ntuple;


File: gsl-ref.info,  Node: Creating ntuples,  Next: Opening an existing ntuple file,  Prev: The ntuple struct,  Up: N-tuples

26.2 Creating ntuples
=====================

 -- Function: gsl_ntuple * gsl_ntuple_create (char * filename, void
          * ntuple_data, size_t size)

     This function creates a new write-only ntuple file ‘filename’ for
     ntuples of size ‘size’ and returns a pointer to the newly created
     ntuple struct.  Any existing file with the same name is truncated
     to zero length and overwritten.  A pointer to memory for the
     current ntuple row ‘ntuple_data’ must be supplied—this is used to
     copy ntuples in and out of the file.


File: gsl-ref.info,  Node: Opening an existing ntuple file,  Next: Writing ntuples,  Prev: Creating ntuples,  Up: N-tuples

26.3 Opening an existing ntuple file
====================================

 -- Function: gsl_ntuple * gsl_ntuple_open (char * filename, void
          * ntuple_data, size_t size)

     This function opens an existing ntuple file ‘filename’ for reading
     and returns a pointer to a corresponding ntuple struct.  The
     ntuples in the file must have size ‘size’.  A pointer to memory for
     the current ntuple row ‘ntuple_data’ must be supplied—this is used
     to copy ntuples in and out of the file.


File: gsl-ref.info,  Node: Writing ntuples,  Next: Reading ntuples,  Prev: Opening an existing ntuple file,  Up: N-tuples

26.4 Writing ntuples
====================

 -- Function: int gsl_ntuple_write (gsl_ntuple * ntuple)

     This function writes the current ntuple ‘ntuple->ntuple_data’ of
     size ‘ntuple->size’ to the corresponding file.

 -- Function: int gsl_ntuple_bookdata (gsl_ntuple * ntuple)

     This function is a synonym for *note gsl_ntuple_write(): 904.


File: gsl-ref.info,  Node: Reading ntuples,  Next: Closing an ntuple file,  Prev: Writing ntuples,  Up: N-tuples

26.5 Reading ntuples
====================

 -- Function: int gsl_ntuple_read (gsl_ntuple * ntuple)

     This function reads the current row of the ntuple file for ‘ntuple’
     and stores the values in ‘ntuple->data’.


File: gsl-ref.info,  Node: Closing an ntuple file,  Next: Histogramming ntuple values,  Prev: Reading ntuples,  Up: N-tuples

26.6 Closing an ntuple file
===========================

 -- Function: int gsl_ntuple_close (gsl_ntuple * ntuple)

     This function closes the ntuple file ‘ntuple’ and frees its
     associated allocated memory.


File: gsl-ref.info,  Node: Histogramming ntuple values,  Next: Examples<19>,  Prev: Closing an ntuple file,  Up: N-tuples

26.7 Histogramming ntuple values
================================

Once an ntuple has been created its contents can be histogrammed in
various ways using the function *note gsl_ntuple_project(): 90b.  Two
user-defined functions must be provided, a function to select events and
a function to compute scalar values.  The selection function and the
value function both accept the ntuple row as a first argument and other
parameters as a second argument.

 -- Type: gsl_ntuple_select_fn

     The `selection function' determines which ntuple rows are selected
     for histogramming.  It is defined by the following struct:

          typedef struct
            {
              int (* function) (void * ntuple_data, void * params);
              void * params;
            } gsl_ntuple_select_fn;

     The struct component ‘function’ should return a non-zero value for
     each ntuple row that is to be included in the histogram.

 -- Type: gsl_ntuple_value_fn

     The `value function' computes scalar values for those ntuple rows
     selected by the selection function:

          typedef struct
            {
              double (* function) (void * ntuple_data, void * params);
              void * params;
            } gsl_ntuple_value_fn;

     In this case the struct component ‘function’ should return the
     value to be added to the histogram for the ntuple row.

 -- Function: int gsl_ntuple_project (gsl_histogram * h, gsl_ntuple
          * ntuple, gsl_ntuple_value_fn * value_func,
          gsl_ntuple_select_fn * select_func)

     This function updates the histogram ‘h’ from the ntuple ‘ntuple’
     using the functions ‘value_func’ and ‘select_func’.  For each
     ntuple row where the selection function ‘select_func’ is non-zero
     the corresponding value of that row is computed using the function
     ‘value_func’ and added to the histogram.  Those ntuple rows where
     ‘select_func’ returns zero are ignored.  New entries are added to
     the histogram, so subsequent calls can be used to accumulate
     further data in the same histogram.


File: gsl-ref.info,  Node: Examples<19>,  Next: References and Further Reading<19>,  Prev: Histogramming ntuple values,  Up: N-tuples

26.8 Examples
=============

The following example programs demonstrate the use of ntuples in
managing a large dataset.  The first program creates a set of 10,000
simulated “events”, each with 3 associated values (x,y,z).  These are
generated from a Gaussian distribution with unit variance, for
demonstration purposes, and written to the ntuple file ‘test.dat’.

     #include <gsl/gsl_ntuple.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>

     struct data
     {
       double x;
       double y;
       double z;
     };

     int
     main (void)
     {
       const gsl_rng_type * T;
       gsl_rng * r;

       struct data ntuple_row;
       int i;

       gsl_ntuple *ntuple
         = gsl_ntuple_create ("test.dat", &ntuple_row,
                              sizeof (ntuple_row));

       gsl_rng_env_setup ();

       T = gsl_rng_default;
       r = gsl_rng_alloc (T);

       for (i = 0; i < 10000; i++)
         {
           ntuple_row.x = gsl_ran_ugaussian (r);
           ntuple_row.y = gsl_ran_ugaussian (r);
           ntuple_row.z = gsl_ran_ugaussian (r);

           gsl_ntuple_write (ntuple);
         }

       gsl_ntuple_close (ntuple);
       gsl_rng_free (r);

       return 0;
     }

The next program analyses the ntuple data in the file ‘test.dat’.  The
analysis procedure is to compute the squared-magnitude of each event,
E^2=x^2+y^2+z^2, and select only those which exceed a lower limit of
1.5.  The selected events are then histogrammed using their E^2 values.

     #include <math.h>
     #include <gsl/gsl_ntuple.h>
     #include <gsl/gsl_histogram.h>

     struct data
     {
       double x;
       double y;
       double z;
     };

     int sel_func (void *ntuple_data, void *params);
     double val_func (void *ntuple_data, void *params);

     int
     main (void)
     {
       struct data ntuple_row;

       gsl_ntuple *ntuple
         = gsl_ntuple_open ("test.dat", &ntuple_row,
                            sizeof (ntuple_row));
       double lower = 1.5;

       gsl_ntuple_select_fn S;
       gsl_ntuple_value_fn V;

       gsl_histogram *h = gsl_histogram_alloc (100);
       gsl_histogram_set_ranges_uniform(h, 0.0, 10.0);

       S.function = &sel_func;
       S.params = &lower;

       V.function = &val_func;
       V.params = 0;

       gsl_ntuple_project (h, ntuple, &V, &S);
       gsl_histogram_fprintf (stdout, h, "%f", "%f");
       gsl_histogram_free (h);
       gsl_ntuple_close (ntuple);

       return 0;
     }

     int
     sel_func (void *ntuple_data, void *params)
     {
       struct data * data = (struct data *) ntuple_data;
       double x, y, z, E2, scale;
       scale = *(double *) params;

       x = data->x;
       y = data->y;
       z = data->z;

       E2 = x * x + y * y + z * z;

       return E2 > scale;
     }

     double
     val_func (void *ntuple_data, void *params)
     {
       (void)(params); /* avoid unused parameter warning */
       struct data * data = (struct data *) ntuple_data;
       double x, y, z;

       x = data->x;
       y = data->y;
       z = data->z;

       return x * x + y * y + z * z;
     }

Fig.  %s shows the distribution of the selected events.  Note the
cut-off at the lower bound.

[image src="ntuple.png"]

Figure: Distribution of selected events


File: gsl-ref.info,  Node: References and Further Reading<19>,  Prev: Examples<19>,  Up: N-tuples

26.9 References and Further Reading
===================================

Further information on the use of ntuples can be found in the
documentation for the CERN packages PAW and HBOOK (available online).


File: gsl-ref.info,  Node: Monte Carlo Integration,  Next: Simulated Annealing,  Prev: N-tuples,  Up: Top

27 Monte Carlo Integration
**************************

This chapter describes routines for multidimensional Monte Carlo
integration.  These include the traditional Monte Carlo method and
adaptive algorithms such as VEGAS and MISER which use importance
sampling and stratified sampling techniques.  Each algorithm computes an
estimate of a multidimensional definite integral of the form,

     I = \int_{x_l}^{x_u} dx \int_{y_l}^{y_u} dy ... f(x, y, ...)

over a hypercubic region ((x_l,x_u), (y_l,y_u), ...) using a fixed
number of function calls.  The routines also provide a statistical
estimate of the error on the result.  This error estimate should be
taken as a guide rather than as a strict error bound—random sampling of
the region may not uncover all the important features of the function,
resulting in an underestimate of the error.

The functions are defined in separate header files for each routine,
‘gsl_monte_plain.h’, ‘gsl_monte_miser.h’ and ‘gsl_monte_vegas.h’.

* Menu:

* Interface::
* PLAIN Monte Carlo::
* MISER::
* VEGAS::
* Examples: Examples<20>.
* References and Further Reading: References and Further Reading<20>.


File: gsl-ref.info,  Node: Interface,  Next: PLAIN Monte Carlo,  Up: Monte Carlo Integration

27.1 Interface
==============

All of the Monte Carlo integration routines use the same general form of
interface.  There is an allocator to allocate memory for control
variables and workspace, a routine to initialize those control
variables, the integrator itself, and a function to free the space when
done.

Each integration function requires a random number generator to be
supplied, and returns an estimate of the integral and its standard
deviation.  The accuracy of the result is determined by the number of
function calls specified by the user.  If a known level of accuracy is
required this can be achieved by calling the integrator several times
and averaging the individual results until the desired accuracy is
obtained.

Random sample points used within the Monte Carlo routines are always
chosen strictly within the integration region, so that endpoint
singularities are automatically avoided.

The function to be integrated has its own datatype, defined in the
header file ‘gsl_monte.h’.

 -- Type: gsl_monte_function

     This data type defines a general function with parameters for Monte
     Carlo integration.

     ‘double (* f) (double * x, size_t dim, void * params)’           this function should return the value
                                                                      f(x,params) for the argument ‘x’ and
                                                                      parameters ‘params’, where ‘x’ is an array
                                                                      of size ‘dim’ giving the coordinates of the
                                                                      point where the function is to be evaluated.
                                                                      
                                                                      
     ‘size_t dim’                                                     the number of dimensions for ‘x’.
                                                                      
                                                                      
     ‘void * params’                                                  a pointer to the parameters of the function.
                                                                      

Here is an example for a quadratic function in two dimensions,

     f(x,y) = a x^2 + b x y + c y^2

with a = 3, b = 2, c = 1.  The following code defines a *note
gsl_monte_function: 914. ‘F’ which you could pass to an integrator:

     struct my_f_params { double a; double b; double c; };

     double
     my_f (double x[], size_t dim, void * p) {
        struct my_f_params * fp = (struct my_f_params *)p;

        if (dim != 2)
           {
             fprintf (stderr, "error: dim != 2");
             abort ();
           }

        return  fp->a * x[0] * x[0]
                  + fp->b * x[0] * x[1]
                    + fp->c * x[1] * x[1];
     }

     gsl_monte_function F;
     struct my_f_params params = { 3.0, 2.0, 1.0 };

     F.f = &my_f;
     F.dim = 2;
     F.params = &params;

The function f(x) can be evaluated using the following macro:

     #define GSL_MONTE_FN_EVAL(F,x)
         (*((F)->f))(x,(F)->dim,(F)->params)


File: gsl-ref.info,  Node: PLAIN Monte Carlo,  Next: MISER,  Prev: Interface,  Up: Monte Carlo Integration

27.2 PLAIN Monte Carlo
======================

The plain Monte Carlo algorithm samples points randomly from the
integration region to estimate the integral and its error.  Using this
algorithm the estimate of the integral E(f; N) for N randomly
distributed points x_i is given by,

     E(f; N) = =  V <f> = (V / N) \sum_i^N f(x_i)

where V is the volume of the integration region.  The error on this
estimate \sigma(E;N) is calculated from the estimated variance of the
mean,

     \sigma^2 (E; N) = (V^2 / N^2) \sum_i^N (f(x_i) -  <f>)^2.

For large N this variance decreases asymptotically as \Var(f)/N, where
\Var(f) is the true variance of the function over the integration
region.  The error estimate itself should decrease as
\sigma(f)/\sqrt{N}.  The familiar law of errors decreasing as 1/\sqrt{N}
applies—to reduce the error by a factor of 10 requires a 100-fold
increase in the number of sample points.

The functions described in this section are declared in the header file
‘gsl_monte_plain.h’.

 -- Type: gsl_monte_plain_state

     This is a workspace for plain Monte Carlo integration

 -- Function: gsl_monte_plain_state * gsl_monte_plain_alloc (size_t dim)

     This function allocates and initializes a workspace for Monte Carlo
     integration in ‘dim’ dimensions.

 -- Function: int gsl_monte_plain_init (gsl_monte_plain_state* s)

     This function initializes a previously allocated integration state.
     This allows an existing workspace to be reused for different
     integrations.

 -- Function: int gsl_monte_plain_integrate (gsl_monte_function * f,
          const double xl[], const double xu[], size_t dim,
          size_t calls, gsl_rng * r, gsl_monte_plain_state * s, double
          * result, double * abserr)

     This routines uses the plain Monte Carlo algorithm to integrate the
     function ‘f’ over the ‘dim’-dimensional hypercubic region defined
     by the lower and upper limits in the arrays ‘xl’ and ‘xu’, each of
     size ‘dim’.  The integration uses a fixed number of function calls
     ‘calls’, and obtains random sampling points using the random number
     generator ‘r’.  A previously allocated workspace ‘s’ must be
     supplied.  The result of the integration is returned in ‘result’,
     with an estimated absolute error ‘abserr’.

 -- Function: void gsl_monte_plain_free (gsl_monte_plain_state * s)

     This function frees the memory associated with the integrator state
     ‘s’.


File: gsl-ref.info,  Node: MISER,  Next: VEGAS,  Prev: PLAIN Monte Carlo,  Up: Monte Carlo Integration

27.3 MISER
==========

The MISER algorithm of Press and Farrar is based on recursive stratified
sampling.  This technique aims to reduce the overall integration error
by concentrating integration points in the regions of highest variance.

The idea of stratified sampling begins with the observation that for two
disjoint regions a and b with Monte Carlo estimates of the integral
E_a(f) and E_b(f) and variances \sigma_a^2(f) and \sigma_b^2(f), the
variance \Var(f) of the combined estimate E(f) = {1\over 2} (E_a(f) +
E_b(f)) is given by,

     \Var(f) = (\sigma_a^2(f) / 4 N_a) + (\sigma_b^2(f) / 4 N_b).

It can be shown that this variance is minimized by distributing the
points such that,

     N_a / (N_a + N_b) = \sigma_a / (\sigma_a + \sigma_b).

Hence the smallest error estimate is obtained by allocating sample
points in proportion to the standard deviation of the function in each
sub-region.

The MISER algorithm proceeds by bisecting the integration region along
one coordinate axis to give two sub-regions at each step.  The direction
is chosen by examining all d possible bisections and selecting the one
which will minimize the combined variance of the two sub-regions.  The
variance in the sub-regions is estimated by sampling with a fraction of
the total number of points available to the current step.  The same
procedure is then repeated recursively for each of the two half-spaces
from the best bisection.  The remaining sample points are allocated to
the sub-regions using the formula for N_a and N_b.  This recursive
allocation of integration points continues down to a user-specified
depth where each sub-region is integrated using a plain Monte Carlo
estimate.  These individual values and their error estimates are then
combined upwards to give an overall result and an estimate of its error.

The functions described in this section are declared in the header file
‘gsl_monte_miser.h’.

 -- Type: gsl_monte_miser_state

     This workspace is used for MISER Monte Carlo integration

 -- Function: gsl_monte_miser_state * gsl_monte_miser_alloc (size_t dim)

     This function allocates and initializes a workspace for Monte Carlo
     integration in ‘dim’ dimensions.  The workspace is used to maintain
     the state of the integration.

 -- Function: int gsl_monte_miser_init (gsl_monte_miser_state* s)

     This function initializes a previously allocated integration state.
     This allows an existing workspace to be reused for different
     integrations.

 -- Function: int gsl_monte_miser_integrate (gsl_monte_function * f,
          const double xl[], const double xu[], size_t dim,
          size_t calls, gsl_rng * r, gsl_monte_miser_state * s, double
          * result, double * abserr)

     This routines uses the MISER Monte Carlo algorithm to integrate the
     function ‘f’ over the ‘dim’-dimensional hypercubic region defined
     by the lower and upper limits in the arrays ‘xl’ and ‘xu’, each of
     size ‘dim’.  The integration uses a fixed number of function calls
     ‘calls’, and obtains random sampling points using the random number
     generator ‘r’.  A previously allocated workspace ‘s’ must be
     supplied.  The result of the integration is returned in ‘result’,
     with an estimated absolute error ‘abserr’.

 -- Function: void gsl_monte_miser_free (gsl_monte_miser_state * s)

     This function frees the memory associated with the integrator state
     ‘s’.

The MISER algorithm has several configurable parameters which can be
changed using the following two functions (1).

 -- Function: void gsl_monte_miser_params_get (const
          gsl_monte_miser_state * s, gsl_monte_miser_params * params)

     This function copies the parameters of the integrator state into
     the user-supplied ‘params’ structure.

 -- Function: void gsl_monte_miser_params_set (gsl_monte_miser_state
          * s, const gsl_monte_miser_params * params)

     This function sets the integrator parameters based on values
     provided in the ‘params’ structure.

Typically the values of the parameters are first read using *note
gsl_monte_miser_params_get(): 921, the necessary changes are made to the
fields of the ‘params’ structure, and the values are copied back into
the integrator state using *note gsl_monte_miser_params_set(): 922.  The
functions use the *note gsl_monte_miser_params: 923. structure which
contains the following fields:

 -- Type: gsl_monte_miser_params

      -- Variable: double estimate_frac

          This parameter specifies the fraction of the currently
          available number of function calls which are allocated to
          estimating the variance at each recursive step.  The default
          value is 0.1.

      -- Variable: size_t min_calls

          This parameter specifies the minimum number of function calls
          required for each estimate of the variance.  If the number of
          function calls allocated to the estimate using *note
          estimate_frac: 924. falls below *note min_calls: 925. then
          *note min_calls: 925. are used instead.  This ensures that
          each estimate maintains a reasonable level of accuracy.  The
          default value of *note min_calls: 925. is ‘16 * dim’.

      -- Variable: size_t min_calls_per_bisection

          This parameter specifies the minimum number of function calls
          required to proceed with a bisection step.  When a recursive
          step has fewer calls available than *note
          min_calls_per_bisection: 926. it performs a plain Monte Carlo
          estimate of the current sub-region and terminates its branch
          of the recursion.  The default value of this parameter is ‘32
          * min_calls’.

      -- Variable: double alpha

          This parameter controls how the estimated variances for the
          two sub-regions of a bisection are combined when allocating
          points.  With recursive sampling the overall variance should
          scale better than 1/N, since the values from the sub-regions
          will be obtained using a procedure which explicitly minimizes
          their variance.  To accommodate this behavior the MISER
          algorithm allows the total variance to depend on a scaling
          parameter \alpha,

               \Var(f) = {\sigma_a \over N_a^\alpha} + {\sigma_b \over N_b^\alpha}.

          The authors of the original paper describing MISER recommend
          the value \alpha = 2 as a good choice, obtained from numerical
          experiments, and this is used as the default value in this
          implementation.

      -- Variable: double dither

          This parameter introduces a random fractional variation of
          size *note dither: 927. into each bisection, which can be used
          to break the symmetry of integrands which are concentrated
          near the exact center of the hypercubic integration region.
          The default value of dither is zero, so no variation is
          introduced.  If needed, a typical value of *note dither: 927.
          is 0.1.

   ---------- Footnotes ----------

   (1) The previous method of accessing these fields directly through
the *note gsl_monte_miser_state: 91c. struct is now deprecated.


File: gsl-ref.info,  Node: VEGAS,  Next: Examples<20>,  Prev: MISER,  Up: Monte Carlo Integration

27.4 VEGAS
==========

The VEGAS algorithm of Lepage is based on importance sampling.  It
samples points from the probability distribution described by the
function |f|, so that the points are concentrated in the regions that
make the largest contribution to the integral.

In general, if the Monte Carlo integral of f is sampled with points
distributed according to a probability distribution described by the
function g, we obtain an estimate E_g(f; N),

     E_g(f; N) = E(f/g; N)

with a corresponding variance,

     \Var_g(f; N) = \Var(f/g; N)

If the probability distribution is chosen as g = |f|/I(|f|) then it can
be shown that the variance V_g(f; N) vanishes, and the error in the
estimate will be zero.  In practice it is not possible to sample from
the exact distribution g for an arbitrary function, so importance
sampling algorithms aim to produce efficient approximations to the
desired distribution.

The VEGAS algorithm approximates the exact distribution by making a
number of passes over the integration region while histogramming the
function f.  Each histogram is used to define a sampling distribution
for the next pass.  Asymptotically this procedure converges to the
desired distribution.  In order to avoid the number of histogram bins
growing like K^d the probability distribution is approximated by a
separable function: g(x_1, x_2, \ldots) = g_1(x_1) g_2(x_2) \ldots so
that the number of bins required is only Kd.  This is equivalent to
locating the peaks of the function from the projections of the integrand
onto the coordinate axes.  The efficiency of VEGAS depends on the
validity of this assumption.  It is most efficient when the peaks of the
integrand are well-localized.  If an integrand can be rewritten in a
form which is approximately separable this will increase the efficiency
of integration with VEGAS.

VEGAS incorporates a number of additional features, and combines both
stratified sampling and importance sampling.  The integration region is
divided into a number of “boxes”, with each box getting a fixed number
of points (the goal is 2).  Each box can then have a fractional number
of bins, but if the ratio of bins-per-box is less than two, Vegas
switches to a kind variance reduction (rather than importance sampling).

 -- Type: gsl_monte_vegas_state

     This workspace is used for VEGAS Monte Carlo integration

 -- Function: gsl_monte_vegas_state * gsl_monte_vegas_alloc (size_t dim)

     This function allocates and initializes a workspace for Monte Carlo
     integration in ‘dim’ dimensions.  The workspace is used to maintain
     the state of the integration.

 -- Function: int gsl_monte_vegas_init (gsl_monte_vegas_state* s)

     This function initializes a previously allocated integration state.
     This allows an existing workspace to be reused for different
     integrations.

 -- Function: int gsl_monte_vegas_integrate (gsl_monte_function * f,
          double xl[], double xu[], size_t dim, size_t calls, gsl_rng
          * r, gsl_monte_vegas_state * s, double * result, double
          * abserr)

     This routines uses the VEGAS Monte Carlo algorithm to integrate the
     function ‘f’ over the ‘dim’-dimensional hypercubic region defined
     by the lower and upper limits in the arrays ‘xl’ and ‘xu’, each of
     size ‘dim’.  The integration uses a fixed number of function calls
     ‘calls’, and obtains random sampling points using the random number
     generator ‘r’.  A previously allocated workspace ‘s’ must be
     supplied.  The result of the integration is returned in ‘result’,
     with an estimated absolute error ‘abserr’.  The result and its
     error estimate are based on a weighted average of independent
     samples.  The chi-squared per degree of freedom for the weighted
     average is returned via the state struct component, ‘s->chisq’, and
     must be consistent with 1 for the weighted average to be reliable.

 -- Function: void gsl_monte_vegas_free (gsl_monte_vegas_state * s)

     This function frees the memory associated with the integrator state
     ‘s’.

The VEGAS algorithm computes a number of independent estimates of the
integral internally, according to the ‘iterations’ parameter described
below, and returns their weighted average.  Random sampling of the
integrand can occasionally produce an estimate where the error is zero,
particularly if the function is constant in some regions.  An estimate
with zero error causes the weighted average to break down and must be
handled separately.  In the original Fortran implementations of VEGAS
the error estimate is made non-zero by substituting a small value
(typically ‘1e-30’).  The implementation in GSL differs from this and
avoids the use of an arbitrary constant—it either assigns the value a
weight which is the average weight of the preceding estimates or
discards it according to the following procedure,

   * current estimate has zero error, weighted average has finite error

     The current estimate is assigned a weight which is the average
     weight of the preceding estimates.

   * current estimate has finite error, previous estimates had zero
     error

     The previous estimates are discarded and the weighted averaging
     procedure begins with the current estimate.

   * current estimate has zero error, previous estimates had zero error

     The estimates are averaged using the arithmetic mean, but no error
     is computed.

The convergence of the algorithm can be tested using the overall
chi-squared value of the results, which is available from the following
function:

 -- Function: double gsl_monte_vegas_chisq (const gsl_monte_vegas_state
          * s)

     This function returns the chi-squared per degree of freedom for the
     weighted estimate of the integral.  The returned value should be
     close to 1.  A value which differs significantly from 1 indicates
     that the values from different iterations are inconsistent.  In
     this case the weighted error will be under-estimated, and further
     iterations of the algorithm are needed to obtain reliable results.

 -- Function: void gsl_monte_vegas_runval (const gsl_monte_vegas_state
          * s, double * result, double * sigma)

     This function returns the raw (unaveraged) values of the integral
     ‘result’ and its error ‘sigma’ from the most recent iteration of
     the algorithm.

The VEGAS algorithm is highly configurable.  Several parameters can be
changed using the following two functions.

 -- Function: void gsl_monte_vegas_params_get (const
          gsl_monte_vegas_state * s, gsl_monte_vegas_params * params)

     This function copies the parameters of the integrator state into
     the user-supplied ‘params’ structure.

 -- Function: void gsl_monte_vegas_params_set (gsl_monte_vegas_state
          * s, const gsl_monte_vegas_params * params)

     This function sets the integrator parameters based on values
     provided in the ‘params’ structure.

Typically the values of the parameters are first read using *note
gsl_monte_vegas_params_get(): 930, the necessary changes are made to the
fields of the ‘params’ structure, and the values are copied back into
the integrator state using *note gsl_monte_vegas_params_set(): 931.  The
functions use the *note gsl_monte_vegas_params: 932. structure which
contains the following fields:

 -- Type: gsl_monte_vegas_params

      -- Variable: double alpha

          The parameter *note alpha: 49a. controls the stiffness of the
          rebinning algorithm.  It is typically set between one and two.
          A value of zero prevents rebinning of the grid.  The default
          value is 1.5.

      -- Variable: size_t iterations

          The number of iterations to perform for each call to the
          routine.  The default value is 5 iterations.

      -- Variable: int stage

          Setting this determines the `stage' of the calculation.
          Normally, ‘stage = 0’ which begins with a new uniform grid and
          empty weighted average.  Calling VEGAS with ‘stage = 1’
          retains the grid from the previous run but discards the
          weighted average, so that one can “tune” the grid using a
          relatively small number of points and then do a large run with
          ‘stage = 1’ on the optimized grid.  Setting ‘stage = 2’ keeps
          the grid and the weighted average from the previous run, but
          may increase (or decrease) the number of histogram bins in the
          grid depending on the number of calls available.  Choosing
          ‘stage = 3’ enters at the main loop, so that nothing is
          changed, and is equivalent to performing additional iterations
          in a previous call.

      -- Variable: int mode

          The possible choices are ‘GSL_VEGAS_MODE_IMPORTANCE’,
          ‘GSL_VEGAS_MODE_STRATIFIED’, ‘GSL_VEGAS_MODE_IMPORTANCE_ONLY’.
          This determines whether VEGAS will use importance sampling or
          stratified sampling, or whether it can pick on its own.  In
          low dimensions VEGAS uses strict stratified sampling (more
          precisely, stratified sampling is chosen if there are fewer
          than 2 bins per box).

      -- Variable: int verbose
      -- Variable: FILE * ostream

          These parameters set the level of information printed by
          VEGAS. All information is written to the stream *note ostream:
          936.  The default setting of *note verbose: 935. is ‘-1’,
          which turns off all output.  A *note verbose: 935. value of
          ‘0’ prints summary information about the weighted average and
          final result, while a value of ‘1’ also displays the grid
          coordinates.  A value of ‘2’ prints information from the
          rebinning procedure for each iteration.

The above fields and the ‘chisq’ value can also be accessed directly in
the *note gsl_monte_vegas_state: 929. but such use is deprecated.


File: gsl-ref.info,  Node: Examples<20>,  Next: References and Further Reading<20>,  Prev: VEGAS,  Up: Monte Carlo Integration

27.5 Examples
=============

The example program below uses the Monte Carlo routines to estimate the
value of the following 3-dimensional integral from the theory of random
walks,

     I = \int_{-pi}^{+pi} {dk_x/(2 pi)}
         \int_{-pi}^{+pi} {dk_y/(2 pi)}
         \int_{-pi}^{+pi} {dk_z/(2 pi)}
          1 / (1 - cos(k_x)cos(k_y)cos(k_z)).

The analytic value of this integral can be shown to be I =
\Gamma(1/4)^4/(4 \pi^3) = 1.393203929685676859....  The integral gives
the mean time spent at the origin by a random walk on a body-centered
cubic lattice in three dimensions.

For simplicity we will compute the integral over the region (0,0,0) to
(\pi,\pi,\pi) and multiply by 8 to obtain the full result.  The integral
is slowly varying in the middle of the region but has integrable
singularities at the corners (0,0,0), (0,\pi,\pi), (\pi,0,\pi) and
(\pi,\pi,0).  The Monte Carlo routines only select points which are
strictly within the integration region and so no special measures are
needed to avoid these singularities.

     #include <stdlib.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_monte.h>
     #include <gsl/gsl_monte_plain.h>
     #include <gsl/gsl_monte_miser.h>
     #include <gsl/gsl_monte_vegas.h>

     /* Computation of the integral,

           I = int (dx dy dz)/(2pi)^3  1/(1-cos(x)cos(y)cos(z))

        over (-pi,-pi,-pi) to (+pi, +pi, +pi).  The exact answer
        is Gamma(1/4)^4/(4 pi^3).  This example is taken from
        C.Itzykson, J.M.Drouffe, "Statistical Field Theory -
        Volume 1", Section 1.1, p21, which cites the original
        paper M.L.Glasser, I.J.Zucker, Proc.Natl.Acad.Sci.USA 74
        1800 (1977) */

     /* For simplicity we compute the integral over the region
        (0,0,0) -> (pi,pi,pi) and multiply by 8 */

     double exact = 1.3932039296856768591842462603255;

     double
     g (double *k, size_t dim, void *params)
     {
       (void)(dim); /* avoid unused parameter warnings */
       (void)(params);
       double A = 1.0 / (M_PI * M_PI * M_PI);
       return A / (1.0 - cos (k[0]) * cos (k[1]) * cos (k[2]));
     }

     void
     display_results (char *title, double result, double error)
     {
       printf ("%s ==================\n", title);
       printf ("result = % .6f\n", result);
       printf ("sigma  = % .6f\n", error);
       printf ("exact  = % .6f\n", exact);
       printf ("error  = % .6f = %.2g sigma\n", result - exact,
               fabs (result - exact) / error);
     }

     int
     main (void)
     {
       double res, err;

       double xl[3] = { 0, 0, 0 };
       double xu[3] = { M_PI, M_PI, M_PI };

       const gsl_rng_type *T;
       gsl_rng *r;

       gsl_monte_function G = { &g, 3, 0 };

       size_t calls = 500000;

       gsl_rng_env_setup ();

       T = gsl_rng_default;
       r = gsl_rng_alloc (T);

       {
         gsl_monte_plain_state *s = gsl_monte_plain_alloc (3);
         gsl_monte_plain_integrate (&G, xl, xu, 3, calls, r, s,
                                    &res, &err);
         gsl_monte_plain_free (s);

         display_results ("plain", res, err);
       }

       {
         gsl_monte_miser_state *s = gsl_monte_miser_alloc (3);
         gsl_monte_miser_integrate (&G, xl, xu, 3, calls, r, s,
                                    &res, &err);
         gsl_monte_miser_free (s);

         display_results ("miser", res, err);
       }

       {
         gsl_monte_vegas_state *s = gsl_monte_vegas_alloc (3);

         gsl_monte_vegas_integrate (&G, xl, xu, 3, 10000, r, s,
                                    &res, &err);
         display_results ("vegas warm-up", res, err);

         printf ("converging...\n");

         do
           {
             gsl_monte_vegas_integrate (&G, xl, xu, 3, calls/5, r, s,
                                        &res, &err);
             printf ("result = % .6f sigma = % .6f "
                     "chisq/dof = %.1f\n", res, err, gsl_monte_vegas_chisq (s));
           }
         while (fabs (gsl_monte_vegas_chisq (s) - 1.0) > 0.5);

         display_results ("vegas final", res, err);

         gsl_monte_vegas_free (s);
       }

       gsl_rng_free (r);

       return 0;
     }

With 500,000 function calls the plain Monte Carlo algorithm achieves a
fractional error of 1%.  The estimated error ‘sigma’ is roughly
consistent with the actual error–the computed result differs from the
true result by about 1.4 standard deviations:

     plain ==================
     result =  1.412209
     sigma  =  0.013436
     exact  =  1.393204
     error  =  0.019005 = 1.4 sigma

The MISER algorithm reduces the error by a factor of four, and also
correctly estimates the error:

     miser ==================
     result =  1.391322
     sigma  =  0.003461
     exact  =  1.393204
     error  = -0.001882 = 0.54 sigma

In the case of the VEGAS algorithm the program uses an initial warm-up
run of 10,000 function calls to prepare, or “warm up”, the grid.  This
is followed by a main run with five iterations of 100,000 function
calls.  The chi-squared per degree of freedom for the five iterations
are checked for consistency with 1, and the run is repeated if the
results have not converged.  In this case the estimates are consistent
on the first pass:

     vegas warm-up ==================
     result =  1.392673
     sigma  =  0.003410
     exact  =  1.393204
     error  = -0.000531 = 0.16 sigma
     converging...
     result =  1.393281 sigma =  0.000362 chisq/dof = 1.5
     vegas final ==================
     result =  1.393281
     sigma  =  0.000362
     exact  =  1.393204
     error  =  0.000077 = 0.21 sigma

If the value of ‘chisq’ had differed significantly from 1 it would
indicate inconsistent results, with a correspondingly underestimated
error.  The final estimate from VEGAS (using a similar number of
function calls) is significantly more accurate than the other two
algorithms.


File: gsl-ref.info,  Node: References and Further Reading<20>,  Prev: Examples<20>,  Up: Monte Carlo Integration

27.6 References and Further Reading
===================================

The MISER algorithm is described in the following article by Press and
Farrar,

   * W.H. Press, G.R. Farrar, `Recursive Stratified Sampling for
     Multidimensional Monte Carlo Integration', Computers in Physics, v4
     (1990), pp190–195.

The VEGAS algorithm is described in the following papers,

   * G.P. Lepage, `A New Algorithm for Adaptive Multidimensional
     Integration', Journal of Computational Physics 27, 192–203, (1978)

   * G.P. Lepage, `VEGAS: An Adaptive Multi-dimensional Integration
     Program', Cornell preprint CLNS 80-447, March 1980


File: gsl-ref.info,  Node: Simulated Annealing,  Next: Ordinary Differential Equations,  Prev: Monte Carlo Integration,  Up: Top

28 Simulated Annealing
**********************

Stochastic search techniques are used when the structure of a space is
not well understood or is not smooth, so that techniques like Newton’s
method (which requires calculating Jacobian derivative matrices) cannot
be used.  In particular, these techniques are frequently used to solve
combinatorial optimization problems, such as the traveling salesman
problem.

The goal is to find a point in the space at which a real valued `energy
function' (or `cost function') is minimized.  Simulated annealing is a
minimization technique which has given good results in avoiding local
minima; it is based on the idea of taking a random walk through the
space at successively lower temperatures, where the probability of
taking a step is given by a Boltzmann distribution.

The functions described in this chapter are declared in the header file
‘gsl_siman.h’.

* Menu:

* Simulated Annealing algorithm::
* Simulated Annealing functions::
* Examples: Examples<21>.
* References and Further Reading: References and Further Reading<21>.


File: gsl-ref.info,  Node: Simulated Annealing algorithm,  Next: Simulated Annealing functions,  Up: Simulated Annealing

28.1 Simulated Annealing algorithm
==================================

The simulated annealing algorithm takes random walks through the problem
space, looking for points with low energies; in these random walks, the
probability of taking a step is determined by the Boltzmann
distribution,

     p = e^{-(E_{i+1} - E_i)/(kT)}

if E_{i+1} > E_i, and p = 1 when E_{i+1} \le E_i.

In other words, a step will occur if the new energy is lower.  If the
new energy is higher, the transition can still occur, and its likelihood
is proportional to the temperature T and inversely proportional to the
energy difference E_{i+1} - E_i.

The temperature T is initially set to a high value, and a random walk is
carried out at that temperature.  Then the temperature is lowered very
slightly according to a `cooling schedule', for example: T \rightarrow
T/\mu_T where \mu_T is slightly greater than 1.

The slight probability of taking a step that gives higher energy is what
allows simulated annealing to frequently get out of local minima.


File: gsl-ref.info,  Node: Simulated Annealing functions,  Next: Examples<21>,  Prev: Simulated Annealing algorithm,  Up: Simulated Annealing

28.2 Simulated Annealing functions
==================================

 -- Function: void gsl_siman_solve (const gsl_rng * r, void * x0_p,
          gsl_siman_Efunc_t Ef, gsl_siman_step_t take_step,
          gsl_siman_metric_t distance, gsl_siman_print_t print_position,
          gsl_siman_copy_t copyfunc,
          gsl_siman_copy_construct_t copy_constructor,
          gsl_siman_destroy_t destructor, size_t element_size,
          gsl_siman_params_t params)

     This function performs a simulated annealing search through a given
     space.  The space is specified by providing the functions ‘Ef’ and
     ‘distance’.  The simulated annealing steps are generated using the
     random number generator ‘r’ and the function ‘take_step’.

     The starting configuration of the system should be given by ‘x0_p’.
     The routine offers two modes for updating configurations, a
     fixed-size mode and a variable-size mode.  In the fixed-size mode
     the configuration is stored as a single block of memory of size
     ‘element_size’.  Copies of this configuration are created, copied
     and destroyed internally using the standard library functions
     ‘malloc()’, ‘memcpy()’ and ‘free()’.  The function pointers
     ‘copyfunc’, ‘copy_constructor’ and ‘destructor’ should be null
     pointers in fixed-size mode.  In the variable-size mode the
     functions ‘copyfunc’, ‘copy_constructor’ and ‘destructor’ are used
     to create, copy and destroy configurations internally.  The
     variable ‘element_size’ should be zero in the variable-size mode.

     The ‘params’ structure (described below) controls the run by
     providing the temperature schedule and other tunable parameters to
     the algorithm.

     On exit the best result achieved during the search is placed in
     ‘x0_p’.  If the annealing process has been successful this should
     be a good approximation to the optimal point in the space.

     If the function pointer ‘print_position’ is not null, a debugging
     log will be printed to ‘stdout’ with the following columns:

          #-iter  #-evals  temperature  position  energy  best_energy

     and the output of the function ‘print_position’ itself.  If
     ‘print_position’ is null then no information is printed.

The simulated annealing routines require several user-specified
functions to define the configuration space and energy function.  The
prototypes for these functions are given below.

 -- Type: gsl_siman_Efunc_t

     This function type should return the energy of a configuration
     ‘xp’:

          double (*gsl_siman_Efunc_t) (void *xp)

 -- Type: gsl_siman_step_t

     This function type should modify the configuration ‘xp’ using a
     random step taken from the generator ‘r’, up to a maximum distance
     of ‘step_size’:

          void (*gsl_siman_step_t) (const gsl_rng *r, void *xp,
                                    double step_size)

 -- Type: gsl_siman_metric_t

     This function type should return the distance between two
     configurations ‘xp’ and ‘yp’:

          double (*gsl_siman_metric_t) (void *xp, void *yp)

 -- Type: gsl_siman_print_t

     This function type should print the contents of the configuration
     ‘xp’:

          void (*gsl_siman_print_t) (void *xp)

 -- Type: gsl_siman_copy_t

     This function type should copy the configuration ‘source’ into
     ‘dest’:

          void (*gsl_siman_copy_t) (void *source, void *dest)

 -- Type: gsl_siman_copy_construct_t

     This function type should create a new copy of the configuration
     ‘xp’:

          void * (*gsl_siman_copy_construct_t) (void *xp)

 -- Type: gsl_siman_destroy_t

     This function type should destroy the configuration ‘xp’, freeing
     its memory:

          void (*gsl_siman_destroy_t) (void *xp)

 -- Type: gsl_siman_params_t

     These are the parameters that control a run of *note
     gsl_siman_solve(): 93d.  This structure contains all the
     information needed to control the search, beyond the energy
     function, the step function and the initial guess.

     ‘int n_tries’                                 The number of points to try for each step.
                                                   
                                                   
     ‘int iters_fixed_T’                           The number of iterations at each temperature.
                                                   
                                                   
     ‘double step_size’                            The maximum step size in the random walk.
                                                   
                                                   
     ‘double k, t_initial, mu_t, t_min’            The parameters of the Boltzmann distribution and cooling
                                                   schedule.
                                                   


File: gsl-ref.info,  Node: Examples<21>,  Next: References and Further Reading<21>,  Prev: Simulated Annealing functions,  Up: Simulated Annealing

28.3 Examples
=============

The simulated annealing package is clumsy, and it has to be because it
is written in C, for C callers, and tries to be polymorphic at the same
time.  But here we provide some examples which can be pasted into your
application with little change and should make things easier.

* Menu:

* Trivial example::
* Traveling Salesman Problem::


File: gsl-ref.info,  Node: Trivial example,  Next: Traveling Salesman Problem,  Up: Examples<21>

28.3.1 Trivial example
----------------------

The first example, in one dimensional Cartesian space, sets up an energy
function which is a damped sine wave; this has many local minima, but
only one global minimum, somewhere between 1.0 and 1.5.  The initial
guess given is 15.5, which is several local minima away from the global
minimum.

     #include <math.h>
     #include <stdlib.h>
     #include <string.h>
     #include <gsl/gsl_siman.h>

     /* set up parameters for this simulated annealing run */

     /* how many points do we try before stepping */
     #define N_TRIES 200

     /* how many iterations for each T? */
     #define ITERS_FIXED_T 1000

     /* max step size in random walk */
     #define STEP_SIZE 1.0

     /* Boltzmann constant */
     #define K 1.0

     /* initial temperature */
     #define T_INITIAL 0.008

     /* damping factor for temperature */
     #define MU_T 1.003
     #define T_MIN 2.0e-6

     gsl_siman_params_t params
       = {N_TRIES, ITERS_FIXED_T, STEP_SIZE,
          K, T_INITIAL, MU_T, T_MIN};

     /* now some functions to test in one dimension */
     double E1(void *xp)
     {
       double x = * ((double *) xp);

       return exp(-pow((x-1.0),2.0))*sin(8*x);
     }

     double M1(void *xp, void *yp)
     {
       double x = *((double *) xp);
       double y = *((double *) yp);

       return fabs(x - y);
     }

     void S1(const gsl_rng * r, void *xp, double step_size)
     {
       double old_x = *((double *) xp);
       double new_x;

       double u = gsl_rng_uniform(r);
       new_x = u * 2 * step_size - step_size + old_x;

       memcpy(xp, &new_x, sizeof(new_x));
     }

     void P1(void *xp)
     {
       printf ("%12g", *((double *) xp));
     }

     int
     main(void)
     {
       const gsl_rng_type * T;
       gsl_rng * r;

       double x_initial = 15.5;

       gsl_rng_env_setup();

       T = gsl_rng_default;
       r = gsl_rng_alloc(T);

       gsl_siman_solve(r, &x_initial, E1, S1, M1, P1,
                       NULL, NULL, NULL,
                       sizeof(double), params);

       gsl_rng_free (r);
       return 0;
     }

Fig.  %s is generated by running ‘siman_test’ in the following way:

     $ ./siman_test | awk '!/^#/ {print $1, $4}'
      | graph -y 1.34 1.4 -W0 -X generation -Y position
      | plot -Tps > siman-test.eps

Fig.  %s is generated by running ‘siman_test’ in the following way:

     $ ./siman_test | awk '!/^#/ {print $1, $5}'
      | graph -y -0.88 -0.83 -W0 -X generation -Y energy
      | plot -Tps > siman-energy.eps

[image src="siman-test.png"]

Figure: Example of a simulated annealing run: at higher temperatures
(early in the plot) you see that the solution can fluctuate, but at
lower temperatures it converges.

[image src="siman-energy.png"]

Figure: Simulated annealing energy vs generation


File: gsl-ref.info,  Node: Traveling Salesman Problem,  Prev: Trivial example,  Up: Examples<21>

28.3.2 Traveling Salesman Problem
---------------------------------

The TSP (`Traveling Salesman Problem') is the classic combinatorial
optimization problem.  I have provided a very simple version of it,
based on the coordinates of twelve cities in the southwestern United
States.  This should maybe be called the `Flying Salesman Problem',
since I am using the great-circle distance between cities, rather than
the driving distance.  Also: I assume the earth is a sphere, so I don’t
use geoid distances.

The *note gsl_siman_solve(): 93d. routine finds a route which is 3490.62
Kilometers long; this is confirmed by an exhaustive search of all
possible routes with the same initial city.

The full code is given below.

     /* siman/siman_tsp.c
      *
      * Copyright (C) 1996, 1997, 1998, 1999, 2000 Mark Galassi
      *
      * This program is free software; you can redistribute it and/or modify
      * it under the terms of the GNU General Public License as published by
      * the Free Software Foundation; either version 3 of the License, or (at
      * your option) any later version.
      *
      * This program is distributed in the hope that it will be useful, but
      * WITHOUT ANY WARRANTY; without even the implied warranty of
      * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
      * General Public License for more details.
      *
      * You should have received a copy of the GNU General Public License
      * along with this program; if not, write to the Free Software
      * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
      */

     #include <config.h>
     #include <math.h>
     #include <string.h>
     #include <stdio.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_siman.h>
     #include <gsl/gsl_ieee_utils.h>

     /* set up parameters for this simulated annealing run */

     #define N_TRIES 200             /* how many points do we try before stepping */
     #define ITERS_FIXED_T 2000      /* how many iterations for each T? */
     #define STEP_SIZE 1.0           /* max step size in random walk */
     #define K 1.0                   /* Boltzmann constant */
     #define T_INITIAL 5000.0        /* initial temperature */
     #define MU_T 1.002              /* damping factor for temperature */
     #define T_MIN 5.0e-1

     gsl_siman_params_t params = {N_TRIES, ITERS_FIXED_T, STEP_SIZE,
                                  K, T_INITIAL, MU_T, T_MIN};

     struct s_tsp_city {
       const char * name;
       double lat, longitude;        /* coordinates */
     };
     typedef struct s_tsp_city Stsp_city;

     void prepare_distance_matrix(void);
     void exhaustive_search(void);
     void print_distance_matrix(void);
     double city_distance(Stsp_city c1, Stsp_city c2);
     double Etsp(void *xp);
     double Mtsp(void *xp, void *yp);
     void Stsp(const gsl_rng * r, void *xp, double step_size);
     void Ptsp(void *xp);

     /* in this table, latitude and longitude are obtained from the US
        Census Bureau, at http://www.census.gov/cgi-bin/gazetteer */

     Stsp_city cities[] = {{"Santa Fe",    35.68,   105.95},
                           {"Phoenix",     33.54,   112.07},
                           {"Albuquerque", 35.12,   106.62},
                           {"Clovis",      34.41,   103.20},
                           {"Durango",     37.29,   107.87},
                           {"Dallas",      32.79,    96.77},
                           {"Tesuque",     35.77,   105.92},
                           {"Grants",      35.15,   107.84},
                           {"Los Alamos",  35.89,   106.28},
                           {"Las Cruces",  32.34,   106.76},
                           {"Cortez",      37.35,   108.58},
                           {"Gallup",      35.52,   108.74}};

     #define N_CITIES (sizeof(cities)/sizeof(Stsp_city))

     double distance_matrix[N_CITIES][N_CITIES];

     /* distance between two cities */
     double city_distance(Stsp_city c1, Stsp_city c2)
     {
       const double earth_radius = 6375.000; /* 6000KM approximately */
       /* sin and cos of lat and long; must convert to radians */
       double sla1 = sin(c1.lat*M_PI/180), cla1 = cos(c1.lat*M_PI/180),
         slo1 = sin(c1.longitude*M_PI/180), clo1 = cos(c1.longitude*M_PI/180);
       double sla2 = sin(c2.lat*M_PI/180), cla2 = cos(c2.lat*M_PI/180),
         slo2 = sin(c2.longitude*M_PI/180), clo2 = cos(c2.longitude*M_PI/180);

       double x1 = cla1*clo1;
       double x2 = cla2*clo2;

       double y1 = cla1*slo1;
       double y2 = cla2*slo2;

       double z1 = sla1;
       double z2 = sla2;

       double dot_product = x1*x2 + y1*y2 + z1*z2;

       double angle = acos(dot_product);

       /* distance is the angle (in radians) times the earth radius */
       return angle*earth_radius;
     }

     /* energy for the travelling salesman problem */
     double Etsp(void *xp)
     {
       /* an array of N_CITIES integers describing the order */
       int *route = (int *) xp;
       double E = 0;
       unsigned int i;

       for (i = 0; i < N_CITIES; ++i) {
         /* use the distance_matrix to optimize this calculation; it had
            better be allocated!! */
         E += distance_matrix[route[i]][route[(i + 1) % N_CITIES]];
       }

       return E;
     }

     double Mtsp(void *xp, void *yp)
     {
       int *route1 = (int *) xp, *route2 = (int *) yp;
       double distance = 0;
       unsigned int i;

       for (i = 0; i < N_CITIES; ++i) {
         distance += ((route1[i] == route2[i]) ? 0 : 1);
       }

       return distance;
     }

     /* take a step through the TSP space */
     void Stsp(const gsl_rng * r, void *xp, double step_size)
     {
       int x1, x2, dummy;
       int *route = (int *) xp;

       step_size = 0 ; /* prevent warnings about unused parameter */

       /* pick the two cities to swap in the matrix; we leave the first
          city fixed */
       x1 = (gsl_rng_get (r) % (N_CITIES-1)) + 1;
       do {
         x2 = (gsl_rng_get (r) % (N_CITIES-1)) + 1;
       } while (x2 == x1);

       dummy = route[x1];
       route[x1] = route[x2];
       route[x2] = dummy;
     }

     void Ptsp(void *xp)
     {
       unsigned int i;
       int *route = (int *) xp;
       printf("  [");
       for (i = 0; i < N_CITIES; ++i) {
         printf(" %d ", route[i]);
       }
       printf("]  ");
     }

     int main(void)
     {
       int x_initial[N_CITIES];
       unsigned int i;

       const gsl_rng * r = gsl_rng_alloc (gsl_rng_env_setup()) ;

       gsl_ieee_env_setup ();

       prepare_distance_matrix();

       /* set up a trivial initial route */
       printf("# initial order of cities:\n");
       for (i = 0; i < N_CITIES; ++i) {
         printf("# \"%s\"\n", cities[i].name);
         x_initial[i] = i;
       }

       printf("# distance matrix is:\n");
       print_distance_matrix();

       printf("# initial coordinates of cities (longitude and latitude)\n");
       /* this can be plotted with */
       /* ./siman_tsp > hhh ; grep city_coord hhh | awk '{print $2 "   " $3}' | xyplot -ps -d "xy" > c.eps */
       for (i = 0; i < N_CITIES+1; ++i) {
         printf("###initial_city_coord: %g %g \"%s\"\n",
                -cities[x_initial[i % N_CITIES]].longitude,
                cities[x_initial[i % N_CITIES]].lat,
                cities[x_initial[i % N_CITIES]].name);
       }

     /*   exhaustive_search(); */

       gsl_siman_solve(r, x_initial, Etsp, Stsp, Mtsp, Ptsp, NULL, NULL, NULL,
                       N_CITIES*sizeof(int), params);

       printf("# final order of cities:\n");
       for (i = 0; i < N_CITIES; ++i) {
         printf("# \"%s\"\n", cities[x_initial[i]].name);
       }

       printf("# final coordinates of cities (longitude and latitude)\n");
       /* this can be plotted with */
       /* ./siman_tsp > hhh ; grep city_coord hhh | awk '{print $2 "   " $3}' | xyplot -ps -d "xy" > c.eps */
       for (i = 0; i < N_CITIES+1; ++i) {
         printf("###final_city_coord: %g %g %s\n",
                -cities[x_initial[i % N_CITIES]].longitude,
                cities[x_initial[i % N_CITIES]].lat,
                cities[x_initial[i % N_CITIES]].name);
       }

       printf("# ");
       fflush(stdout);
     #if 0
       system("date");
     #endif /* 0 */
       fflush(stdout);

       return 0;
     }

     void prepare_distance_matrix()
     {
       unsigned int i, j;
       double dist;

       for (i = 0; i < N_CITIES; ++i) {
         for (j = 0; j < N_CITIES; ++j) {
           if (i == j) {
             dist = 0;
           } else {
             dist = city_distance(cities[i], cities[j]);
           }
           distance_matrix[i][j] = dist;
         }
       }
     }

     void print_distance_matrix()
     {
       unsigned int i, j;

       for (i = 0; i < N_CITIES; ++i) {
         printf("# ");
         for (j = 0; j < N_CITIES; ++j) {
           printf("%15.8f   ", distance_matrix[i][j]);
         }
         printf("\n");
       }
     }

     /* [only works for 12] search the entire space for solutions */
     static double best_E = 1.0e100, second_E = 1.0e100, third_E = 1.0e100;
     static int best_route[N_CITIES];
     static int second_route[N_CITIES];
     static int third_route[N_CITIES];
     static void do_all_perms(int *route, int n);

     void exhaustive_search()
     {
       static int initial_route[N_CITIES] = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11};
       printf("\n# ");
       fflush(stdout);
     #if 0
       system("date");
     #endif
       fflush(stdout);
       do_all_perms(initial_route, 1);
       printf("\n# ");
       fflush(stdout);
     #if 0
       system("date");
     #endif /* 0 */
       fflush(stdout);

       printf("# exhaustive best route: ");
       Ptsp(best_route);
       printf("\n# its energy is: %g\n", best_E);

       printf("# exhaustive second_best route: ");
       Ptsp(second_route);
       printf("\n# its energy is: %g\n", second_E);

       printf("# exhaustive third_best route: ");
       Ptsp(third_route);
       printf("\n# its energy is: %g\n", third_E);
     }

     /* James Theiler's recursive algorithm for generating all routes */
     static void do_all_perms(int *route, int n)
     {
       if (n == (N_CITIES-1)) {
         /* do it! calculate the energy/cost for that route */
         double E;
         E = Etsp(route);            /* TSP energy function */
         /* now save the best 3 energies and routes */
         if (E < best_E) {
           third_E = second_E;
           memcpy(third_route, second_route, N_CITIES*sizeof(*route));
           second_E = best_E;
           memcpy(second_route, best_route, N_CITIES*sizeof(*route));
           best_E = E;
           memcpy(best_route, route, N_CITIES*sizeof(*route));
         } else if (E < second_E) {
           third_E = second_E;
           memcpy(third_route, second_route, N_CITIES*sizeof(*route));
           second_E = E;
           memcpy(second_route, route, N_CITIES*sizeof(*route));
         } else if (E < third_E) {
           third_E = E;
           memcpy(route, third_route, N_CITIES*sizeof(*route));
         }
       } else {
         int new_route[N_CITIES];
         unsigned int j;
         int swap_tmp;
         memcpy(new_route, route, N_CITIES*sizeof(*route));
         for (j = n; j < N_CITIES; ++j) {
           swap_tmp = new_route[j];
           new_route[j] = new_route[n];
           new_route[n] = swap_tmp;
           do_all_perms(new_route, n+1);
         }
       }
     }

Below are some plots generated in the following way:

     $ ./siman_tsp > tsp.output
     $ grep -v "^#" tsp.output
      | awk '{print $1, $NF}'
      | graph -y 3300 6500 -W0 -X generation -Y distance
         -L "TSP - 12 southwest cities"
      | plot -Tps > 12-cities.eps
     $ grep initial_city_coord tsp.output
       | awk '{print $2, $3}'
       | graph -X "longitude (- means west)" -Y "latitude"
          -L "TSP - initial-order" -f 0.03 -S 1 0.1
       | plot -Tps > initial-route.eps
     $ grep final_city_coord tsp.output
       | awk '{print $2, $3}'
       | graph -X "longitude (- means west)" -Y "latitude"
          -L "TSP - final-order" -f 0.03 -S 1 0.1
       | plot -Tps > final-route.eps

This is the output showing the initial order of the cities; longitude is
negative, since it is west and I want the plot to look like a map:

     # initial coordinates of cities (longitude and latitude)
     ###initial_city_coord: -105.95 35.68 Santa Fe
     ###initial_city_coord: -112.07 33.54 Phoenix
     ###initial_city_coord: -106.62 35.12 Albuquerque
     ###initial_city_coord: -103.2 34.41 Clovis
     ###initial_city_coord: -107.87 37.29 Durango
     ###initial_city_coord: -96.77 32.79 Dallas
     ###initial_city_coord: -105.92 35.77 Tesuque
     ###initial_city_coord: -107.84 35.15 Grants
     ###initial_city_coord: -106.28 35.89 Los Alamos
     ###initial_city_coord: -106.76 32.34 Las Cruces
     ###initial_city_coord: -108.58 37.35 Cortez
     ###initial_city_coord: -108.74 35.52 Gallup
     ###initial_city_coord: -105.95 35.68 Santa Fe

The optimal route turns out to be:

     # final coordinates of cities (longitude and latitude)
     ###final_city_coord: -105.95 35.68 Santa Fe
     ###final_city_coord: -103.2 34.41 Clovis
     ###final_city_coord: -96.77 32.79 Dallas
     ###final_city_coord: -106.76 32.34 Las Cruces
     ###final_city_coord: -112.07 33.54 Phoenix
     ###final_city_coord: -108.74 35.52 Gallup
     ###final_city_coord: -108.58 37.35 Cortez
     ###final_city_coord: -107.87 37.29 Durango
     ###final_city_coord: -107.84 35.15 Grants
     ###final_city_coord: -106.62 35.12 Albuquerque
     ###final_city_coord: -106.28 35.89 Los Alamos
     ###final_city_coord: -105.92 35.77 Tesuque
     ###final_city_coord: -105.95 35.68 Santa Fe

[image src="siman-initial-route.png"]

Figure: Initial route for the 12 southwestern cities Flying Salesman
Problem.

[image src="siman-final-route.png"]

Figure: Final (optimal) route for the 12 southwestern cities Flying
Salesman Problem.

Here’s a plot of the cost function (energy) versus generation (point in
the calculation at which a new temperature is set) for this problem:

[image src="siman-12-cities.png"]

Figure: Example of a simulated annealing run for the 12 southwestern
cities Flying Salesman Problem.


File: gsl-ref.info,  Node: References and Further Reading<21>,  Prev: Examples<21>,  Up: Simulated Annealing

28.4 References and Further Reading
===================================

Further information is available in the following book,

   * `Modern Heuristic Techniques for Combinatorial Problems', Colin R.
     Reeves (ed.), McGraw-Hill, 1995 (ISBN 0-07-709239-2).


File: gsl-ref.info,  Node: Ordinary Differential Equations,  Next: Interpolation,  Prev: Simulated Annealing,  Up: Top

29 Ordinary Differential Equations
**********************************

This chapter describes functions for solving ordinary differential
equation (ODE) initial value problems.  The library provides a variety
of low-level methods, such as Runge-Kutta and Bulirsch-Stoer routines,
and higher-level components for adaptive step-size control.  The
components can be combined by the user to achieve the desired solution,
with full access to any intermediate steps.  A driver object can be used
as a high level wrapper for easy use of low level functions.

These functions are declared in the header file ‘gsl_odeiv2.h’.  This is
a new interface in version 1.15 and uses the prefix ‘gsl_odeiv2’ for all
functions.  It is recommended over the previous ‘gsl_odeiv’
implementation defined in ‘gsl_odeiv.h’ The old interface has been
retained under the original name for backwards compatibility.

* Menu:

* Defining the ODE System::
* Stepping Functions::
* Adaptive Step-size Control::
* Evolution::
* Driver::
* Examples: Examples<22>.
* References and Further Reading: References and Further Reading<22>.


File: gsl-ref.info,  Node: Defining the ODE System,  Next: Stepping Functions,  Up: Ordinary Differential Equations

29.1 Defining the ODE System
============================

The routines solve the general n-dimensional first-order system,

     dy_i(t)/dt = f_i(t, y_1(t), ..., y_n(t))

for i = 1, \dots, n.  The stepping functions rely on the vector of
derivatives f_i and the Jacobian matrix,

     J_{ij} = df_i(t,y(t)) / dy_j

A system of equations is defined using the *note gsl_odeiv2_system: 94f.
datatype.

 -- Type: gsl_odeiv2_system

     This data type defines a general ODE system with arbitrary
     parameters.

     ‘int (* function) (double t, const double y[], double dydt[], void
     * params)’

          This function should store the vector elements f_i(t,y,params)
          in the array ‘dydt’, for arguments (‘t’, ‘y’) and parameters
          ‘params’.

          The function should return ‘GSL_SUCCESS’ if the calculation
          was completed successfully.  Any other return value indicates
          an error.  A special return value ‘GSL_EBADFUNC’ causes
          ‘gsl_odeiv2’ routines to immediately stop and return.  If
          ‘function’ is modified (for example contents of ‘params’), the
          user must call an appropriate reset function (*note
          gsl_odeiv2_driver_reset(): 950, *note
          gsl_odeiv2_evolve_reset(): 951. or *note
          gsl_odeiv2_step_reset(): 952.) before continuing.  Use return
          values distinct from standard GSL error codes to distinguish
          your function as the source of the error.

     ‘int (* jacobian) (double t, const double y[], double * dfdy,
     double dfdt[], void * params)’

          This function should store the vector of derivative elements

               df_i(t,y,params)/dt

          in the array ‘dfdt’ and the Jacobian matrix J_{ij} in the
          array ‘dfdy’, regarded as a row-ordered matrix ‘J(i,j) =
          dfdy[i * dimension + j]’ where ‘dimension’ is the dimension of
          the system.

          Not all of the stepper algorithms of ‘gsl_odeiv2’ make use of
          the Jacobian matrix, so it may not be necessary to provide
          this function (the ‘jacobian’ element of the struct can be
          replaced by a null pointer for those algorithms).

          The function should return ‘GSL_SUCCESS’ if the calculation
          was completed successfully.  Any other return value indicates
          an error.  A special return value ‘GSL_EBADFUNC’ causes
          ‘gsl_odeiv2’ routines to immediately stop and return.  If
          ‘jacobian’ is modified (for example contents of ‘params’), the
          user must call an appropriate reset function (*note
          gsl_odeiv2_driver_reset(): 950, *note
          gsl_odeiv2_evolve_reset(): 951. or *note
          gsl_odeiv2_step_reset(): 952.) before continuing.  Use return
          values distinct from standard GSL error codes to distinguish
          your function as the source of the error.

     ‘size_t dimension’

          This is the dimension of the system of equations.

     ‘void * params’

          This is a pointer to the arbitrary parameters of the system.


File: gsl-ref.info,  Node: Stepping Functions,  Next: Adaptive Step-size Control,  Prev: Defining the ODE System,  Up: Ordinary Differential Equations

29.2 Stepping Functions
=======================

The lowest level components are the `stepping functions' which advance a
solution from time t to t+h for a fixed step-size h and estimate the
resulting local error.

 -- Type: gsl_odeiv2_step

     This contains internal parameters for a stepping function.

 -- Function: gsl_odeiv2_step * gsl_odeiv2_step_alloc (const
          gsl_odeiv2_step_type * T, size_t dim)

     This function returns a pointer to a newly allocated instance of a
     stepping function of type ‘T’ for a system of ‘dim’ dimensions.
     Please note that if you use a stepper method that requires access
     to a driver object, it is advisable to use a driver allocation
     method, which automatically allocates a stepper, too.

 -- Function: int gsl_odeiv2_step_reset (gsl_odeiv2_step * s)

     This function resets the stepping function ‘s’.  It should be used
     whenever the next use of ‘s’ will not be a continuation of a
     previous step.

 -- Function: void gsl_odeiv2_step_free (gsl_odeiv2_step * s)

     This function frees all the memory associated with the stepping
     function ‘s’.

 -- Function: const char * gsl_odeiv2_step_name (const gsl_odeiv2_step
          * s)

     This function returns a pointer to the name of the stepping
     function.  For example:

          printf ("step method is '%s'\n", gsl_odeiv2_step_name (s));

     would print something like ‘step method is 'rkf45'’.

 -- Function: unsigned int gsl_odeiv2_step_order (const gsl_odeiv2_step
          * s)

     This function returns the order of the stepping function on the
     previous step.  The order can vary if the stepping function itself
     is adaptive.

 -- Function: int gsl_odeiv2_step_set_driver (gsl_odeiv2_step * s, const
          gsl_odeiv2_driver * d)

     This function sets a pointer of the driver object ‘d’ for stepper
     ‘s’, to allow the stepper to access control (and evolve) object
     through the driver object.  This is a requirement for some
     steppers, to get the desired error level for internal iteration of
     stepper.  Allocation of a driver object calls this function
     automatically.

 -- Function: int gsl_odeiv2_step_apply (gsl_odeiv2_step * s, double t,
          double h, double y[], double yerr[], const double dydt_in[],
          double dydt_out[], const gsl_odeiv2_system * sys)

     This function applies the stepping function ‘s’ to the system of
     equations defined by ‘sys’, using the step-size ‘h’ to advance the
     system from time ‘t’ and state ‘y’ to time ‘t’ + ‘h’.  The new
     state of the system is stored in ‘y’ on output, with an estimate of
     the absolute error in each component stored in ‘yerr’.  If the
     argument ‘dydt_in’ is not null it should point an array containing
     the derivatives for the system at time ‘t’ on input.  This is
     optional as the derivatives will be computed internally if they are
     not provided, but allows the reuse of existing derivative
     information.  On output the new derivatives of the system at time
     ‘t’ + ‘h’ will be stored in ‘dydt_out’ if it is not null.

     The stepping function returns ‘GSL_FAILURE’ if it is unable to
     compute the requested step.  Also, if the user-supplied functions
     defined in the system ‘sys’ return a status other than
     ‘GSL_SUCCESS’ the step will be aborted.  In that case, the elements
     of ‘y’ will be restored to their pre-step values and the error code
     from the user-supplied function will be returned.  Failure may be
     due to a singularity in the system or too large step-size ‘h’.  In
     that case the step should be attempted again with a smaller
     step-size, e.g.  ‘h’ / 2.

     If the driver object is not appropriately set via *note
     gsl_odeiv2_step_set_driver(): 959. for those steppers that need it,
     the stepping function returns ‘GSL_EFAULT’.  If the user-supplied
     functions defined in the system ‘sys’ returns ‘GSL_EBADFUNC’, the
     function returns immediately with the same return code.  In this
     case the user must call *note gsl_odeiv2_step_reset(): 952. before
     calling this function again.

The following algorithms are available,

 -- Type: gsl_odeiv2_step_type

      -- Variable: gsl_odeiv2_step_rk2

          Explicit embedded Runge-Kutta (2, 3) method.

      -- Variable: gsl_odeiv2_step_rk4

          Explicit 4th order (classical) Runge-Kutta.  Error estimation
          is carried out by the step doubling method.  For more
          efficient estimate of the error, use the embedded methods
          described below.

      -- Variable: gsl_odeiv2_step_rkf45

          Explicit embedded Runge-Kutta-Fehlberg (4, 5) method.  This
          method is a good general-purpose integrator.

      -- Variable: gsl_odeiv2_step_rkck

          Explicit embedded Runge-Kutta Cash-Karp (4, 5) method.

      -- Variable: gsl_odeiv2_step_rk8pd

          Explicit embedded Runge-Kutta Prince-Dormand (8, 9) method.

      -- Variable: gsl_odeiv2_step_rk1imp

          Implicit Gaussian first order Runge-Kutta.  Also known as
          implicit Euler or backward Euler method.  Error estimation is
          carried out by the step doubling method.  This algorithm
          requires the Jacobian and access to the driver object via
          *note gsl_odeiv2_step_set_driver(): 959.

      -- Variable: gsl_odeiv2_step_rk2imp

          Implicit Gaussian second order Runge-Kutta.  Also known as
          implicit mid-point rule.  Error estimation is carried out by
          the step doubling method.  This stepper requires the Jacobian
          and access to the driver object via *note
          gsl_odeiv2_step_set_driver(): 959.

      -- Variable: gsl_odeiv2_step_rk4imp

          Implicit Gaussian 4th order Runge-Kutta.  Error estimation is
          carried out by the step doubling method.  This algorithm
          requires the Jacobian and access to the driver object via
          *note gsl_odeiv2_step_set_driver(): 959.

      -- Variable: gsl_odeiv2_step_bsimp

          Implicit Bulirsch-Stoer method of Bader and Deuflhard.  The
          method is generally suitable for stiff problems.  This stepper
          requires the Jacobian.

      -- Variable: gsl_odeiv2_step_msadams

          A variable-coefficient linear multistep Adams method in
          Nordsieck form.  This stepper uses explicit Adams-Bashforth
          (predictor) and implicit Adams-Moulton (corrector) methods in
          P(EC)^m functional iteration mode.  Method order varies
          dynamically between 1 and 12.  This stepper requires the
          access to the driver object via *note
          gsl_odeiv2_step_set_driver(): 959.

      -- Variable: gsl_odeiv2_step_msbdf

          A variable-coefficient linear multistep backward
          differentiation formula (BDF) method in Nordsieck form.  This
          stepper uses the explicit BDF formula as predictor and
          implicit BDF formula as corrector.  A modified Newton
          iteration method is used to solve the system of non-linear
          equations.  Method order varies dynamically between 1 and 5.
          The method is generally suitable for stiff problems.  This
          stepper requires the Jacobian and the access to the driver
          object via *note gsl_odeiv2_step_set_driver(): 959.


File: gsl-ref.info,  Node: Adaptive Step-size Control,  Next: Evolution,  Prev: Stepping Functions,  Up: Ordinary Differential Equations

29.3 Adaptive Step-size Control
===============================

The control function examines the proposed change to the solution
produced by a stepping function and attempts to determine the optimal
step-size for a user-specified level of error.

 -- Type: gsl_odeiv2_control

     This is a workspace for controlling step size.

 -- Type: gsl_odeiv2_control_type

     This specifies the control type.

 -- Function: gsl_odeiv2_control * gsl_odeiv2_control_standard_new
          (double eps_abs, double eps_rel, double a_y, double a_dydt)

     The standard control object is a four parameter heuristic based on
     absolute and relative errors ‘eps_abs’ and ‘eps_rel’, and scaling
     factors ‘a_y’ and ‘a_dydt’ for the system state y(t) and
     derivatives y'(t) respectively.

     The step-size adjustment procedure for this method begins by
     computing the desired error level D_i for each component,

          D_i = eps_abs + eps_rel * (a_y |y_i| + a_dydt h |y\prime_i|)

     and comparing it with the observed error E_i = |yerr_i|.  If the
     observed error ‘E’ exceeds the desired error level ‘D’ by more than
     10% for any component then the method reduces the step-size by an
     appropriate factor,

          h_{new} = h_{old} * S * (E/D)^{-1/q}

     where q is the consistency order of the method (e.g.  q=4 for 4(5)
     embedded RK), and S is a safety factor of 0.9.  The ratio E/D is
     taken to be the maximum of the ratios E_i/D_i.

     If the observed error E is less than 50% of the desired error level
     ‘D’ for the maximum ratio E_i/D_i then the algorithm takes the
     opportunity to increase the step-size to bring the error in line
     with the desired level,

          h_{new} = h_{old} * S * (E/D)^{-1/(q+1)}

     This encompasses all the standard error scaling methods.  To avoid
     uncontrolled changes in the stepsize, the overall scaling factor is
     limited to the range 1/5 to 5.

 -- Function: gsl_odeiv2_control * gsl_odeiv2_control_y_new
          (double eps_abs, double eps_rel)

     This function creates a new control object which will keep the
     local error on each step within an absolute error of ‘eps_abs’ and
     relative error of ‘eps_rel’ with respect to the solution y_i(t).
     This is equivalent to the standard control object with ‘a_y’ = 1
     and ‘a_dydt’ = 0.

 -- Function: gsl_odeiv2_control * gsl_odeiv2_control_yp_new
          (double eps_abs, double eps_rel)

     This function creates a new control object which will keep the
     local error on each step within an absolute error of ‘eps_abs’ and
     relative error of ‘eps_rel’ with respect to the derivatives of the
     solution y'_i(t).  This is equivalent to the standard control
     object with ‘a_y’ = 0 and ‘a_dydt’ = 1.

 -- Function: gsl_odeiv2_control * gsl_odeiv2_control_scaled_new
          (double eps_abs, double eps_rel, double a_y, double a_dydt,
          const double scale_abs[], size_t dim)

     This function creates a new control object which uses the same
     algorithm as *note gsl_odeiv2_control_standard_new(): 96a. but with
     an absolute error which is scaled for each component by the array
     ‘scale_abs’.  The formula for D_i for this control object is,

          D_i = eps_abs * s_i + eps_rel * (a_y |y_i| + a_dydt h |y\prime_i|)

     where s_i is the i-th component of the array ‘scale_abs’.  The same
     error control heuristic is used by the Matlab ODE suite.

 -- Function: gsl_odeiv2_control * gsl_odeiv2_control_alloc (const
          gsl_odeiv2_control_type * T)

     This function returns a pointer to a newly allocated instance of a
     control function of type ‘T’.  This function is only needed for
     defining new types of control functions.  For most purposes the
     standard control functions described above should be sufficient.

 -- Function: int gsl_odeiv2_control_init (gsl_odeiv2_control * c,
          double eps_abs, double eps_rel, double a_y, double a_dydt)

     This function initializes the control function ‘c’ with the
     parameters ‘eps_abs’ (absolute error), ‘eps_rel’ (relative error),
     ‘a_y’ (scaling factor for y) and ‘a_dydt’ (scaling factor for
     derivatives).

 -- Function: void gsl_odeiv2_control_free (gsl_odeiv2_control * c)

     This function frees all the memory associated with the control
     function ‘c’.

 -- Function: int gsl_odeiv2_control_hadjust (gsl_odeiv2_control * c,
          gsl_odeiv2_step * s, const double y[], const double yerr[],
          const double dydt[], double * h)

     This function adjusts the step-size ‘h’ using the control function
     ‘c’, and the current values of ‘y’, ‘yerr’ and ‘dydt’.  The
     stepping function ‘step’ is also needed to determine the order of
     the method.  If the error in the y-values ‘yerr’ is found to be too
     large then the step-size ‘h’ is reduced and the function returns
     ‘GSL_ODEIV_HADJ_DEC’.  If the error is sufficiently small then ‘h’
     may be increased and ‘GSL_ODEIV_HADJ_INC’ is returned.  The
     function returns ‘GSL_ODEIV_HADJ_NIL’ if the step-size is
     unchanged.  The goal of the function is to estimate the largest
     step-size which satisfies the user-specified accuracy requirements
     for the current point.

 -- Function: const char * gsl_odeiv2_control_name (const
          gsl_odeiv2_control * c)

     This function returns a pointer to the name of the control
     function.  For example:

          printf ("control method is '%s'\n", gsl_odeiv2_control_name (c));

     would print something like ‘control method is 'standard'’

 -- Function: int gsl_odeiv2_control_errlevel (gsl_odeiv2_control * c,
          const double y, const double dydt, const double h, const
          size_t ind, double * errlev)

     This function calculates the desired error level of the ‘ind’-th
     component to ‘errlev’.  It requires the value (‘y’) and value of
     the derivative (‘dydt’) of the component, and the current step size
     ‘h’.

 -- Function: int gsl_odeiv2_control_set_driver (gsl_odeiv2_control * c,
          const gsl_odeiv2_driver * d)

     This function sets a pointer of the driver object ‘d’ for control
     object ‘c’.


File: gsl-ref.info,  Node: Evolution,  Next: Driver,  Prev: Adaptive Step-size Control,  Up: Ordinary Differential Equations

29.4 Evolution
==============

The evolution function combines the results of a stepping function and
control function to reliably advance the solution forward one step using
an acceptable step-size.

 -- Type: gsl_odeiv2_evolve

     This workspace contains parameters for controlling the evolution
     function

 -- Function: gsl_odeiv2_evolve * gsl_odeiv2_evolve_alloc (size_t dim)

     This function returns a pointer to a newly allocated instance of an
     evolution function for a system of ‘dim’ dimensions.

 -- Function: int gsl_odeiv2_evolve_apply (gsl_odeiv2_evolve * e,
          gsl_odeiv2_control * con, gsl_odeiv2_step * step, const
          gsl_odeiv2_system * sys, double * t, double t1, double * h,
          double y[])

     This function advances the system (‘e’, ‘sys’) from time ‘t’ and
     position ‘y’ using the stepping function ‘step’.  The new time and
     position are stored in ‘t’ and ‘y’ on output.

     The initial step-size is taken as ‘h’.  The control function ‘con’
     is applied to check whether the local error estimated by the
     stepping function ‘step’ using step-size ‘h’ exceeds the required
     error tolerance.  If the error is too high, the step is retried by
     calling ‘step’ with a decreased step-size.  This process is
     continued until an acceptable step-size is found.  An estimate of
     the local error for the step can be obtained from the components of
     the array ‘e->yerr[]’.

     If the user-supplied functions defined in the system ‘sys’ returns
     ‘GSL_EBADFUNC’, the function returns immediately with the same
     return code.  In this case the user must call *note
     gsl_odeiv2_step_reset(): 952. and *note gsl_odeiv2_evolve_reset():
     951. before calling this function again.

     Otherwise, if the user-supplied functions defined in the system
     ‘sys’ or the stepping function ‘step’ return a status other than
     ‘GSL_SUCCESS’, the step is retried with a decreased step-size.  If
     the step-size decreases below machine precision, a status of
     ‘GSL_FAILURE’ is returned if the user functions returned
     ‘GSL_SUCCESS’.  Otherwise the value returned by user function is
     returned.  If no acceptable step can be made, ‘t’ and ‘y’ will be
     restored to their pre-step values and ‘h’ contains the final
     attempted step-size.

     If the step is successful the function returns a suggested
     step-size for the next step in ‘h’.  The maximum time ‘t1’ is
     guaranteed not to be exceeded by the time-step.  On the final
     time-step the value of ‘t’ will be set to ‘t1’ exactly.

 -- Function: int gsl_odeiv2_evolve_apply_fixed_step (gsl_odeiv2_evolve
          * e, gsl_odeiv2_control * con, gsl_odeiv2_step * step, const
          gsl_odeiv2_system * sys, double * t, const double h,
          double y[])

     This function advances the ODE-system (‘e’, ‘sys’, ‘con’) from time
     ‘t’ and position ‘y’ using the stepping function ‘step’ by a
     specified step size ‘h’.  If the local error estimated by the
     stepping function exceeds the desired error level, the step is not
     taken and the function returns ‘GSL_FAILURE’.  Otherwise the value
     returned by user function is returned.

 -- Function: int gsl_odeiv2_evolve_reset (gsl_odeiv2_evolve * e)

     This function resets the evolution function ‘e’.  It should be used
     whenever the next use of ‘e’ will not be a continuation of a
     previous step.

 -- Function: void gsl_odeiv2_evolve_free (gsl_odeiv2_evolve * e)

     This function frees all the memory associated with the evolution
     function ‘e’.

 -- Function: int gsl_odeiv2_evolve_set_driver (gsl_odeiv2_evolve * e,
          const gsl_odeiv2_driver * d)

     This function sets a pointer of the driver object ‘d’ for evolve
     object ‘e’.

If a system has discontinuous changes in the derivatives at known
points, it is advisable to evolve the system between each discontinuity
in sequence.  For example, if a step-change in an external driving force
occurs at times t_a, t_b and t_c then evolution should be carried out
over the ranges (t_0,t_a), (t_a,t_b), (t_b,t_c), and (t_c,t_1)
separately and not directly over the range (t_0,t_1).


File: gsl-ref.info,  Node: Driver,  Next: Examples<22>,  Prev: Evolution,  Up: Ordinary Differential Equations

29.5 Driver
===========

The driver object is a high level wrapper that combines the evolution,
control and stepper objects for easy use.

 -- Function: gsl_odeiv2_driver * gsl_odeiv2_driver_alloc_y_new (const
          gsl_odeiv2_system * sys, const gsl_odeiv2_step_type * T, const
          double hstart, const double epsabs, const double epsrel)
 -- Function: gsl_odeiv2_driver * gsl_odeiv2_driver_alloc_yp_new (const
          gsl_odeiv2_system * sys, const gsl_odeiv2_step_type * T, const
          double hstart, const double epsabs, const double epsrel)
 -- Function: gsl_odeiv2_driver * gsl_odeiv2_driver_alloc_standard_new
          (const gsl_odeiv2_system * sys, const gsl_odeiv2_step_type
          * T, const double hstart, const double epsabs, const
          double epsrel, const double a_y, const double a_dydt)
 -- Function: gsl_odeiv2_driver * gsl_odeiv2_driver_alloc_scaled_new
          (const gsl_odeiv2_system * sys, const gsl_odeiv2_step_type
          * T, const double hstart, const double epsabs, const
          double epsrel, const double a_y, const double a_dydt, const
          double scale_abs[])

     These functions return a pointer to a newly allocated instance of a
     driver object.  The functions automatically allocate and initialise
     the evolve, control and stepper objects for ODE system ‘sys’ using
     stepper type ‘T’.  The initial step size is given in ‘hstart’.  The
     rest of the arguments follow the syntax and semantics of the
     control functions with same name (‘gsl_odeiv2_control_*_new’).

 -- Function: int gsl_odeiv2_driver_set_hmin (gsl_odeiv2_driver * d,
          const double hmin)

     The function sets a minimum for allowed step size ‘hmin’ for driver
     ‘d’.  Default value is 0.

 -- Function: int gsl_odeiv2_driver_set_hmax (gsl_odeiv2_driver * d,
          const double hmax)

     The function sets a maximum for allowed step size ‘hmax’ for driver
     ‘d’.  Default value is ‘GSL_DBL_MAX’.

 -- Function: int gsl_odeiv2_driver_set_nmax (gsl_odeiv2_driver * d,
          const unsigned long int nmax)

     The function sets a maximum for allowed number of steps ‘nmax’ for
     driver ‘d’.  Default value of 0 sets no limit for steps.

 -- Function: int gsl_odeiv2_driver_apply (gsl_odeiv2_driver * d, double
          * t, const double t1, double y[])

     This function evolves the driver system ‘d’ from ‘t’ to ‘t1’.
     Initially vector ‘y’ should contain the values of dependent
     variables at point ‘t’.  If the function is unable to complete the
     calculation, an error code from *note gsl_odeiv2_evolve_apply():
     978. is returned, and ‘t’ and ‘y’ contain the values from last
     successful step.

     If maximum number of steps is reached, a value of ‘GSL_EMAXITER’ is
     returned.  If the step size drops below minimum value, the function
     returns with ‘GSL_ENOPROG’.  If the user-supplied functions defined
     in the system ‘sys’ returns ‘GSL_EBADFUNC’, the function returns
     immediately with the same return code.  In this case the user must
     call *note gsl_odeiv2_driver_reset(): 950. before calling this
     function again.

 -- Function: int gsl_odeiv2_driver_apply_fixed_step (gsl_odeiv2_driver
          * d, double * t, const double h, const unsigned long int n,
          double y[])

     This function evolves the driver system ‘d’ from ‘t’ with ‘n’ steps
     of size ‘h’.  If the function is unable to complete the
     calculation, an error code from *note
     gsl_odeiv2_evolve_apply_fixed_step(): 979. is returned, and ‘t’ and
     ‘y’ contain the values from last successful step.

 -- Function: int gsl_odeiv2_driver_reset (gsl_odeiv2_driver * d)

     This function resets the evolution and stepper objects.

 -- Function: int gsl_odeiv2_driver_reset_hstart (gsl_odeiv2_driver * d,
          const double hstart)

     The routine resets the evolution and stepper objects and sets new
     initial step size to ‘hstart’.  This function can be used e.g.  to
     change the direction of integration.

 -- Function: int gsl_odeiv2_driver_free (gsl_odeiv2_driver * d)

     This function frees the driver object, and the related evolution,
     stepper and control objects.


File: gsl-ref.info,  Node: Examples<22>,  Next: References and Further Reading<22>,  Prev: Driver,  Up: Ordinary Differential Equations

29.6 Examples
=============

The following program solves the second-order nonlinear Van der Pol
oscillator equation,

     u''(t) + \mu u'(t) (u(t)^2 - 1) + u(t) = 0

This can be converted into a first order system suitable for use with
the routines described in this chapter by introducing a separate
variable for the velocity, v = u'(t),

     u' = v
     v' = -u + \mu v (1-u^2)

The program begins by defining functions for these derivatives and their
Jacobian.  The main function uses driver level functions to solve the
problem.  The program evolves the solution from (u, v) = (1, 0) at t = 0
to t = 100.  The step-size h is automatically adjusted by the controller
to maintain an absolute accuracy of 10^{-6} in the function values (u,
v).  The loop in the example prints the solution at the points t_i = 1,
2, \dots, 100.

     #include <stdio.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_odeiv2.h>

     int
     func (double t, const double y[], double f[],
           void *params)
     {
       (void)(t); /* avoid unused parameter warning */
       double mu = *(double *)params;
       f[0] = y[1];
       f[1] = -y[0] - mu*y[1]*(y[0]*y[0] - 1);
       return GSL_SUCCESS;
     }

     int
     jac (double t, const double y[], double *dfdy,
          double dfdt[], void *params)
     {
       (void)(t); /* avoid unused parameter warning */
       double mu = *(double *)params;
       gsl_matrix_view dfdy_mat
         = gsl_matrix_view_array (dfdy, 2, 2);
       gsl_matrix * m = &dfdy_mat.matrix;
       gsl_matrix_set (m, 0, 0, 0.0);
       gsl_matrix_set (m, 0, 1, 1.0);
       gsl_matrix_set (m, 1, 0, -2.0*mu*y[0]*y[1] - 1.0);
       gsl_matrix_set (m, 1, 1, -mu*(y[0]*y[0] - 1.0));
       dfdt[0] = 0.0;
       dfdt[1] = 0.0;
       return GSL_SUCCESS;
     }

     int
     main (void)
     {
       double mu = 10;
       gsl_odeiv2_system sys = {func, jac, 2, &mu};

       gsl_odeiv2_driver * d =
         gsl_odeiv2_driver_alloc_y_new (&sys, gsl_odeiv2_step_rk8pd,
                                       1e-6, 1e-6, 0.0);
       int i;
       double t = 0.0, t1 = 100.0;
       double y[2] = { 1.0, 0.0 };

       for (i = 1; i <= 100; i++)
         {
           double ti = i * t1 / 100.0;
           int status = gsl_odeiv2_driver_apply (d, &t, ti, y);

           if (status != GSL_SUCCESS)
             {
               printf ("error, return value=%d\n", status);
               break;
             }

           printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
         }

       gsl_odeiv2_driver_free (d);
       return 0;
     }

The user can work with the lower level functions directly, as in the
following example.  In this case an intermediate result is printed after
each successful step instead of equidistant time points.

     int
     main (void)
     {
       const gsl_odeiv2_step_type * T
         = gsl_odeiv2_step_rk8pd;

       gsl_odeiv2_step * s
         = gsl_odeiv2_step_alloc (T, 2);
       gsl_odeiv2_control * c
         = gsl_odeiv2_control_y_new (1e-6, 0.0);
       gsl_odeiv2_evolve * e
         = gsl_odeiv2_evolve_alloc (2);

       double mu = 10;
       gsl_odeiv2_system sys = {func, jac, 2, &mu};

       double t = 0.0, t1 = 100.0;
       double h = 1e-6;
       double y[2] = { 1.0, 0.0 };

       while (t < t1)
         {
           int status = gsl_odeiv2_evolve_apply (e, c, s,
                                                &sys,
                                                &t, t1,
                                                &h, y);

           if (status != GSL_SUCCESS)
               break;

           printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
         }

       gsl_odeiv2_evolve_free (e);
       gsl_odeiv2_control_free (c);
       gsl_odeiv2_step_free (s);
       return 0;
     }

For functions with multiple parameters, the appropriate information can
be passed in through the ‘params’ argument in *note gsl_odeiv2_system:
94f. definition (‘mu’ in this example) by using a pointer to a struct.

[image src="ode-vdp.png"]

Figure: Numerical solution of the Van der Pol oscillator equation using
Prince-Dormand 8th order Runge-Kutta.

It is also possible to work with a non-adaptive integrator, using only
the stepping function itself, *note
gsl_odeiv2_driver_apply_fixed_step(): 985. or *note
gsl_odeiv2_evolve_apply_fixed_step(): 979.  The following program uses
the driver level function, with fourth-order Runge-Kutta stepping
function with a fixed stepsize of 0.001.

     int
     main (void)
     {
       double mu = 10;
       gsl_odeiv2_system sys = { func, jac, 2, &mu };

       gsl_odeiv2_driver *d =
         gsl_odeiv2_driver_alloc_y_new (&sys, gsl_odeiv2_step_rk4,
                                        1e-3, 1e-8, 1e-8);

       double t = 0.0;
       double y[2] = { 1.0, 0.0 };
       int i, s;

       for (i = 0; i < 100; i++)
         {
           s = gsl_odeiv2_driver_apply_fixed_step (d, &t, 1e-3, 1000, y);

           if (s != GSL_SUCCESS)
             {
               printf ("error: driver returned %d\n", s);
               break;
             }

           printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
         }

       gsl_odeiv2_driver_free (d);
       return s;
     }


File: gsl-ref.info,  Node: References and Further Reading<22>,  Prev: Examples<22>,  Up: Ordinary Differential Equations

29.7 References and Further Reading
===================================

   * Ascher, U.M., Petzold, L.R., `Computer Methods for Ordinary
     Differential and Differential-Algebraic Equations', SIAM,
     Philadelphia, 1998.

   * Hairer, E., Norsett, S. P., Wanner, G., `Solving Ordinary
     Differential Equations I: Nonstiff Problems', Springer, Berlin,
     1993.

   * Hairer, E., Wanner, G., `Solving Ordinary Differential Equations
     II: Stiff and Differential-Algebraic Problems', Springer, Berlin,
     1996.

Many of the basic Runge-Kutta formulas can be found in the Handbook of
Mathematical Functions,

   * Abramowitz & Stegun (eds.), `Handbook of Mathematical Functions',
     Section 25.5.

The implicit Bulirsch-Stoer algorithm ‘bsimp’ is described in the
following paper,

   * G. Bader and P. Deuflhard, “A Semi-Implicit Mid-Point Rule for
     Stiff Systems of Ordinary Differential Equations.”, Numer.: Math.:
     41, 373–398, 1983.

The Adams and BDF multistep methods ‘msadams’ and ‘msbdf’ are based on
the following articles,

   * G. D. Byrne and A. C. Hindmarsh, “A Polyalgorithm for the Numerical
     Solution of Ordinary Differential Equations.”, ACM Trans.  Math.
     Software, 1, 71–96, 1975.

   * P. N. Brown, G. D. Byrne and A. C. Hindmarsh, “VODE: A
     Variable-coefficient ODE Solver.”, SIAM J. Sci.  Stat.  Comput.
     10, 1038–1051, 1989.

   * A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban, D.
     E. Shumaker and C. S. Woodward, “SUNDIALS: Suite of Nonlinear and
     Differential/Algebraic Equation Solvers.”, ACM Trans.  Math.
     Software 31, 363–396, 2005.


File: gsl-ref.info,  Node: Interpolation,  Next: Numerical Differentiation,  Prev: Ordinary Differential Equations,  Up: Top

30 Interpolation
****************

This chapter describes functions for performing interpolation.  The
library provides a variety of interpolation methods, including Cubic,
Akima, and Steffen splines.  The interpolation types are
interchangeable, allowing different methods to be used without
recompiling.  Interpolations can be defined for both normal and periodic
boundary conditions.  Additional functions are available for computing
derivatives and integrals of interpolating functions.  Routines are
provided for interpolating both one and two dimensional datasets.

These interpolation methods produce curves that pass through each
datapoint.  To interpolate noisy data with a smoothing curve see *note
Basis Splines: 98d.

The functions described in this section are declared in the header files
‘gsl_interp.h’ and ‘gsl_spline.h’.

* Menu:

* Introduction to 1D Interpolation::
* 1D Interpolation Functions::
* 1D Interpolation Types::
* 1D Index Look-up and Acceleration::
* 1D Evaluation of Interpolating Functions::
* 1D Higher-level Interface::
* 1D Interpolation Example Programs::
* Introduction to 2D Interpolation::
* 2D Interpolation Functions::
* 2D Interpolation Grids::
* 2D Interpolation Types::
* 2D Evaluation of Interpolating Functions::
* 2D Higher-level Interface::
* 2D Interpolation Example programs::
* References and Further Reading: References and Further Reading<23>.


File: gsl-ref.info,  Node: Introduction to 1D Interpolation,  Next: 1D Interpolation Functions,  Up: Interpolation

30.1 Introduction to 1D Interpolation
=====================================

Given a set of data points (x_1, y_1) \dots (x_n, y_n) the routines
described in this section compute a continuous interpolating function
y(x) such that y(x_i) = y_i.  The interpolation is piecewise smooth, and
its behavior at the end-points is determined by the type of
interpolation used.


File: gsl-ref.info,  Node: 1D Interpolation Functions,  Next: 1D Interpolation Types,  Prev: Introduction to 1D Interpolation,  Up: Interpolation

30.2 1D Interpolation Functions
===============================

The interpolation function for a given dataset is stored in a *note
gsl_interp: 990. object.  These are created by the following functions.

 -- Type: gsl_interp

     Workspace for 1D interpolation

 -- Function: gsl_interp * gsl_interp_alloc (const gsl_interp_type * T,
          size_t size)

     This function returns a pointer to a newly allocated interpolation
     object of type ‘T’ for ‘size’ data-points.

 -- Function: int gsl_interp_init (gsl_interp * interp, const
          double xa[], const double ya[], size_t size)

     This function initializes the interpolation object ‘interp’ for the
     data (‘xa’, ‘ya’) where ‘xa’ and ‘ya’ are arrays of size ‘size’.
     The interpolation object (*note gsl_interp: 990.) does not save the
     data arrays ‘xa’ and ‘ya’ and only stores the static state computed
     from the data.  The ‘xa’ data array is always assumed to be
     strictly ordered, with increasing x values; the behavior for other
     arrangements is not defined.

 -- Function: void gsl_interp_free (gsl_interp * interp)

     This function frees the interpolation object ‘interp’.


File: gsl-ref.info,  Node: 1D Interpolation Types,  Next: 1D Index Look-up and Acceleration,  Prev: 1D Interpolation Functions,  Up: Interpolation

30.3 1D Interpolation Types
===========================

The interpolation library provides the following interpolation types:

 -- Type: gsl_interp_type

      -- Variable: gsl_interp_linear

          Linear interpolation.  This interpolation method does not
          require any additional memory.

      -- Variable: gsl_interp_polynomial

          Polynomial interpolation.  This method should only be used for
          interpolating small numbers of points because polynomial
          interpolation introduces large oscillations, even for
          well-behaved datasets.  The number of terms in the
          interpolating polynomial is equal to the number of points.

      -- Variable: gsl_interp_cspline

          Cubic spline with natural boundary conditions.  The resulting
          curve is piecewise cubic on each interval, with matching first
          and second derivatives at the supplied data-points.  The
          second derivative is chosen to be zero at the first point and
          last point.

      -- Variable: gsl_interp_cspline_periodic

          Cubic spline with periodic boundary conditions.  The resulting
          curve is piecewise cubic on each interval, with matching first
          and second derivatives at the supplied data-points.  The
          derivatives at the first and last points are also matched.
          Note that the last point in the data must have the same
          y-value as the first point, otherwise the resulting periodic
          interpolation will have a discontinuity at the boundary.

      -- Variable: gsl_interp_akima

          Non-rounded Akima spline with natural boundary conditions.
          This method uses the non-rounded corner algorithm of Wodicka.

      -- Variable: gsl_interp_akima_periodic

          Non-rounded Akima spline with periodic boundary conditions.
          This method uses the non-rounded corner algorithm of Wodicka.

      -- Variable: gsl_interp_steffen

          Steffen’s method guarantees the monotonicity of the
          interpolating function between the given data points.
          Therefore, minima and maxima can only occur exactly at the
          data points, and there can never be spurious oscillations
          between data points.  The interpolated function is piecewise
          cubic in each interval.  The resulting curve and its first
          derivative are guaranteed to be continuous, but the second
          derivative may be discontinuous.

The following related functions are available:

 -- Function: const char * gsl_interp_name (const gsl_interp * interp)

     This function returns the name of the interpolation type used by
     ‘interp’.  For example:

          printf ("interp uses '%s' interpolation.\n", gsl_interp_name (interp));

     would print something like:

          interp uses 'cspline' interpolation.

 -- Function: unsigned int gsl_interp_min_size (const gsl_interp
          * interp)
 -- Function: unsigned int gsl_interp_type_min_size (const
          gsl_interp_type * T)

     These functions return the minimum number of points required by the
     interpolation object ‘interp’ or interpolation type ‘T’.  For
     example, Akima spline interpolation requires a minimum of 5 points.


File: gsl-ref.info,  Node: 1D Index Look-up and Acceleration,  Next: 1D Evaluation of Interpolating Functions,  Prev: 1D Interpolation Types,  Up: Interpolation

30.4 1D Index Look-up and Acceleration
======================================

The state of searches can be stored in a *note gsl_interp_accel: 9a1.
object, which is a kind of iterator for interpolation lookups.

 -- Type: gsl_interp_accel

     This workspace stores state variables for interpolation lookups.
     It caches the previous value of an index lookup.  When the
     subsequent interpolation point falls in the same interval its index
     value can be returned immediately.

 -- Function: size_t gsl_interp_bsearch (const double x_array[],
          double x, size_t index_lo, size_t index_hi)

     This function returns the index i of the array ‘x_array’ such that
     ‘x_array[i] <= x < x_array[i+1]’.  The index is searched for in the
     range [‘index_lo’, ‘index_hi’].  An inline version of this function
     is used when ‘HAVE_INLINE’ is defined.

 -- Function: gsl_interp_accel * gsl_interp_accel_alloc (void)

     This function returns a pointer to an accelerator object, which is
     a kind of iterator for interpolation lookups.  It tracks the state
     of lookups, thus allowing for application of various acceleration
     strategies.

 -- Function: size_t gsl_interp_accel_find (gsl_interp_accel * a, const
          double x_array[], size_t size, double x)

     This function performs a lookup action on the data array ‘x_array’
     of size ‘size’, using the given accelerator ‘a’.  This is how
     lookups are performed during evaluation of an interpolation.  The
     function returns an index i such that ‘x_array[i] <= x <
     x_array[i+1]’.  An inline version of this function is used when
     ‘HAVE_INLINE’ is defined.

 -- Function: int gsl_interp_accel_reset (gsl_interp_accel * acc);

     This function reinitializes the accelerator object ‘acc’.  It
     should be used when the cached information is no longer
     applicable—for example, when switching to a new dataset.

 -- Function: void gsl_interp_accel_free (gsl_interp_accel* acc)

     This function frees the accelerator object ‘acc’.


File: gsl-ref.info,  Node: 1D Evaluation of Interpolating Functions,  Next: 1D Higher-level Interface,  Prev: 1D Index Look-up and Acceleration,  Up: Interpolation

30.5 1D Evaluation of Interpolating Functions
=============================================

 -- Function: double gsl_interp_eval (const gsl_interp * interp, const
          double xa[], const double ya[], double x, gsl_interp_accel
          * acc)
 -- Function: int gsl_interp_eval_e (const gsl_interp * interp, const
          double xa[], const double ya[], double x, gsl_interp_accel
          * acc, double * y)

     These functions return the interpolated value of ‘y’ for a given
     point ‘x’, using the interpolation object ‘interp’, data arrays
     ‘xa’ and ‘ya’ and the accelerator ‘acc’.  When ‘x’ is outside the
     range of ‘xa’, the error code *note GSL_EDOM: 28. is returned with
     a value of *note GSL_NAN: 3c. for ‘y’.

 -- Function: double gsl_interp_eval_deriv (const gsl_interp * interp,
          const double xa[], const double ya[], double x,
          gsl_interp_accel * acc)
 -- Function: int gsl_interp_eval_deriv_e (const gsl_interp * interp,
          const double xa[], const double ya[], double x,
          gsl_interp_accel * acc, double * d)

     These functions return the derivative ‘d’ of an interpolated
     function for a given point ‘x’, using the interpolation object
     ‘interp’, data arrays ‘xa’ and ‘ya’ and the accelerator ‘acc’.

 -- Function: double gsl_interp_eval_deriv2 (const gsl_interp * interp,
          const double xa[], const double ya[], double x,
          gsl_interp_accel * acc)
 -- Function: int gsl_interp_eval_deriv2_e (const gsl_interp * interp,
          const double xa[], const double ya[], double x,
          gsl_interp_accel * acc, double * d2)

     These functions return the second derivative ‘d2’ of an
     interpolated function for a given point ‘x’, using the
     interpolation object ‘interp’, data arrays ‘xa’ and ‘ya’ and the
     accelerator ‘acc’.

 -- Function: double gsl_interp_eval_integ (const gsl_interp * interp,
          const double xa[], const double ya[], double a, double b,
          gsl_interp_accel * acc)
 -- Function: int gsl_interp_eval_integ_e (const gsl_interp * interp,
          const double xa[], const double ya[], double a, double b,
          gsl_interp_accel * acc, double * result)

     These functions return the numerical integral ‘result’ of an
     interpolated function over the range [‘a’, ‘b’], using the
     interpolation object ‘interp’, data arrays ‘xa’ and ‘ya’ and the
     accelerator ‘acc’.


File: gsl-ref.info,  Node: 1D Higher-level Interface,  Next: 1D Interpolation Example Programs,  Prev: 1D Evaluation of Interpolating Functions,  Up: Interpolation

30.6 1D Higher-level Interface
==============================

The functions described in the previous sections required the user to
supply pointers to the x and y arrays on each call.  The following
functions are equivalent to the corresponding *note gsl_interp: 990.
functions but maintain a copy of this data in the *note gsl_spline: 9b0.
object.  This removes the need to pass both ‘xa’ and ‘ya’ as arguments
on each evaluation.  These functions are defined in the header file
‘gsl_spline.h’.

 -- Type: gsl_spline

     This workspace provides a higher level interface for the *note
     gsl_interp: 990. object

 -- Function: gsl_spline * gsl_spline_alloc (const gsl_interp_type * T,
          size_t size)

 -- Function: int gsl_spline_init (gsl_spline * spline, const
          double xa[], const double ya[], size_t size)

 -- Function: void gsl_spline_free (gsl_spline * spline)

 -- Function: const char * gsl_spline_name (const gsl_spline * spline)

 -- Function: unsigned int gsl_spline_min_size (const gsl_spline
          * spline)

 -- Function: double gsl_spline_eval (const gsl_spline * spline,
          double x, gsl_interp_accel * acc)
 -- Function: int gsl_spline_eval_e (const gsl_spline * spline,
          double x, gsl_interp_accel * acc, double * y)

 -- Function: double gsl_spline_eval_deriv (const gsl_spline * spline,
          double x, gsl_interp_accel * acc)
 -- Function: int gsl_spline_eval_deriv_e (const gsl_spline * spline,
          double x, gsl_interp_accel * acc, double * d)

 -- Function: double gsl_spline_eval_deriv2 (const gsl_spline * spline,
          double x, gsl_interp_accel * acc)
 -- Function: int gsl_spline_eval_deriv2_e (const gsl_spline * spline,
          double x, gsl_interp_accel * acc, double * d2)

 -- Function: double gsl_spline_eval_integ (const gsl_spline * spline,
          double a, double b, gsl_interp_accel * acc)
 -- Function: int gsl_spline_eval_integ_e (const gsl_spline * spline,
          double a, double b, gsl_interp_accel * acc, double * result)


File: gsl-ref.info,  Node: 1D Interpolation Example Programs,  Next: Introduction to 2D Interpolation,  Prev: 1D Higher-level Interface,  Up: Interpolation

30.7 1D Interpolation Example Programs
======================================

The following program demonstrates the use of the interpolation and
spline functions.  It computes a cubic spline interpolation of the
10-point dataset (x_i, y_i) where x_i = i + \sin(i)/2 and y_i = i +
\cos(i^2) for i = 0 \dots 9.

     #include <stdlib.h>
     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_spline.h>

     int
     main (void)
     {
       int i;
       double xi, yi, x[10], y[10];

       printf ("#m=0,S=17\n");

       for (i = 0; i < 10; i++)
         {
           x[i] = i + 0.5 * sin (i);
           y[i] = i + cos (i * i);
           printf ("%g %g\n", x[i], y[i]);
         }

       printf ("#m=1,S=0\n");

       {
         gsl_interp_accel *acc
           = gsl_interp_accel_alloc ();
         gsl_spline *spline
           = gsl_spline_alloc (gsl_interp_cspline, 10);

         gsl_spline_init (spline, x, y, 10);

         for (xi = x[0]; xi < x[9]; xi += 0.01)
           {
             yi = gsl_spline_eval (spline, xi, acc);
             printf ("%g %g\n", xi, yi);
           }
         gsl_spline_free (spline);
         gsl_interp_accel_free (acc);
       }
       return 0;
     }

The output is designed to be used with the GNU plotutils ‘graph’
program:

     $ ./a.out > interp.dat
     $ graph -T ps < interp.dat > interp.ps

[image src="interp.png"]

Figure: Cubic spline interpolation

Fig.  %s shows a smooth interpolation of the original points.  The
interpolation method can be changed simply by varying the first argument
of *note gsl_spline_alloc(): 9b1.

The next program demonstrates a periodic cubic spline with 4 data
points.  Note that the first and last points must be supplied with the
same y-value for a periodic spline.

     #include <stdlib.h>
     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_spline.h>

     int
     main (void)
     {
       int N = 4;
       double x[4] = {0.00, 0.10,  0.27,  0.30};
       double y[4] = {0.15, 0.70, -0.10,  0.15};
                  /* Note: y[0] == y[3] for periodic data */

       gsl_interp_accel *acc = gsl_interp_accel_alloc ();
       const gsl_interp_type *t = gsl_interp_cspline_periodic;
       gsl_spline *spline = gsl_spline_alloc (t, N);

       int i; double xi, yi;

       printf ("#m=0,S=5\n");
       for (i = 0; i < N; i++)
         {
           printf ("%g %g\n", x[i], y[i]);
         }

       printf ("#m=1,S=0\n");
       gsl_spline_init (spline, x, y, N);

       for (i = 0; i <= 100; i++)
         {
           xi = (1 - i / 100.0) * x[0] + (i / 100.0) * x[N-1];
           yi = gsl_spline_eval (spline, xi, acc);
           printf ("%g %g\n", xi, yi);
         }

       gsl_spline_free (spline);
       gsl_interp_accel_free (acc);
       return 0;
     }

The output can be plotted with GNU ‘graph’:

     $ ./a.out > interp.dat
     $ graph -T ps < interp.dat > interp.ps

[image src="interpp.png"]

Figure: Periodic cubic spline interpolation

Fig.  %s shows a periodic interpolation of the original points.  The
slope of the fitted curve is the same at the beginning and end of the
data, and the second derivative is also.

The next program illustrates the difference between the cubic spline,
Akima, and Steffen interpolation types on a difficult dataset.

     #include <stdio.h>
     #include <stdlib.h>
     #include <math.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_spline.h>

     int
     main(void)
     {
       size_t i;
       const size_t N = 9;

       /* this dataset is taken from
        * J. M. Hyman, Accurate Monotonicity preserving cubic interpolation,
        * SIAM J. Sci. Stat. Comput. 4, 4, 1983. */
       const double x[] = { 7.99, 8.09, 8.19, 8.7, 9.2,
                            10.0, 12.0, 15.0, 20.0 };
       const double y[] = { 0.0, 2.76429e-5, 4.37498e-2,
                            0.169183, 0.469428, 0.943740,
                            0.998636, 0.999919, 0.999994 };

       gsl_interp_accel *acc = gsl_interp_accel_alloc();
       gsl_spline *spline_cubic = gsl_spline_alloc(gsl_interp_cspline, N);
       gsl_spline *spline_akima = gsl_spline_alloc(gsl_interp_akima, N);
       gsl_spline *spline_steffen = gsl_spline_alloc(gsl_interp_steffen, N);

       gsl_spline_init(spline_cubic, x, y, N);
       gsl_spline_init(spline_akima, x, y, N);
       gsl_spline_init(spline_steffen, x, y, N);

       for (i = 0; i < N; ++i)
         printf("%g %g\n", x[i], y[i]);

       printf("\n\n");

       for (i = 0; i <= 100; ++i)
         {
           double xi = (1 - i / 100.0) * x[0] + (i / 100.0) * x[N-1];
           double yi_cubic = gsl_spline_eval(spline_cubic, xi, acc);
           double yi_akima = gsl_spline_eval(spline_akima, xi, acc);
           double yi_steffen = gsl_spline_eval(spline_steffen, xi, acc);

           printf("%g %g %g %g\n", xi, yi_cubic, yi_akima, yi_steffen);
         }

       gsl_spline_free(spline_cubic);
       gsl_spline_free(spline_akima);
       gsl_spline_free(spline_steffen);
       gsl_interp_accel_free(acc);

       return 0;
     }

[image src="interp_compare.png"]

Figure: Comparison of different 1D interpolation methods

The output is shown in Fig.  %s.  The cubic method exhibits a local
maxima between the 6th and 7th data points and continues oscillating for
the rest of the data.  Akima also shows a local maxima but recovers and
follows the data well after the 7th grid point.  Steffen preserves
monotonicity in all intervals and does not exhibit oscillations, at the
expense of having a discontinuous second derivative.


File: gsl-ref.info,  Node: Introduction to 2D Interpolation,  Next: 2D Interpolation Functions,  Prev: 1D Interpolation Example Programs,  Up: Interpolation

30.8 Introduction to 2D Interpolation
=====================================

Given a set of x coordinates x_1,...,x_m and a set of y coordinates
y_1,...,y_n, each in increasing order, plus a set of function values
z_{ij} for each grid point (x_i,y_j), the routines described in this
section compute a continuous interpolation function z(x,y) such that
z(x_i,y_j) = z_{ij}.


File: gsl-ref.info,  Node: 2D Interpolation Functions,  Next: 2D Interpolation Grids,  Prev: Introduction to 2D Interpolation,  Up: Interpolation

30.9 2D Interpolation Functions
===============================

The interpolation function for a given dataset is stored in a *note
gsl_interp2d: 9c4. object.  These are created by the following
functions.

 -- Type: gsl_interp2d

     Workspace for 2D interpolation

 -- Function: gsl_interp2d * gsl_interp2d_alloc (const gsl_interp2d_type
          * T, const size_t xsize, const size_t ysize)

     This function returns a pointer to a newly allocated interpolation
     object of type ‘T’ for ‘xsize’ grid points in the x direction and
     ‘ysize’ grid points in the y direction.

 -- Function: int gsl_interp2d_init (gsl_interp2d * interp, const
          double xa[], const double ya[], const double za[], const
          size_t xsize, const size_t ysize)

     This function initializes the interpolation object ‘interp’ for the
     data (‘xa’, ‘ya’, ‘za’) where ‘xa’ and ‘ya’ are arrays of the x and
     y grid points of size ‘xsize’ and ‘ysize’ respectively, and ‘za’ is
     an array of function values of size ‘xsize’ * ‘ysize’.  The
     interpolation object (*note gsl_interp2d: 9c4.) does not save the
     data arrays ‘xa’, ‘ya’, and ‘za’ and only stores the static state
     computed from the data.  The ‘xa’ and ‘ya’ data arrays are always
     assumed to be strictly ordered, with increasing x,y values; the
     behavior for other arrangements is not defined.

 -- Function: void gsl_interp2d_free (gsl_interp2d * interp)

     This function frees the interpolation object ‘interp’.


File: gsl-ref.info,  Node: 2D Interpolation Grids,  Next: 2D Interpolation Types,  Prev: 2D Interpolation Functions,  Up: Interpolation

30.10 2D Interpolation Grids
============================

The 2D interpolation routines access the function values z_{ij} with the
following ordering:

     z_{ij} = za[j*xsize + i]

with i = 0,...,xsize-1 and j = 0,...,ysize-1.  However, for ease of use,
the following functions are provided to add and retrieve elements from
the function grid without requiring knowledge of the internal ordering.

 -- Function: int gsl_interp2d_set (const gsl_interp2d * interp,
          double za[], const size_t i, const size_t j, const double z)

     This function sets the value z_{ij} for grid point (‘i’, ‘j’) of
     the array ‘za’ to ‘z’.

 -- Function: double gsl_interp2d_get (const gsl_interp2d * interp,
          const double za[], const size_t i, const size_t j)

     This function returns the value z_{ij} for grid point (‘i’, ‘j’)
     stored in the array ‘za’.

 -- Function: size_t gsl_interp2d_idx (const gsl_interp2d * interp,
          const size_t i, const size_t j)

     This function returns the index corresponding to the grid point
     (‘i’, ‘j’).  The index is given by j*xsize + i.


File: gsl-ref.info,  Node: 2D Interpolation Types,  Next: 2D Evaluation of Interpolating Functions,  Prev: 2D Interpolation Grids,  Up: Interpolation

30.11 2D Interpolation Types
============================

 -- Type: gsl_interp2d_type

     The interpolation library provides the following 2D interpolation
     types:

      -- Variable: gsl_interp2d_bilinear

          Bilinear interpolation.  This interpolation method does not
          require any additional memory.

      -- Variable: gsl_interp2d_bicubic

          Bicubic interpolation.

 -- Function: const char * gsl_interp2d_name (const gsl_interp2d
          * interp)

     This function returns the name of the interpolation type used by
     ‘interp’.  For example:

          printf ("interp uses '%s' interpolation.\n", gsl_interp2d_name (interp));

     would print something like:

          interp uses 'bilinear' interpolation.

 -- Function: unsigned int gsl_interp2d_min_size (const gsl_interp2d
          * interp)
 -- Function: unsigned int gsl_interp2d_type_min_size (const
          gsl_interp2d_type * T)

     These functions return the minimum number of points required by the
     interpolation object ‘interp’ or interpolation type ‘T’.  For
     example, bicubic interpolation requires a minimum of 4 points.


File: gsl-ref.info,  Node: 2D Evaluation of Interpolating Functions,  Next: 2D Higher-level Interface,  Prev: 2D Interpolation Types,  Up: Interpolation

30.12 2D Evaluation of Interpolating Functions
==============================================

 -- Function: double gsl_interp2d_eval (const gsl_interp2d * interp,
          const double xa[], const double ya[], const double za[], const
          double x, const double y, gsl_interp_accel * xacc,
          gsl_interp_accel * yacc)
 -- Function: int gsl_interp2d_eval_e (const gsl_interp2d * interp,
          const double xa[], const double ya[], const double za[], const
          double x, const double y, gsl_interp_accel * xacc,
          gsl_interp_accel * yacc, double * z)

     These functions return the interpolated value of ‘z’ for a given
     point (‘x’, ‘y’), using the interpolation object ‘interp’, data
     arrays ‘xa’, ‘ya’, and ‘za’ and the accelerators ‘xacc’ and ‘yacc’.
     When ‘x’ is outside the range of ‘xa’ or ‘y’ is outside the range
     of ‘ya’, the error code *note GSL_EDOM: 28. is returned.

 -- Function: double gsl_interp2d_eval_extrap (const gsl_interp2d
          * interp, const double xa[], const double ya[], const
          double za[], const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc)
 -- Function: int gsl_interp2d_eval_extrap_e (const gsl_interp2d
          * interp, const double xa[], const double ya[], const
          double za[], const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc, double * z)

     These functions return the interpolated value of ‘z’ for a given
     point (‘x’, ‘y’), using the interpolation object ‘interp’, data
     arrays ‘xa’, ‘ya’, and ‘za’ and the accelerators ‘xacc’ and ‘yacc’.
     The functions perform no bounds checking, so when ‘x’ is outside
     the range of ‘xa’ or ‘y’ is outside the range of ‘ya’,
     extrapolation is performed.

 -- Function: double gsl_interp2d_eval_deriv_x (const gsl_interp2d
          * interp, const double xa[], const double ya[], const
          double za[], const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc)
 -- Function: int gsl_interp2d_eval_deriv_x_e (const gsl_interp2d
          * interp, const double xa[], const double ya[], const
          double za[], const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc, double * d)

     These functions return the interpolated value ‘d’ = \partial z /
     \partial x for a given point (‘x’, ‘y’), using the interpolation
     object ‘interp’, data arrays ‘xa’, ‘ya’, and ‘za’ and the
     accelerators ‘xacc’ and ‘yacc’.  When ‘x’ is outside the range of
     ‘xa’ or ‘y’ is outside the range of ‘ya’, the error code *note
     GSL_EDOM: 28. is returned.

 -- Function: double gsl_interp2d_eval_deriv_y (const gsl_interp2d
          * interp, const double xa[], const double ya[], const
          double za[], const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc)
 -- Function: int gsl_interp2d_eval_deriv_y_e (const gsl_interp2d
          * interp, const double xa[], const double ya[], const
          double za[], const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc, double * d)

     These functions return the interpolated value ‘d’ = \partial z /
     \partial y for a given point (‘x’, ‘y’), using the interpolation
     object ‘interp’, data arrays ‘xa’, ‘ya’, and ‘za’ and the
     accelerators ‘xacc’ and ‘yacc’.  When ‘x’ is outside the range of
     ‘xa’ or ‘y’ is outside the range of ‘ya’, the error code *note
     GSL_EDOM: 28. is returned.

 -- Function: double gsl_interp2d_eval_deriv_xx (const gsl_interp2d
          * interp, const double xa[], const double ya[], const
          double za[], const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc)
 -- Function: int gsl_interp2d_eval_deriv_xx_e (const gsl_interp2d
          * interp, const double xa[], const double ya[], const
          double za[], const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc, double * d)

     These functions return the interpolated value ‘d’ = \partial^2 z /
     \partial x^2 for a given point (‘x’, ‘y’), using the interpolation
     object ‘interp’, data arrays ‘xa’, ‘ya’, and ‘za’ and the
     accelerators ‘xacc’ and ‘yacc’.  When ‘x’ is outside the range of
     ‘xa’ or ‘y’ is outside the range of ‘ya’, the error code *note
     GSL_EDOM: 28. is returned.

 -- Function: double gsl_interp2d_eval_deriv_yy (const gsl_interp2d
          * interp, const double xa[], const double ya[], const
          double za[], const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc)
 -- Function: int gsl_interp2d_eval_deriv_yy_e (const gsl_interp2d
          * interp, const double xa[], const double ya[], const
          double za[], const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc, double * d)

     These functions return the interpolated value ‘d’ = \partial^2 z /
     \partial y^2 for a given point (‘x’, ‘y’), using the interpolation
     object ‘interp’, data arrays ‘xa’, ‘ya’, and ‘za’ and the
     accelerators ‘xacc’ and ‘yacc’.  When ‘x’ is outside the range of
     ‘xa’ or ‘y’ is outside the range of ‘ya’, the error code *note
     GSL_EDOM: 28. is returned.

 -- Function: double gsl_interp2d_eval_deriv_xy (const gsl_interp2d
          * interp, const double xa[], const double ya[], const
          double za[], const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc)
 -- Function: int gsl_interp2d_eval_deriv_xy_e (const gsl_interp2d
          * interp, const double xa[], const double ya[], const
          double za[], const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc, double * d)

     These functions return the interpolated value ‘d’ = \partial^2 z /
     \partial x \partial y for a given point (‘x’, ‘y’), using the
     interpolation object ‘interp’, data arrays ‘xa’, ‘ya’, and ‘za’ and
     the accelerators ‘xacc’ and ‘yacc’.  When ‘x’ is outside the range
     of ‘xa’ or ‘y’ is outside the range of ‘ya’, the error code *note
     GSL_EDOM: 28. is returned.


File: gsl-ref.info,  Node: 2D Higher-level Interface,  Next: 2D Interpolation Example programs,  Prev: 2D Evaluation of Interpolating Functions,  Up: Interpolation

30.13 2D Higher-level Interface
===============================

The functions described in the previous sections required the user to
supply pointers to the x, y, and z arrays on each call.  The following
functions are equivalent to the corresponding ‘gsl_interp2d’ functions
but maintain a copy of this data in the *note gsl_spline2d: 9e3. object.
This removes the need to pass ‘xa’, ‘ya’, and ‘za’ as arguments on each
evaluation.  These functions are defined in the header file
‘gsl_spline2d.h’.

 -- Type: gsl_spline2d

     This workspace provides a higher level interface for the *note
     gsl_interp2d: 9c4. object

 -- Function: gsl_spline2d * gsl_spline2d_alloc (const gsl_interp2d_type
          * T, size_t xsize, size_t ysize)

 -- Function: int gsl_spline2d_init (gsl_spline2d * spline, const
          double xa[], const double ya[], const double za[],
          size_t xsize, size_t ysize)

 -- Function: void gsl_spline2d_free (gsl_spline2d * spline)

 -- Function: const char * gsl_spline2d_name (const gsl_spline2d
          * spline)

 -- Function: unsigned int gsl_spline2d_min_size (const gsl_spline2d
          * spline)

 -- Function: double gsl_spline2d_eval (const gsl_spline2d * spline,
          const double x, const double y, gsl_interp_accel * xacc,
          gsl_interp_accel * yacc)
 -- Function: int gsl_spline2d_eval_e (const gsl_spline2d * spline,
          const double x, const double y, gsl_interp_accel * xacc,
          gsl_interp_accel * yacc, double * z)

 -- Function: double gsl_spline2d_eval_deriv_x (const gsl_spline2d
          * spline, const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc)
 -- Function: int gsl_spline2d_eval_deriv_x_e (const gsl_spline2d
          * spline, const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc, double * d)

 -- Function: double gsl_spline2d_eval_deriv_y (const gsl_spline2d
          * spline, const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc)
 -- Function: int gsl_spline2d_eval_deriv_y_e (const gsl_spline2d
          * spline, const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc, double * d)

 -- Function: double gsl_spline2d_eval_deriv_xx (const gsl_spline2d
          * spline, const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc)
 -- Function: int gsl_spline2d_eval_deriv_xx_e (const gsl_spline2d
          * spline, const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc, double * d)

 -- Function: double gsl_spline2d_eval_deriv_yy (const gsl_spline2d
          * spline, const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc)
 -- Function: int gsl_spline2d_eval_deriv_yy_e (const gsl_spline2d
          * spline, const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc, double * d)

 -- Function: double gsl_spline2d_eval_deriv_xy (const gsl_spline2d
          * spline, const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc)
 -- Function: int gsl_spline2d_eval_deriv_xy_e (const gsl_spline2d
          * spline, const double x, const double y, gsl_interp_accel
          * xacc, gsl_interp_accel * yacc, double * d)

 -- Function: int gsl_spline2d_set (const gsl_spline2d * spline,
          double za[], const size_t i, const size_t j, const double z)

 -- Function: double gsl_spline2d_get (const gsl_spline2d * spline,
          const double za[], const size_t i, const size_t j)

     This function returns the value z_{ij} for grid point (‘i’, ‘j’)
     stored in the array ‘za’.


File: gsl-ref.info,  Node: 2D Interpolation Example programs,  Next: References and Further Reading<23>,  Prev: 2D Higher-level Interface,  Up: Interpolation

30.14 2D Interpolation Example programs
=======================================

The following example performs bilinear interpolation on the unit
square, using z values of (0,1,0.5,1) going clockwise around the square.

     #include <stdio.h>
     #include <stdlib.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_interp2d.h>
     #include <gsl/gsl_spline2d.h>

     int
     main()
     {
       const gsl_interp2d_type *T = gsl_interp2d_bilinear;
       const size_t N = 100;             /* number of points to interpolate */
       const double xa[] = { 0.0, 1.0 }; /* define unit square */
       const double ya[] = { 0.0, 1.0 };
       const size_t nx = sizeof(xa) / sizeof(double); /* x grid points */
       const size_t ny = sizeof(ya) / sizeof(double); /* y grid points */
       double *za = malloc(nx * ny * sizeof(double));
       gsl_spline2d *spline = gsl_spline2d_alloc(T, nx, ny);
       gsl_interp_accel *xacc = gsl_interp_accel_alloc();
       gsl_interp_accel *yacc = gsl_interp_accel_alloc();
       size_t i, j;

       /* set z grid values */
       gsl_spline2d_set(spline, za, 0, 0, 0.0);
       gsl_spline2d_set(spline, za, 0, 1, 1.0);
       gsl_spline2d_set(spline, za, 1, 1, 0.5);
       gsl_spline2d_set(spline, za, 1, 0, 1.0);

       /* initialize interpolation */
       gsl_spline2d_init(spline, xa, ya, za, nx, ny);

       /* interpolate N values in x and y and print out grid for plotting */
       for (i = 0; i < N; ++i)
         {
           double xi = i / (N - 1.0);

           for (j = 0; j < N; ++j)
             {
               double yj = j / (N - 1.0);
               double zij = gsl_spline2d_eval(spline, xi, yj, xacc, yacc);

               printf("%f %f %f\n", xi, yj, zij);
             }
           printf("\n");
         }

       gsl_spline2d_free(spline);
       gsl_interp_accel_free(xacc);
       gsl_interp_accel_free(yacc);
       free(za);

       return 0;
     }

The results of the interpolation are shown in Fig.  %s, where the
corners are labeled with their fixed z values.

[image src="interp2d.png"]

Figure: 2D interpolation example


File: gsl-ref.info,  Node: References and Further Reading<23>,  Prev: 2D Interpolation Example programs,  Up: Interpolation

30.15 References and Further Reading
====================================

Descriptions of the interpolation algorithms and further references can
be found in the following publications:

   * C.W. Ueberhuber, `Numerical Computation (Volume 1), Chapter 9
     “Interpolation”', Springer (1997), ISBN 3-540-62058-3.

   * D.M. Young, R.T. Gregory, `A Survey of Numerical Mathematics
     (Volume 1), Chapter 6.8', Dover (1988), ISBN 0-486-65691-8.

   * M. Steffen, `A simple method for monotonic interpolation in one
     dimension', Astron.  Astrophys.  239, 443-450, 1990.


File: gsl-ref.info,  Node: Numerical Differentiation,  Next: Chebyshev Approximations,  Prev: Interpolation,  Up: Top

31 Numerical Differentiation
****************************

The functions described in this chapter compute numerical derivatives by
finite differencing.  An adaptive algorithm is used to find the best
choice of finite difference and to estimate the error in the derivative.
These functions are declared in the header file ‘gsl_deriv.h’.

* Menu:

* Functions::
* Examples: Examples<23>.
* References and Further Reading: References and Further Reading<24>.


File: gsl-ref.info,  Node: Functions,  Next: Examples<23>,  Up: Numerical Differentiation

31.1 Functions
==============

 -- Function: int gsl_deriv_central (const gsl_function * f, double x,
          double h, double * result, double * abserr)

     This function computes the numerical derivative of the function ‘f’
     at the point ‘x’ using an adaptive central difference algorithm
     with a step-size of ‘h’.  The derivative is returned in ‘result’
     and an estimate of its absolute error is returned in ‘abserr’.

     The initial value of ‘h’ is used to estimate an optimal step-size,
     based on the scaling of the truncation error and round-off error in
     the derivative calculation.  The derivative is computed using a
     5-point rule for equally spaced abscissae at x - h, x - h/2, x, x +
     h/2, x+h, with an error estimate taken from the difference between
     the 5-point rule and the corresponding 3-point rule x-h, x, x+h.
     Note that the value of the function at x does not contribute to the
     derivative calculation, so only 4-points are actually used.

 -- Function: int gsl_deriv_forward (const gsl_function * f, double x,
          double h, double * result, double * abserr)

     This function computes the numerical derivative of the function ‘f’
     at the point ‘x’ using an adaptive forward difference algorithm
     with a step-size of ‘h’.  The function is evaluated only at points
     greater than ‘x’, and never at ‘x’ itself.  The derivative is
     returned in ‘result’ and an estimate of its absolute error is
     returned in ‘abserr’.  This function should be used if f(x) has a
     discontinuity at ‘x’, or is undefined for values less than ‘x’.

     The initial value of ‘h’ is used to estimate an optimal step-size,
     based on the scaling of the truncation error and round-off error in
     the derivative calculation.  The derivative at x is computed using
     an “open” 4-point rule for equally spaced abscissae at x+h/4, x +
     h/2, x + 3h/4, x+h, with an error estimate taken from the
     difference between the 4-point rule and the corresponding 2-point
     rule x+h/2, x+h.

 -- Function: int gsl_deriv_backward (const gsl_function * f, double x,
          double h, double * result, double * abserr)

     This function computes the numerical derivative of the function ‘f’
     at the point ‘x’ using an adaptive backward difference algorithm
     with a step-size of ‘h’.  The function is evaluated only at points
     less than ‘x’, and never at ‘x’ itself.  The derivative is returned
     in ‘result’ and an estimate of its absolute error is returned in
     ‘abserr’.  This function should be used if f(x) has a discontinuity
     at ‘x’, or is undefined for values greater than ‘x’.

     This function is equivalent to calling *note gsl_deriv_forward():
     9fe. with a negative step-size.


File: gsl-ref.info,  Node: Examples<23>,  Next: References and Further Reading<24>,  Prev: Functions,  Up: Numerical Differentiation

31.2 Examples
=============

The following code estimates the derivative of the function f(x) =
x^{3/2} at x = 2 and at x = 0.  The function f(x) is undefined for x < 0
so the derivative at x=0 is computed using *note gsl_deriv_forward():
9fe.

     #include <stdio.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_deriv.h>

     double f (double x, void * params)
     {
       (void)(params); /* avoid unused parameter warning */
       return pow (x, 1.5);
     }

     int
     main (void)
     {
       gsl_function F;
       double result, abserr;

       F.function = &f;
       F.params = 0;

       printf ("f(x) = x^(3/2)\n");

       gsl_deriv_central (&F, 2.0, 1e-8, &result, &abserr);
       printf ("x = 2.0\n");
       printf ("f'(x) = %.10f +/- %.10f\n", result, abserr);
       printf ("exact = %.10f\n\n", 1.5 * sqrt(2.0));

       gsl_deriv_forward (&F, 0.0, 1e-8, &result, &abserr);
       printf ("x = 0.0\n");
       printf ("f'(x) = %.10f +/- %.10f\n", result, abserr);
       printf ("exact = %.10f\n", 0.0);

       return 0;
     }

Here is the output of the program,

     f(x) = x^(3/2)
     x = 2.0
     f'(x) = 2.1213203120 +/- 0.0000005006
     exact = 2.1213203436

     x = 0.0
     f'(x) = 0.0000000160 +/- 0.0000000339
     exact = 0.0000000000


File: gsl-ref.info,  Node: References and Further Reading<24>,  Prev: Examples<23>,  Up: Numerical Differentiation

31.3 References and Further Reading
===================================

The algorithms used by these functions are described in the following
sources:

   * Abramowitz and Stegun, `Handbook of Mathematical Functions',
     Section 25.3.4, and Table 25.5 (Coefficients for Differentiation).

   * S.D. Conte and Carl de Boor, `Elementary Numerical Analysis: An
     Algorithmic Approach', McGraw-Hill, 1972.


File: gsl-ref.info,  Node: Chebyshev Approximations,  Next: Series Acceleration,  Prev: Numerical Differentiation,  Up: Top

32 Chebyshev Approximations
***************************

This chapter describes routines for computing Chebyshev approximations
to univariate functions.  A Chebyshev approximation is a truncation of
the series f(x) = \sum c_n T_n(x), where the Chebyshev polynomials
T_n(x) = \cos(n \arccos x) provide an orthogonal basis of polynomials on
the interval [-1,1] with the weight function 1 / \sqrt{1-x^2}.  The
first few Chebyshev polynomials are, T_0(x) = 1, T_1(x) = x, T_2(x) = 2
x^2 - 1.  For further information see Abramowitz & Stegun, Chapter 22.

The functions described in this chapter are declared in the header file
‘gsl_chebyshev.h’.

* Menu:

* Definitions::
* Creation and Calculation of Chebyshev Series::
* Auxiliary Functions::
* Chebyshev Series Evaluation::
* Derivatives and Integrals::
* Examples: Examples<24>.
* References and Further Reading: References and Further Reading<25>.


File: gsl-ref.info,  Node: Definitions,  Next: Creation and Calculation of Chebyshev Series,  Up: Chebyshev Approximations

32.1 Definitions
================

 -- Type: gsl_cheb_series

     A Chebyshev series is stored using the following structure:

          typedef struct
          {
            double * c;   /* coefficients  c[0] .. c[order] */
            int order;    /* order of expansion             */
            double a;     /* lower interval point           */
            double b;     /* upper interval point           */
            ...
          } gsl_cheb_series

The approximation is made over the range [a,b] using ‘order’ + 1 terms,
including the coefficient c[0].  The series is computed using the
following convention,

     f(x) = (c_0 / 2) + \sum_{n=1} c_n T_n(x)

which is needed when accessing the coefficients directly.


File: gsl-ref.info,  Node: Creation and Calculation of Chebyshev Series,  Next: Auxiliary Functions,  Prev: Definitions,  Up: Chebyshev Approximations

32.2 Creation and Calculation of Chebyshev Series
=================================================

 -- Function: gsl_cheb_series * gsl_cheb_alloc (const size_t n)

     This function allocates space for a Chebyshev series of order ‘n’
     and returns a pointer to a new *note gsl_cheb_series: a05. struct.

 -- Function: void gsl_cheb_free (gsl_cheb_series * cs)

     This function frees a previously allocated Chebyshev series ‘cs’.

 -- Function: int gsl_cheb_init (gsl_cheb_series * cs, const
          gsl_function * f, const double a, const double b)

     This function computes the Chebyshev approximation ‘cs’ for the
     function ‘f’ over the range (a,b) to the previously specified
     order.  The computation of the Chebyshev approximation is an O(n^2)
     process, and requires n function evaluations.


File: gsl-ref.info,  Node: Auxiliary Functions,  Next: Chebyshev Series Evaluation,  Prev: Creation and Calculation of Chebyshev Series,  Up: Chebyshev Approximations

32.3 Auxiliary Functions
========================

The following functions provide information about an existing Chebyshev
series.

 -- Function: size_t gsl_cheb_order (const gsl_cheb_series * cs)

     This function returns the order of Chebyshev series ‘cs’.

 -- Function: size_t gsl_cheb_size (const gsl_cheb_series * cs)
 -- Function: double * gsl_cheb_coeffs (const gsl_cheb_series * cs)

     These functions return the size of the Chebyshev coefficient array
     ‘c[]’ and a pointer to its location in memory for the Chebyshev
     series ‘cs’.


File: gsl-ref.info,  Node: Chebyshev Series Evaluation,  Next: Derivatives and Integrals,  Prev: Auxiliary Functions,  Up: Chebyshev Approximations

32.4 Chebyshev Series Evaluation
================================

 -- Function: double gsl_cheb_eval (const gsl_cheb_series * cs,
          double x)

     This function evaluates the Chebyshev series ‘cs’ at a given point
     ‘x’.

 -- Function: int gsl_cheb_eval_err (const gsl_cheb_series * cs, const
          double x, double * result, double * abserr)

     This function computes the Chebyshev series ‘cs’ at a given point
     ‘x’, estimating both the series ‘result’ and its absolute error
     ‘abserr’.  The error estimate is made from the first neglected term
     in the series.

 -- Function: double gsl_cheb_eval_n (const gsl_cheb_series * cs,
          size_t order, double x)

     This function evaluates the Chebyshev series ‘cs’ at a given point
     ‘x’, to (at most) the given order ‘order’.

 -- Function: int gsl_cheb_eval_n_err (const gsl_cheb_series * cs, const
          size_t order, const double x, double * result, double
          * abserr)

     This function evaluates a Chebyshev series ‘cs’ at a given point
     ‘x’, estimating both the series ‘result’ and its absolute error
     ‘abserr’, to (at most) the given order ‘order’.  The error estimate
     is made from the first neglected term in the series.


File: gsl-ref.info,  Node: Derivatives and Integrals,  Next: Examples<24>,  Prev: Chebyshev Series Evaluation,  Up: Chebyshev Approximations

32.5 Derivatives and Integrals
==============================

The following functions allow a Chebyshev series to be differentiated or
integrated, producing a new Chebyshev series.  Note that the error
estimate produced by evaluating the derivative series will be
underestimated due to the contribution of higher order terms being
neglected.

 -- Function: int gsl_cheb_calc_deriv (gsl_cheb_series * deriv, const
          gsl_cheb_series * cs)

     This function computes the derivative of the series ‘cs’, storing
     the derivative coefficients in the previously allocated ‘deriv’.
     The two series ‘cs’ and ‘deriv’ must have been allocated with the
     same order.

 -- Function: int gsl_cheb_calc_integ (gsl_cheb_series * integ, const
          gsl_cheb_series * cs)

     This function computes the integral of the series ‘cs’, storing the
     integral coefficients in the previously allocated ‘integ’.  The two
     series ‘cs’ and ‘integ’ must have been allocated with the same
     order.  The lower limit of the integration is taken to be the left
     hand end of the range ‘a’.


File: gsl-ref.info,  Node: Examples<24>,  Next: References and Further Reading<25>,  Prev: Derivatives and Integrals,  Up: Chebyshev Approximations

32.6 Examples
=============

The following example program computes Chebyshev approximations to a
step function.  This is an extremely difficult approximation to make,
due to the discontinuity, and was chosen as an example where
approximation error is visible.  For smooth functions the Chebyshev
approximation converges extremely rapidly and errors would not be
visible.

     #include <stdio.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_chebyshev.h>

     double
     f (double x, void *p)
     {
       (void)(p); /* avoid unused parameter warning */

       if (x < 0.5)
         return 0.25;
       else
         return 0.75;
     }

     int
     main (void)
     {
       int i, n = 10000;

       gsl_cheb_series *cs = gsl_cheb_alloc (40);

       gsl_function F;

       F.function = f;
       F.params = 0;

       gsl_cheb_init (cs, &F, 0.0, 1.0);

       for (i = 0; i < n; i++)
         {
           double x = i / (double)n;
           double r10 = gsl_cheb_eval_n (cs, 10, x);
           double r40 = gsl_cheb_eval (cs, x);
           printf ("%g %g %g %g\n",
                   x, GSL_FN_EVAL (&F, x), r10, r40);
         }

       gsl_cheb_free (cs);

       return 0;
     }

Fig.  %s shows output from the program with the original function, 10-th
order approximation and 40-th order approximation, all sampled at
intervals of 0.001 in x.

[image src="cheb.png"]

Figure: Chebyshev approximations to a step function


File: gsl-ref.info,  Node: References and Further Reading<25>,  Prev: Examples<24>,  Up: Chebyshev Approximations

32.7 References and Further Reading
===================================

The following paper describes the use of Chebyshev series,

   * R. Broucke, “Ten Subroutines for the Manipulation of Chebyshev
     Series [C1] (Algorithm 446)”.  `Communications of the ACM' 16(4),
     254–256 (1973)


File: gsl-ref.info,  Node: Series Acceleration,  Next: Wavelet Transforms,  Prev: Chebyshev Approximations,  Up: Top

33 Series Acceleration
**********************

The functions described in this chapter accelerate the convergence of a
series using the Levin u-transform.  This method takes a small number of
terms from the start of a series and uses a systematic approximation to
compute an extrapolated value and an estimate of its error.  The
u-transform works for both convergent and divergent series, including
asymptotic series.

These functions are declared in the header file ‘gsl_sum.h’.

* Menu:

* Acceleration functions::
* Acceleration functions without error estimation::
* Examples: Examples<25>.
* References and Further Reading: References and Further Reading<26>.


File: gsl-ref.info,  Node: Acceleration functions,  Next: Acceleration functions without error estimation,  Up: Series Acceleration

33.1 Acceleration functions
===========================

The following functions compute the full Levin u-transform of a series
with its error estimate.  The error estimate is computed by propagating
rounding errors from each term through to the final extrapolation.

These functions are intended for summing analytic series where each term
is known to high accuracy, and the rounding errors are assumed to
originate from finite precision.  They are taken to be relative errors
of order ‘GSL_DBL_EPSILON’ for each term.

The calculation of the error in the extrapolated value is an O(N^2)
process, which is expensive in time and memory.  A faster but less
reliable method which estimates the error from the convergence of the
extrapolated value is described in the next section.  For the method
described here a full table of intermediate values and derivatives
through to O(N) must be computed and stored, but this does give a
reliable error estimate.

 -- Type: gsl_sum_levin_u_workspace

     Workspace for a Leven u-transform.

 -- Function: gsl_sum_levin_u_workspace * gsl_sum_levin_u_alloc
          (size_t n)

     This function allocates a workspace for a Levin u-transform of ‘n’
     terms.  The size of the workspace is O(2n^2 + 3n).

 -- Function: void gsl_sum_levin_u_free (gsl_sum_levin_u_workspace * w)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: int gsl_sum_levin_u_accel (const double * array,
          size_t array_size, gsl_sum_levin_u_workspace * w, double
          * sum_accel, double * abserr)

     This function takes the terms of a series in ‘array’ of size
     ‘array_size’ and computes the extrapolated limit of the series
     using a Levin u-transform.  Additional working space must be
     provided in ‘w’.  The extrapolated sum is stored in ‘sum_accel’,
     with an estimate of the absolute error stored in ‘abserr’.  The
     actual term-by-term sum is returned in ‘w->sum_plain’.  The
     algorithm calculates the truncation error (the difference between
     two successive extrapolations) and round-off error (propagated from
     the individual terms) to choose an optimal number of terms for the
     extrapolation.  All the terms of the series passed in through
     ‘array’ should be non-zero.


File: gsl-ref.info,  Node: Acceleration functions without error estimation,  Next: Examples<25>,  Prev: Acceleration functions,  Up: Series Acceleration

33.2 Acceleration functions without error estimation
====================================================

The functions described in this section compute the Levin u-transform of
series and attempt to estimate the error from the “truncation error” in
the extrapolation, the difference between the final two approximations.
Using this method avoids the need to compute an intermediate table of
derivatives because the error is estimated from the behavior of the
extrapolated value itself.  Consequently this algorithm is an O(N)
process and only requires O(N) terms of storage.  If the series
converges sufficiently fast then this procedure can be acceptable.  It
is appropriate to use this method when there is a need to compute many
extrapolations of series with similar convergence properties at
high-speed.  For example, when numerically integrating a function
defined by a parameterized series where the parameter varies only
slightly.  A reliable error estimate should be computed first using the
full algorithm described above in order to verify the consistency of the
results.

 -- Type: gsl_sum_levin_utrunc_workspace

     Workspace for a Levin u-transform without error estimation

 -- Function: gsl_sum_levin_utrunc_workspace *
          gsl_sum_levin_utrunc_alloc (size_t n)

     This function allocates a workspace for a Levin u-transform of ‘n’
     terms, without error estimation.  The size of the workspace is
     O(3n).

 -- Function: void gsl_sum_levin_utrunc_free
          (gsl_sum_levin_utrunc_workspace * w)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: int gsl_sum_levin_utrunc_accel (const double * array,
          size_t array_size, gsl_sum_levin_utrunc_workspace * w, double
          * sum_accel, double * abserr_trunc)

     This function takes the terms of a series in ‘array’ of size
     ‘array_size’ and computes the extrapolated limit of the series
     using a Levin u-transform.  Additional working space must be
     provided in ‘w’.  The extrapolated sum is stored in ‘sum_accel’.
     The actual term-by-term sum is returned in ‘w->sum_plain’.  The
     algorithm terminates when the difference between two successive
     extrapolations reaches a minimum or is sufficiently small.  The
     difference between these two values is used as estimate of the
     error and is stored in ‘abserr_trunc’.  To improve the reliability
     of the algorithm the extrapolated values are replaced by moving
     averages when calculating the truncation error, smoothing out any
     fluctuations.


File: gsl-ref.info,  Node: Examples<25>,  Next: References and Further Reading<26>,  Prev: Acceleration functions without error estimation,  Up: Series Acceleration

33.3 Examples
=============

The following code calculates an estimate of \zeta(2) = \pi^2 / 6 using
the series,

     \zeta(2) = 1 + 1/2^2 + 1/3^2 + 1/4^2 + \dots

After ‘N’ terms the error in the sum is O(1/N), making direct summation
of the series converge slowly.

     #include <stdio.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_sum.h>

     #define N 20

     int
     main (void)
     {
       double t[N];
       double sum_accel, err;
       double sum = 0;
       int n;

       gsl_sum_levin_u_workspace * w
         = gsl_sum_levin_u_alloc (N);

       const double zeta_2 = M_PI * M_PI / 6.0;

       /* terms for zeta(2) = \sum_{n=1}^{\infty} 1/n^2 */

       for (n = 0; n < N; n++)
         {
           double np1 = n + 1.0;
           t[n] = 1.0 / (np1 * np1);
           sum += t[n];
         }

       gsl_sum_levin_u_accel (t, N, w, &sum_accel, &err);

       printf ("term-by-term sum = % .16f using %d terms\n",
               sum, N);

       printf ("term-by-term sum = % .16f using %zu terms\n",
               w->sum_plain, w->terms_used);

       printf ("exact value      = % .16f\n", zeta_2);
       printf ("accelerated sum  = % .16f using %zu terms\n",
               sum_accel, w->terms_used);

       printf ("estimated error  = % .16f\n", err);
       printf ("actual error     = % .16f\n",
               sum_accel - zeta_2);

       gsl_sum_levin_u_free (w);
       return 0;
     }

The output below shows that the Levin u-transform is able to obtain an
estimate of the sum to 1 part in 10^{10} using the first eleven terms of
the series.  The error estimate returned by the function is also
accurate, giving the correct number of significant digits.

     term-by-term sum =  1.5961632439130233 using 20 terms
     term-by-term sum =  1.5759958390005426 using 13 terms
     exact value      =  1.6449340668482264
     accelerated sum  =  1.6449340669228176 using 13 terms
     estimated error  =  0.0000000000888360
     actual error     =  0.0000000000745912

Note that a direct summation of this series would require 10^{10} terms
to achieve the same precision as the accelerated sum does in 13 terms.


File: gsl-ref.info,  Node: References and Further Reading<26>,  Prev: Examples<25>,  Up: Series Acceleration

33.4 References and Further Reading
===================================

The algorithms used by these functions are described in the following
papers,

   * T. Fessler, W.F. Ford, D.A. Smith, HURRY: An acceleration algorithm
     for scalar sequences and series `ACM Transactions on Mathematical
     Software', 9(3):346–354, 1983.  and Algorithm 602 9(3):355–357,
     1983.

The theory of the u-transform was presented by Levin,

   * D. Levin, Development of Non-Linear Transformations for Improving
     Convergence of Sequences, `Intern.: J.: Computer Math.'
     B3:371–388, 1973.

A review paper on the Levin Transform is available online,

   * Herbert H. H. Homeier, Scalar Levin-Type Sequence Transformations,
     ‘http://arxiv.org/abs/math/0005209’


File: gsl-ref.info,  Node: Wavelet Transforms,  Next: Discrete Hankel Transforms,  Prev: Series Acceleration,  Up: Top

34 Wavelet Transforms
*********************

This chapter describes functions for performing Discrete Wavelet
Transforms (DWTs).  The library includes wavelets for real data in both
one and two dimensions.  The wavelet functions are declared in the
header files ‘gsl_wavelet.h’ and ‘gsl_wavelet2d.h’.

* Menu:

* Definitions: Definitions<2>.
* Initialization::
* Transform Functions::
* Examples: Examples<26>.
* References and Further Reading: References and Further Reading<27>.


File: gsl-ref.info,  Node: Definitions<2>,  Next: Initialization,  Up: Wavelet Transforms

34.1 Definitions
================

The continuous wavelet transform and its inverse are defined by the
relations,

     w(s, \tau) = \int_{-\infty}^\infty f(t) * \psi^*_{s,\tau}(t) dt

and,

     f(t) = \int_0^\infty ds \int_{-\infty}^\infty w(s, \tau) * \psi_{s,\tau}(t) d\tau

where the basis functions \psi_{s,\tau} are obtained by scaling and
translation from a single function, referred to as the `mother wavelet'.

The discrete version of the wavelet transform acts on equally-spaced
samples, with fixed scaling and translation steps (s, \tau).  The
frequency and time axes are sampled `dyadically' on scales of 2^j
through a level parameter j.

The resulting family of functions \{\psi_{j,n}\} constitutes an
orthonormal basis for square-integrable signals.  The discrete wavelet
transform is an O(N) algorithm, and is also referred to as the `fast
wavelet transform'.


File: gsl-ref.info,  Node: Initialization,  Next: Transform Functions,  Prev: Definitions<2>,  Up: Wavelet Transforms

34.2 Initialization
===================

 -- Type: gsl_wavelet

     This structure contains the filter coefficients defining the
     wavelet and any associated offset parameters.

 -- Function: gsl_wavelet * gsl_wavelet_alloc (const gsl_wavelet_type
          * T, size_t k)

     This function allocates and initializes a wavelet object of type
     ‘T’.  The parameter ‘k’ selects the specific member of the wavelet
     family.  A null pointer is returned if insufficient memory is
     available or if a unsupported member is selected.

The following wavelet types are implemented:

 -- Type: gsl_wavelet_type

      -- Variable: gsl_wavelet_daubechies
      -- Variable: gsl_wavelet_daubechies_centered

          This is the Daubechies wavelet family of maximum phase with
          k/2 vanishing moments.  The implemented wavelets are k=4, 6,
          \dots, 20, with ‘k’ even.

      -- Variable: gsl_wavelet_haar
      -- Variable: gsl_wavelet_haar_centered

          This is the Haar wavelet.  The only valid choice of k for the
          Haar wavelet is k=2.

      -- Variable: gsl_wavelet_bspline
      -- Variable: gsl_wavelet_bspline_centered

          This is the biorthogonal B-spline wavelet family of order
          (i,j).  The implemented values of k = 100*i + j are 103, 105,
          202, 204, 206, 208, 301, 303, 305 307, 309.

The centered forms of the wavelets align the coefficients of the various
sub-bands on edges.  Thus the resulting visualization of the
coefficients of the wavelet transform in the phase plane is easier to
understand.

 -- Function: const char * gsl_wavelet_name (const gsl_wavelet * w)

     This function returns a pointer to the name of the wavelet family
     for ‘w’.

 -- Function: void gsl_wavelet_free (gsl_wavelet * w)

     This function frees the wavelet object ‘w’.

 -- Type: gsl_wavelet_workspace

     This structure contains scratch space of the same size as the input
     data and is used to hold intermediate results during the transform.

 -- Function: gsl_wavelet_workspace * gsl_wavelet_workspace_alloc
          (size_t n)

     This function allocates a workspace for the discrete wavelet
     transform.  To perform a one-dimensional transform on ‘n’ elements,
     a workspace of size ‘n’ must be provided.  For two-dimensional
     transforms of ‘n’-by-‘n’ matrices it is sufficient to allocate a
     workspace of size ‘n’, since the transform operates on individual
     rows and columns.  A null pointer is returned if insufficient
     memory is available.

 -- Function: void gsl_wavelet_workspace_free (gsl_wavelet_workspace
          * work)

     This function frees the allocated workspace ‘work’.


File: gsl-ref.info,  Node: Transform Functions,  Next: Examples<26>,  Prev: Initialization,  Up: Wavelet Transforms

34.3 Transform Functions
========================

This sections describes the actual functions performing the discrete
wavelet transform.  Note that the transforms use periodic boundary
conditions.  If the signal is not periodic in the sample length then
spurious coefficients will appear at the beginning and end of each level
of the transform.

* Menu:

* Wavelet transforms in one dimension::
* Wavelet transforms in two dimension::


File: gsl-ref.info,  Node: Wavelet transforms in one dimension,  Next: Wavelet transforms in two dimension,  Up: Transform Functions

34.3.1 Wavelet transforms in one dimension
------------------------------------------

 -- Function: int gsl_wavelet_transform (const gsl_wavelet * w, double
          * data, size_t stride, size_t n, gsl_wavelet_direction dir,
          gsl_wavelet_workspace * work)
 -- Function: int gsl_wavelet_transform_forward (const gsl_wavelet * w,
          double * data, size_t stride, size_t n, gsl_wavelet_workspace
          * work)
 -- Function: int gsl_wavelet_transform_inverse (const gsl_wavelet * w,
          double * data, size_t stride, size_t n, gsl_wavelet_workspace
          * work)

     These functions compute in-place forward and inverse discrete
     wavelet transforms of length ‘n’ with stride ‘stride’ on the array
     ‘data’.  The length of the transform ‘n’ is restricted to powers of
     two.  For the ‘transform’ version of the function the argument
     ‘dir’ can be either ‘forward’ (+1) or ‘backward’ (-1).  A workspace
     ‘work’ of length ‘n’ must be provided.

     For the forward transform, the elements of the original array are
     replaced by the discrete wavelet transform f_i \rightarrow w_{j,k}
     in a packed triangular storage layout, where ‘j’ is the index of
     the level j = 0 \dots J-1 and ‘k’ is the index of the coefficient
     within each level, k = 0 \dots 2^j - 1.  The total number of levels
     is J = \log_2(n).  The output data has the following form,

          (s_{-1,0}, d_{0,0}, d_{1,0}, d_{1,1}, d_{2,0},\cdots, d_{j,k},\cdots, d_{J-1,2^{J-1} - 1})

     where the first element is the smoothing coefficient s_{-1,0},
     followed by the detail coefficients d_{j,k} for each level j.  The
     backward transform inverts these coefficients to obtain the
     original data.

     These functions return a status of ‘GSL_SUCCESS’ upon successful
     completion.  *note GSL_EINVAL: 2b. is returned if ‘n’ is not an
     integer power of 2 or if insufficient workspace is provided.


File: gsl-ref.info,  Node: Wavelet transforms in two dimension,  Prev: Wavelet transforms in one dimension,  Up: Transform Functions

34.3.2 Wavelet transforms in two dimension
------------------------------------------

The library provides functions to perform two-dimensional discrete
wavelet transforms on square matrices.  The matrix dimensions must be an
integer power of two.  There are two possible orderings of the rows and
columns in the two-dimensional wavelet transform, referred to as the
“standard” and “non-standard” forms.

The “standard” transform performs a complete discrete wavelet transform
on the rows of the matrix, followed by a separate complete discrete
wavelet transform on the columns of the resulting row-transformed
matrix.  This procedure uses the same ordering as a two-dimensional
Fourier transform.

The “non-standard” transform is performed in interleaved passes on the
rows and columns of the matrix for each level of the transform.  The
first level of the transform is applied to the matrix rows, and then to
the matrix columns.  This procedure is then repeated across the rows and
columns of the data for the subsequent levels of the transform, until
the full discrete wavelet transform is complete.  The non-standard form
of the discrete wavelet transform is typically used in image analysis.

The functions described in this section are declared in the header file
‘gsl_wavelet2d.h’.

 -- Function: int gsl_wavelet2d_transform (const gsl_wavelet * w, double
          * data, size_t tda, size_t size1, size_t size2,
          gsl_wavelet_direction dir, gsl_wavelet_workspace * work)
 -- Function: int gsl_wavelet2d_transform_forward (const gsl_wavelet
          * w, double * data, size_t tda, size_t size1, size_t size2,
          gsl_wavelet_workspace * work)
 -- Function: int gsl_wavelet2d_transform_inverse (const gsl_wavelet
          * w, double * data, size_t tda, size_t size1, size_t size2,
          gsl_wavelet_workspace * work)

     These functions compute two-dimensional in-place forward and
     inverse discrete wavelet transforms in standard form on the array
     ‘data’ stored in row-major form with dimensions ‘size1’ and ‘size2’
     and physical row length ‘tda’.  The dimensions must be equal
     (square matrix) and are restricted to powers of two.  For the
     ‘transform’ version of the function the argument ‘dir’ can be
     either ‘forward’ (+1) or ‘backward’ (-1).  A workspace ‘work’ of
     the appropriate size must be provided.  On exit, the appropriate
     elements of the array ‘data’ are replaced by their two-dimensional
     wavelet transform.

     The functions return a status of ‘GSL_SUCCESS’ upon successful
     completion.  *note GSL_EINVAL: 2b. is returned if ‘size1’ and
     ‘size2’ are not equal and integer powers of 2, or if insufficient
     workspace is provided.

 -- Function: int gsl_wavelet2d_transform_matrix (const gsl_wavelet * w,
          gsl_matrix * m, gsl_wavelet_direction dir,
          gsl_wavelet_workspace * work)
 -- Function: int gsl_wavelet2d_transform_matrix_forward (const
          gsl_wavelet * w, gsl_matrix * m, gsl_wavelet_workspace * work)
 -- Function: int gsl_wavelet2d_transform_matrix_inverse (const
          gsl_wavelet * w, gsl_matrix * m, gsl_wavelet_workspace * work)

     These functions compute the two-dimensional in-place wavelet
     transform on a matrix ‘m’.

 -- Function: int gsl_wavelet2d_nstransform (const gsl_wavelet * w,
          double * data, size_t tda, size_t size1, size_t size2,
          gsl_wavelet_direction dir, gsl_wavelet_workspace * work)
 -- Function: int gsl_wavelet2d_nstransform_forward (const gsl_wavelet
          * w, double * data, size_t tda, size_t size1, size_t size2,
          gsl_wavelet_workspace * work)
 -- Function: int gsl_wavelet2d_nstransform_inverse (const gsl_wavelet
          * w, double * data, size_t tda, size_t size1, size_t size2,
          gsl_wavelet_workspace * work)

     These functions compute the two-dimensional wavelet transform in
     non-standard form.

 -- Function: int gsl_wavelet2d_nstransform_matrix (const gsl_wavelet
          * w, gsl_matrix * m, gsl_wavelet_direction dir,
          gsl_wavelet_workspace * work)
 -- Function: int gsl_wavelet2d_nstransform_matrix_forward (const
          gsl_wavelet * w, gsl_matrix * m, gsl_wavelet_workspace * work)
 -- Function: int gsl_wavelet2d_nstransform_matrix_inverse (const
          gsl_wavelet * w, gsl_matrix * m, gsl_wavelet_workspace * work)

     These functions compute the non-standard form of the
     two-dimensional in-place wavelet transform on a matrix ‘m’.


File: gsl-ref.info,  Node: Examples<26>,  Next: References and Further Reading<27>,  Prev: Transform Functions,  Up: Wavelet Transforms

34.4 Examples
=============

The following program demonstrates the use of the one-dimensional
wavelet transform functions.  It computes an approximation to an input
signal (of length 256) using the 20 largest components of the wavelet
transform, while setting the others to zero.

     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_sort.h>
     #include <gsl/gsl_wavelet.h>

     int
     main (int argc, char **argv)
     {
       (void)(argc); /* avoid unused parameter warning */
       int i, n = 256, nc = 20;
       double *orig_data = malloc (n * sizeof (double));
       double *data = malloc (n * sizeof (double));
       double *abscoeff = malloc (n * sizeof (double));
       size_t *p = malloc (n * sizeof (size_t));

       FILE * f;
       gsl_wavelet *w;
       gsl_wavelet_workspace *work;

       w = gsl_wavelet_alloc (gsl_wavelet_daubechies, 4);
       work = gsl_wavelet_workspace_alloc (n);

       f = fopen (argv[1], "r");
       for (i = 0; i < n; i++)
         {
           fscanf (f, "%lg", &orig_data[i]);
           data[i] = orig_data[i];
         }
       fclose (f);

       gsl_wavelet_transform_forward (w, data, 1, n, work);

       for (i = 0; i < n; i++)
         {
           abscoeff[i] = fabs (data[i]);
         }

       gsl_sort_index (p, abscoeff, 1, n);

       for (i = 0; (i + nc) < n; i++)
         data[p[i]] = 0;

       gsl_wavelet_transform_inverse (w, data, 1, n, work);

       for (i = 0; i < n; i++)
         {
           printf ("%g %g\n", orig_data[i], data[i]);
         }

       gsl_wavelet_free (w);
       gsl_wavelet_workspace_free (work);

       free (data);
       free (orig_data);
       free (abscoeff);
       free (p);
       return 0;
     }

The output can be used with the GNU plotutils ‘graph’ program:

     $ ./a.out ecg.dat > dwt.txt
     $ graph -T ps -x 0 256 32 -h 0.3 -a dwt.txt > dwt.ps

Fig.  %s shows an original and compressed version of a sample ECG
recording from the MIT-BIH Arrhythmia Database, part of the PhysioNet
archive of public-domain of medical datasets.

[image src="dwt.png"]

Figure: Original (upper) and wavelet-compressed (lower) ECG signals,
using the 20 largest components of the Daubechies(4) discrete wavelet
transform.


File: gsl-ref.info,  Node: References and Further Reading<27>,  Prev: Examples<26>,  Up: Wavelet Transforms

34.5 References and Further Reading
===================================

The mathematical background to wavelet transforms is covered in the
original lectures by Daubechies,

   * Ingrid Daubechies.  Ten Lectures on Wavelets.  `CBMS-NSF Regional
     Conference Series in Applied Mathematics' (1992), SIAM, ISBN
     0898712742.

An easy to read introduction to the subject with an emphasis on the
application of the wavelet transform in various branches of science is,

   * Paul S. Addison.  `The Illustrated Wavelet Transform Handbook'.
     Institute of Physics Publishing (2002), ISBN 0750306920.

For extensive coverage of signal analysis by wavelets, wavelet packets
and local cosine bases see,

   * S. G. Mallat.  `A wavelet tour of signal processing' (Second
     edition).  Academic Press (1999), ISBN 012466606X.

The concept of multiresolution analysis underlying the wavelet transform
is described in,

   * S. G. Mallat.  Multiresolution Approximations and Wavelet
     Orthonormal Bases of L^2(R). `Transactions of the American
     Mathematical Society', 315(1), 1989, 69–87.

   * S. G. Mallat.  A Theory for Multiresolution Signal
     Decomposition—The Wavelet Representation.  `IEEE Transactions on
     Pattern Analysis and Machine Intelligence', 11, 1989, 674–693.

The coefficients for the individual wavelet families implemented by the
library can be found in the following papers,

   * I. Daubechies.  Orthonormal Bases of Compactly Supported Wavelets.
     `Communications on Pure and Applied Mathematics', 41 (1988)
     909–996.

   * A. Cohen, I. Daubechies, and J.-C. Feauveau.  Biorthogonal Bases of
     Compactly Supported Wavelets.  `Communications on Pure and Applied
     Mathematics', 45 (1992) 485–560.

The PhysioNet archive of physiological datasets can be found online at
‘http://www.physionet.org/’ and is described in the following paper,

   * Goldberger et al.  PhysioBank, PhysioToolkit, and PhysioNet:
     Components of a New Research Resource for Complex Physiologic
     Signals.  `Circulation' 101(23):e215-e220 2000.


File: gsl-ref.info,  Node: Discrete Hankel Transforms,  Next: One Dimensional Root-Finding,  Prev: Wavelet Transforms,  Up: Top

35 Discrete Hankel Transforms
*****************************

This chapter describes functions for performing Discrete Hankel
Transforms (DHTs).  The functions are declared in the header file
‘gsl_dht.h’.

* Menu:

* Definitions: Definitions<3>.
* Functions: Functions<2>.
* References and Further Reading: References and Further Reading<28>.


File: gsl-ref.info,  Node: Definitions<3>,  Next: Functions<2>,  Up: Discrete Hankel Transforms

35.1 Definitions
================

The discrete Hankel transform acts on a vector of sampled data, where
the samples are assumed to have been taken at points related to the
zeros of a Bessel function of fixed order; compare this to the case of
the discrete Fourier transform, where samples are taken at points
related to the zeroes of the sine or cosine function.

Starting with its definition, the Hankel transform (or Bessel transform)
of order \nu of a function f with \nu > -1/2 is defined as (see Johnson,
1987 and Lemoine, 1994)

     F_\nu(u) = \int_0^\infty f(t) J_\nu(u t) t dt

If the integral exists, F_\nu is called the Hankel transformation of f.
The reverse transform is given by

     f(t) = \int_0^\infty F_\nu(u) J_\nu(u t) u du

where \int_0^\infty f(t) t^{1/2} dt must exist and be absolutely
convergent, and where f(t) satisfies Dirichlet’s conditions (of limited
total fluctuations) in the interval [0,\infty].

Now the discrete Hankel transform works on a discrete function f, which
is sampled on points n=1...M located at positions
t_n=(j_{\nu,n}/j_{\nu,M}) X in real space and at u_n=j_{\nu,n}/X in
reciprocal space.  Here, j_{\nu,m} are the m-th zeros of the Bessel
function J_\nu(x) arranged in ascending order.  Moreover, the discrete
functions are assumed to be band limited, so f(t_n)=0 and F(u_n)=0 for
n>M. Accordingly, the function f is defined on the interval [0,X].

Following the work of Johnson, 1987 and Lemoine, 1994, the discrete
Hankel transform is given by

     F_\nu(u_m) = (2 X^2 / j_(\nu,M)^2)
           \sum_{k=1}^{M-1} f(j_(\nu,k) X/j_(\nu,M))
               (J_\nu(j_(\nu,m) j_(\nu,k) / j_(\nu,M)) / J_(\nu+1)(j_(\nu,k))^2).

It is this discrete expression which defines the discrete Hankel
transform calculated by GSL. In GSL, forward and backward transforms are
defined equally and calculate F_\nu(u_m).  Following Johnson, the
backward transform reads

     f(t_k) = (2 / X^2)
           \sum_{m=1}^{M-1} F(j_(\nu,m)/X)
               (J_\nu(j_(\nu,m) j_(\nu,k) / j_(\nu,M)) / J_(\nu+1)(j_(\nu,m))^2).

Obviously, using the forward transform instead of the backward transform
gives an additional factor X^4/j_{\nu,M}^2=t_m^2/u_m^2.

The kernel in the summation above defines the matrix of the \nu-Hankel
transform of size M-1.  The coefficients of this matrix, being dependent
on \nu and M, must be precomputed and stored; the *note gsl_dht: a51.
object encapsulates this data.  The allocation function *note
gsl_dht_alloc(): a52. returns a *note gsl_dht: a51. object which must be
properly initialized with *note gsl_dht_init(): a53. before it can be
used to perform transforms on data sample vectors, for fixed \nu and M,
using the *note gsl_dht_apply(): a54. function.  The implementation
allows to define the length X of the fundamental interval, for
convenience, while discrete Hankel transforms are often defined on the
unit interval instead of [0,X].

Notice that by assumption f(t) vanishes at the endpoints of the
interval, consistent with the inversion formula and the sampling formula
given above.  Therefore, this transform corresponds to an orthogonal
expansion in eigenfunctions of the Dirichlet problem for the Bessel
differential equation.


File: gsl-ref.info,  Node: Functions<2>,  Next: References and Further Reading<28>,  Prev: Definitions<3>,  Up: Discrete Hankel Transforms

35.2 Functions
==============

 -- Type: gsl_dht

     Workspace for computing discrete Hankel transforms

 -- Function: gsl_dht * gsl_dht_alloc (size_t size)

     This function allocates a Discrete Hankel transform object of size
     ‘size’.

 -- Function: int gsl_dht_init (gsl_dht * t, double nu, double xmax)

     This function initializes the transform ‘t’ for the given values of
     ‘nu’ and ‘xmax’.

 -- Function: gsl_dht * gsl_dht_new (size_t size, double nu,
          double xmax)

     This function allocates a Discrete Hankel transform object of size
     ‘size’ and initializes it for the given values of ‘nu’ and ‘xmax’.

 -- Function: void gsl_dht_free (gsl_dht * t)

     This function frees the transform ‘t’.

 -- Function: int gsl_dht_apply (const gsl_dht * t, double * f_in,
          double * f_out)

     This function applies the transform ‘t’ to the array ‘f_in’ whose
     size is equal to the size of the transform.  The result is stored
     in the array ‘f_out’ which must be of the same length.

     Applying this function to its output gives the original data
     multiplied by (X^2/j_{\nu,M})^2, up to numerical errors.

 -- Function: double gsl_dht_x_sample (const gsl_dht * t, int n)

     This function returns the value of the ‘n’-th sample point in the
     unit interval, {({j_{\nu,n+1}} / {j_{\nu,M}}}) X. These are the
     points where the function f(t) is assumed to be sampled.

 -- Function: double gsl_dht_k_sample (const gsl_dht * t, int n)

     This function returns the value of the ‘n’-th sample point in
     “k-space”, {{j_{\nu,n+1}} / X}.


File: gsl-ref.info,  Node: References and Further Reading<28>,  Prev: Functions<2>,  Up: Discrete Hankel Transforms

35.3 References and Further Reading
===================================

The algorithms used by these functions are described in the following
papers,

   * 
       8. Fisk Johnson, Comp.: Phys.: Comm.: 43, 181 (1987).

   * 
       4. Lemoine, J. Chem.: Phys.: 101, 3936 (1994).


File: gsl-ref.info,  Node: One Dimensional Root-Finding,  Next: One Dimensional Minimization,  Prev: Discrete Hankel Transforms,  Up: Top

36 One Dimensional Root-Finding
*******************************

This chapter describes routines for finding roots of arbitrary
one-dimensional functions.  The library provides low level components
for a variety of iterative solvers and convergence tests.  These can be
combined by the user to achieve the desired solution, with full access
to the intermediate steps of the iteration.  Each class of methods uses
the same framework, so that you can switch between solvers at runtime
without needing to recompile your program.  Each instance of a solver
keeps track of its own state, allowing the solvers to be used in
multi-threaded programs.

The header file ‘gsl_roots.h’ contains prototypes for the root finding
functions and related declarations.

* Menu:

* Overview::
* Caveats::
* Initializing the Solver::
* Providing the function to solve::
* Search Bounds and Guesses::
* Iteration::
* Search Stopping Parameters::
* Root Bracketing Algorithms::
* Root Finding Algorithms using Derivatives::
* Examples: Examples<27>.
* References and Further Reading: References and Further Reading<29>.


File: gsl-ref.info,  Node: Overview,  Next: Caveats,  Up: One Dimensional Root-Finding

36.1 Overview
=============

One-dimensional root finding algorithms can be divided into two classes,
`root bracketing' and `root polishing'.  Algorithms which proceed by
bracketing a root are guaranteed to converge.  Bracketing algorithms
begin with a bounded region known to contain a root.  The size of this
bounded region is reduced, iteratively, until it encloses the root to a
desired tolerance.  This provides a rigorous error estimate for the
location of the root.

The technique of `root polishing' attempts to improve an initial guess
to the root.  These algorithms converge only if started “close enough”
to a root, and sacrifice a rigorous error bound for speed.  By
approximating the behavior of a function in the vicinity of a root they
attempt to find a higher order improvement of an initial guess.  When
the behavior of the function is compatible with the algorithm and a good
initial guess is available a polishing algorithm can provide rapid
convergence.

In GSL both types of algorithm are available in similar frameworks.  The
user provides a high-level driver for the algorithms, and the library
provides the individual functions necessary for each of the steps.
There are three main phases of the iteration.  The steps are,

   * initialize solver state, ‘s’, for algorithm ‘T’

   * update ‘s’ using the iteration ‘T’

   * test ‘s’ for convergence, and repeat iteration if necessary

The state for bracketing solvers is held in a *note gsl_root_fsolver:
a5e. struct.  The updating procedure uses only function evaluations (not
derivatives).  The state for root polishing solvers is held in a *note
gsl_root_fdfsolver: a5f. struct.  The updates require both the function
and its derivative (hence the name ‘fdf’) to be supplied by the user.


File: gsl-ref.info,  Node: Caveats,  Next: Initializing the Solver,  Prev: Overview,  Up: One Dimensional Root-Finding

36.2 Caveats
============

Note that root finding functions can only search for one root at a time.
When there are several roots in the search area, the first root to be
found will be returned; however it is difficult to predict which of the
roots this will be.  `In most cases, no error will be reported if you
try to find a root in an area where there is more than one.'

Care must be taken when a function may have a multiple root (such as
f(x) = (x-x_0)^2 or f(x) = (x-x_0)^3.  It is not possible to use
root-bracketing algorithms on even-multiplicity roots.  For these
algorithms the initial interval must contain a zero-crossing, where the
function is negative at one end of the interval and positive at the
other end.  Roots with even-multiplicity do not cross zero, but only
touch it instantaneously.  Algorithms based on root bracketing will
still work for odd-multiplicity roots (e.g.  cubic, quintic, …).  Root
polishing algorithms generally work with higher multiplicity roots, but
at a reduced rate of convergence.  In these cases the `Steffenson
algorithm' can be used to accelerate the convergence of multiple roots.

While it is not absolutely required that f have a root within the search
region, numerical root finding functions should not be used haphazardly
to check for the `existence' of roots.  There are better ways to do
this.  Because it is easy to create situations where numerical root
finders can fail, it is a bad idea to throw a root finder at a function
you do not know much about.  In general it is best to examine the
function visually by plotting before searching for a root.


File: gsl-ref.info,  Node: Initializing the Solver,  Next: Providing the function to solve,  Prev: Caveats,  Up: One Dimensional Root-Finding

36.3 Initializing the Solver
============================

 -- Type: gsl_root_fsolver

     This is a workspace for finding roots using methods which do not
     require derivatives.

 -- Type: gsl_root_fdfsolver

     This is a workspace for finding roots using methods which require
     derivatives.

 -- Function: gsl_root_fsolver * gsl_root_fsolver_alloc (const
          gsl_root_fsolver_type * T)

     This function returns a pointer to a newly allocated instance of a
     solver of type ‘T’.  For example, the following code creates an
     instance of a bisection solver:

          const gsl_root_fsolver_type * T = gsl_root_fsolver_bisection;
          gsl_root_fsolver * s = gsl_root_fsolver_alloc (T);

     If there is insufficient memory to create the solver then the
     function returns a null pointer and the error handler is invoked
     with an error code of *note GSL_ENOMEM: 2a.

 -- Function: gsl_root_fdfsolver * gsl_root_fdfsolver_alloc (const
          gsl_root_fdfsolver_type * T)

     This function returns a pointer to a newly allocated instance of a
     derivative-based solver of type ‘T’.  For example, the following
     code creates an instance of a Newton-Raphson solver:

          const gsl_root_fdfsolver_type * T = gsl_root_fdfsolver_newton;
          gsl_root_fdfsolver * s = gsl_root_fdfsolver_alloc (T);

     If there is insufficient memory to create the solver then the
     function returns a null pointer and the error handler is invoked
     with an error code of *note GSL_ENOMEM: 2a.

 -- Function: int gsl_root_fsolver_set (gsl_root_fsolver * s,
          gsl_function * f, double x_lower, double x_upper)

     This function initializes, or reinitializes, an existing solver ‘s’
     to use the function ‘f’ and the initial search interval [‘x_lower’,
     ‘x_upper’].

 -- Function: int gsl_root_fdfsolver_set (gsl_root_fdfsolver * s,
          gsl_function_fdf * fdf, double root)

     This function initializes, or reinitializes, an existing solver ‘s’
     to use the function and derivative ‘fdf’ and the initial guess
     ‘root’.

 -- Function: void gsl_root_fsolver_free (gsl_root_fsolver * s)
 -- Function: void gsl_root_fdfsolver_free (gsl_root_fdfsolver * s)

     These functions free all the memory associated with the solver ‘s’.

 -- Function: const char * gsl_root_fsolver_name (const gsl_root_fsolver
          * s)
 -- Function: const char * gsl_root_fdfsolver_name (const
          gsl_root_fdfsolver * s)

     These functions return a pointer to the name of the solver.  For
     example:

          printf ("s is a '%s' solver\n", gsl_root_fsolver_name (s));

     would print something like ‘s is a 'bisection' solver’.


File: gsl-ref.info,  Node: Providing the function to solve,  Next: Search Bounds and Guesses,  Prev: Initializing the Solver,  Up: One Dimensional Root-Finding

36.4 Providing the function to solve
====================================

You must provide a continuous function of one variable for the root
finders to operate on, and, sometimes, its first derivative.  In order
to allow for general parameters the functions are defined by the
following data types:

 -- Type: gsl_function

     This data type defines a general function with parameters.

     ‘double (* function) (double x, void * params)’

          this function should return the value f(x,params) for argument
          ‘x’ and parameters ‘params’

     ‘void * params’

          a pointer to the parameters of the function

Here is an example for the general quadratic function,

     f(x) = a x^2 + b x + c

with a = 3, b = 2, c = 1.  The following code defines a *note
gsl_function: a6c. ‘F’ which you could pass to a root finder as a
function pointer:

     struct my_f_params { double a; double b; double c; };

     double
     my_f (double x, void * p)
       {
         struct my_f_params * params = (struct my_f_params *)p;
         double a = (params->a);
         double b = (params->b);
         double c = (params->c);

         return  (a * x + b) * x + c;
       }

     gsl_function F;
     struct my_f_params params = { 3.0, 2.0, 1.0 };

     F.function = &my_f;
     F.params = &params;

The function f(x) can be evaluated using the macro ‘GSL_FN_EVAL(&F,x)’
defined in ‘gsl_math.h’.

 -- Type: gsl_function_fdf

     This data type defines a general function with parameters and its
     first derivative.

     ‘double (* f) (double x, void * params)’

          this function should return the value of f(x,params) for
          argument ‘x’ and parameters ‘params’

     ‘double (* df) (double x, void * params)’

          this function should return the value of the derivative of ‘f’
          with respect to ‘x’, f'(x,params), for argument ‘x’ and
          parameters ‘params’

     ‘void (* fdf) (double x, void * params, double * f, double * df)’

          this function should set the values of the function ‘f’ to
          f(x,params) and its derivative ‘df’ to f'(x,params) for
          argument ‘x’ and parameters ‘params’.  This function provides
          an optimization of the separate functions for f(x) and
          f'(x)—it is always faster to compute the function and its
          derivative at the same time.

     ‘void * params’

          a pointer to the parameters of the function

Here is an example where f(x) = \exp(2x):

     double
     my_f (double x, void * params)
     {
        return exp (2 * x);
     }

     double
     my_df (double x, void * params)
     {
        return 2 * exp (2 * x);
     }

     void
     my_fdf (double x, void * params,
             double * f, double * df)
     {
        double t = exp (2 * x);

        *f = t;
        *df = 2 * t;   /* uses existing value */
     }

     gsl_function_fdf FDF;

     FDF.f = &my_f;
     FDF.df = &my_df;
     FDF.fdf = &my_fdf;
     FDF.params = 0;

The function f(x) can be evaluated using the macro
‘GSL_FN_FDF_EVAL_F(&FDF,x)’ and the derivative f'(x) can be evaluated
using the macro ‘GSL_FN_FDF_EVAL_DF(&FDF,x)’.  Both the function y =
f(x) and its derivative dy = f'(x) can be evaluated at the same time
using the macro ‘GSL_FN_FDF_EVAL_F_DF(&FDF,x,y,dy)’.  The macro stores
f(x) in its ‘y’ argument and f'(x) in its ‘dy’ argument—both of these
should be pointers to ‘double’.


File: gsl-ref.info,  Node: Search Bounds and Guesses,  Next: Iteration,  Prev: Providing the function to solve,  Up: One Dimensional Root-Finding

36.5 Search Bounds and Guesses
==============================

You provide either search bounds or an initial guess; this section
explains how search bounds and guesses work and how function arguments
control them.

A guess is simply an x value which is iterated until it is within the
desired precision of a root.  It takes the form of a ‘double’.

Search bounds are the endpoints of an interval which is iterated until
the length of the interval is smaller than the requested precision.  The
interval is defined by two values, the lower limit and the upper limit.
Whether the endpoints are intended to be included in the interval or not
depends on the context in which the interval is used.


File: gsl-ref.info,  Node: Iteration,  Next: Search Stopping Parameters,  Prev: Search Bounds and Guesses,  Up: One Dimensional Root-Finding

36.6 Iteration
==============

The following functions drive the iteration of each algorithm.  Each
function performs one iteration to update the state of any solver of the
corresponding type.  The same functions work for all solvers so that
different methods can be substituted at runtime without modifications to
the code.

 -- Function: int gsl_root_fsolver_iterate (gsl_root_fsolver * s)
 -- Function: int gsl_root_fdfsolver_iterate (gsl_root_fdfsolver * s)

     These functions perform a single iteration of the solver ‘s’.  If
     the iteration encounters an unexpected problem then an error code
     will be returned,

     ‘GSL_EBADFUNC’

          the iteration encountered a singular point where the function
          or its derivative evaluated to ‘Inf’ or ‘NaN’.

     ‘GSL_EZERODIV’

          the derivative of the function vanished at the iteration
          point, preventing the algorithm from continuing without a
          division by zero.

The solver maintains a current best estimate of the root at all times.
The bracketing solvers also keep track of the current best interval
bounding the root.  This information can be accessed with the following
auxiliary functions,

 -- Function: double gsl_root_fsolver_root (const gsl_root_fsolver * s)
 -- Function: double gsl_root_fdfsolver_root (const gsl_root_fdfsolver
          * s)

     These functions return the current estimate of the root for the
     solver ‘s’.

 -- Function: double gsl_root_fsolver_x_lower (const gsl_root_fsolver
          * s)
 -- Function: double gsl_root_fsolver_x_upper (const gsl_root_fsolver
          * s)

     These functions return the current bracketing interval for the
     solver ‘s’.


File: gsl-ref.info,  Node: Search Stopping Parameters,  Next: Root Bracketing Algorithms,  Prev: Iteration,  Up: One Dimensional Root-Finding

36.7 Search Stopping Parameters
===============================

A root finding procedure should stop when one of the following
conditions is true:

   * A root has been found to within the user-specified precision.

   * A user-specified maximum number of iterations has been reached.

   * An error has occurred.

The handling of these conditions is under user control.  The functions
below allow the user to test the precision of the current result in
several standard ways.

 -- Function: int gsl_root_test_interval (double x_lower,
          double x_upper, double epsabs, double epsrel)

     This function tests for the convergence of the interval [‘x_lower’,
     ‘x_upper’] with absolute error ‘epsabs’ and relative error
     ‘epsrel’.  The test returns ‘GSL_SUCCESS’ if the following
     condition is achieved,

          |a - b| < epsabs + epsrel min(|a|,|b|)

     when the interval x = [a,b] does not include the origin.  If the
     interval includes the origin then \min(|a|,|b|) is replaced by zero
     (which is the minimum value of |x| over the interval).  This
     ensures that the relative error is accurately estimated for roots
     close to the origin.

     This condition on the interval also implies that any estimate of
     the root r in the interval satisfies the same condition with
     respect to the true root r^*,

          |r - r^*| < epsabs + epsrel r^*

     assuming that the true root r^* is contained within the interval.

 -- Function: int gsl_root_test_delta (double x1, double x0,
          double epsabs, double epsrel)

     This function tests for the convergence of the sequence ‘x0’, ‘x1’
     with absolute error ‘epsabs’ and relative error ‘epsrel’.  The test
     returns ‘GSL_SUCCESS’ if the following condition is achieved,

          |x_1 - x_0| < epsabs + epsrel |x_1|

     and returns ‘GSL_CONTINUE’ otherwise.

 -- Function: int gsl_root_test_residual (double f, double epsabs)

     This function tests the residual value ‘f’ against the absolute
     error bound ‘epsabs’.  The test returns ‘GSL_SUCCESS’ if the
     following condition is achieved,

          |f| < epsabs

     and returns ‘GSL_CONTINUE’ otherwise.  This criterion is suitable
     for situations where the precise location of the root, x, is
     unimportant provided a value can be found where the residual,
     |f(x)|, is small enough.


File: gsl-ref.info,  Node: Root Bracketing Algorithms,  Next: Root Finding Algorithms using Derivatives,  Prev: Search Stopping Parameters,  Up: One Dimensional Root-Finding

36.8 Root Bracketing Algorithms
===============================

The root bracketing algorithms described in this section require an
initial interval which is guaranteed to contain a root—if a and b are
the endpoints of the interval then f(a) must differ in sign from f(b).
This ensures that the function crosses zero at least once in the
interval.  If a valid initial interval is used then these algorithm
cannot fail, provided the function is well-behaved.

Note that a bracketing algorithm cannot find roots of even degree, since
these do not cross the x-axis.

 -- Type: gsl_root_fsolver_type

      -- Variable: gsl_root_fsolver_bisection

          The `bisection algorithm' is the simplest method of bracketing
          the roots of a function.  It is the slowest algorithm provided
          by the library, with linear convergence.

          On each iteration, the interval is bisected and the value of
          the function at the midpoint is calculated.  The sign of this
          value is used to determine which half of the interval does not
          contain a root.  That half is discarded to give a new, smaller
          interval containing the root.  This procedure can be continued
          indefinitely until the interval is sufficiently small.

          At any time the current estimate of the root is taken as the
          midpoint of the interval.

      -- Variable: gsl_root_fsolver_falsepos

          The `false position algorithm' is a method of finding roots
          based on linear interpolation.  Its convergence is linear, but
          it is usually faster than bisection.

          On each iteration a line is drawn between the endpoints
          (a,f(a)) and (b,f(b)) and the point where this line crosses
          the x-axis taken as a “midpoint”.  The value of the function
          at this point is calculated and its sign is used to determine
          which side of the interval does not contain a root.  That side
          is discarded to give a new, smaller interval containing the
          root.  This procedure can be continued indefinitely until the
          interval is sufficiently small.

          The best estimate of the root is taken from the linear
          interpolation of the interval on the current iteration.

      -- Variable: gsl_root_fsolver_brent

          The `Brent-Dekker method' (referred to here as `Brent’s
          method') combines an interpolation strategy with the bisection
          algorithm.  This produces a fast algorithm which is still
          robust.

          On each iteration Brent’s method approximates the function
          using an interpolating curve.  On the first iteration this is
          a linear interpolation of the two endpoints.  For subsequent
          iterations the algorithm uses an inverse quadratic fit to the
          last three points, for higher accuracy.  The intercept of the
          interpolating curve with the x-axis is taken as a guess for
          the root.  If it lies within the bounds of the current
          interval then the interpolating point is accepted, and used to
          generate a smaller interval.  If the interpolating point is
          not accepted then the algorithm falls back to an ordinary
          bisection step.

          The best estimate of the root is taken from the most recent
          interpolation or bisection.


File: gsl-ref.info,  Node: Root Finding Algorithms using Derivatives,  Next: Examples<27>,  Prev: Root Bracketing Algorithms,  Up: One Dimensional Root-Finding

36.9 Root Finding Algorithms using Derivatives
==============================================

The root polishing algorithms described in this section require an
initial guess for the location of the root.  There is no absolute
guarantee of convergence—the function must be suitable for this
technique and the initial guess must be sufficiently close to the root
for it to work.  When these conditions are satisfied then convergence is
quadratic.

These algorithms make use of both the function and its derivative.

 -- Type: gsl_root_fdfsolver_type

      -- Variable: gsl_root_fdfsolver_newton

          Newton’s Method is the standard root-polishing algorithm.  The
          algorithm begins with an initial guess for the location of the
          root.  On each iteration, a line tangent to the function f is
          drawn at that position.  The point where this line crosses the
          x-axis becomes the new guess.  The iteration is defined by the
          following sequence,

               x_{i+1} = x_i - f(x_i)/f'(x_i)

          Newton’s method converges quadratically for single roots, and
          linearly for multiple roots.

      -- Variable: gsl_root_fdfsolver_secant

          The `secant method' is a simplified version of Newton’s method
          which does not require the computation of the derivative on
          every step.

          On its first iteration the algorithm begins with Newton’s
          method, using the derivative to compute a first step,

               x_1 = x_0 - f(x_0)/f'(x_0)

          Subsequent iterations avoid the evaluation of the derivative
          by replacing it with a numerical estimate, the slope of the
          line through the previous two points,

               x_{i+1} = x_i f(x_i) / f'_{est} where
                f'_{est} = (f(x_i) - f(x_{i-1})/(x_i - x_{i-1})

          When the derivative does not change significantly in the
          vicinity of the root the secant method gives a useful saving.
          Asymptotically the secant method is faster than Newton’s
          method whenever the cost of evaluating the derivative is more
          than 0.44 times the cost of evaluating the function itself.
          As with all methods of computing a numerical derivative the
          estimate can suffer from cancellation errors if the separation
          of the points becomes too small.

          On single roots, the method has a convergence of order (1 +
          \sqrt 5)/2 (approximately 1.62).  It converges linearly for
          multiple roots.

      -- Variable: gsl_root_fdfsolver_steffenson

          The `Steffenson Method' (1) provides the fastest convergence
          of all the routines.  It combines the basic Newton algorithm
          with an Aitken “delta-squared” acceleration.  If the Newton
          iterates are x_i then the acceleration procedure generates a
          new sequence R_i,

               R_i = x_i - (x_{i+1} - x_i)^2 / (x_{i+2} - 2 x_{i+1} + x_{i})

          which converges faster than the original sequence under
          reasonable conditions.  The new sequence requires three terms
          before it can produce its first value so the method returns
          accelerated values on the second and subsequent iterations.
          On the first iteration it returns the ordinary Newton
          estimate.  The Newton iterate is also returned if the
          denominator of the acceleration term ever becomes zero.

          As with all acceleration procedures this method can become
          unstable if the function is not well-behaved.

   ---------- Footnotes ----------

   (1) J.F. Steffensen (1873–1961).  The spelling used in the name of
the function is slightly incorrect, but has been preserved to avoid
incompatibility.


File: gsl-ref.info,  Node: Examples<27>,  Next: References and Further Reading<29>,  Prev: Root Finding Algorithms using Derivatives,  Up: One Dimensional Root-Finding

36.10 Examples
==============

For any root finding algorithm we need to prepare the function to be
solved.  For this example we will use the general quadratic equation
described earlier.  We first need a header file (‘demo_fn.h’) to define
the function parameters,

     struct quadratic_params
       {
         double a, b, c;
       };

     double quadratic (double x, void *params);
     double quadratic_deriv (double x, void *params);
     void quadratic_fdf (double x, void *params,
                         double *y, double *dy);

We place the function definitions in a separate file (‘demo_fn.c’),

     double
     quadratic (double x, void *params)
     {
       struct quadratic_params *p
         = (struct quadratic_params *) params;

       double a = p->a;
       double b = p->b;
       double c = p->c;

       return (a * x + b) * x + c;
     }

     double
     quadratic_deriv (double x, void *params)
     {
       struct quadratic_params *p
         = (struct quadratic_params *) params;

       double a = p->a;
       double b = p->b;

       return 2.0 * a * x + b;
     }

     void
     quadratic_fdf (double x, void *params,
                    double *y, double *dy)
     {
       struct quadratic_params *p
         = (struct quadratic_params *) params;

       double a = p->a;
       double b = p->b;
       double c = p->c;

       *y = (a * x + b) * x + c;
       *dy = 2.0 * a * x + b;
     }

The first program uses the function solver *note gsl_root_fsolver_brent:
a7e. for Brent’s method and the general quadratic defined above to solve
the following equation,

     x^2 - 5 = 0

with solution x = \sqrt 5 = 2.236068...

     #include <stdio.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_roots.h>

     #include "demo_fn.h"
     #include "demo_fn.c"

     int
     main (void)
     {
       int status;
       int iter = 0, max_iter = 100;
       const gsl_root_fsolver_type *T;
       gsl_root_fsolver *s;
       double r = 0, r_expected = sqrt (5.0);
       double x_lo = 0.0, x_hi = 5.0;
       gsl_function F;
       struct quadratic_params params = {1.0, 0.0, -5.0};

       F.function = &quadratic;
       F.params = &params;

       T = gsl_root_fsolver_brent;
       s = gsl_root_fsolver_alloc (T);
       gsl_root_fsolver_set (s, &F, x_lo, x_hi);

       printf ("using %s method\n",
               gsl_root_fsolver_name (s));

       printf ("%5s [%9s, %9s] %9s %10s %9s\n",
               "iter", "lower", "upper", "root",
               "err", "err(est)");

       do
         {
           iter++;
           status = gsl_root_fsolver_iterate (s);
           r = gsl_root_fsolver_root (s);
           x_lo = gsl_root_fsolver_x_lower (s);
           x_hi = gsl_root_fsolver_x_upper (s);
           status = gsl_root_test_interval (x_lo, x_hi,
                                            0, 0.001);

           if (status == GSL_SUCCESS)
             printf ("Converged:\n");

           printf ("%5d [%.7f, %.7f] %.7f %+.7f %.7f\n",
                   iter, x_lo, x_hi,
                   r, r - r_expected,
                   x_hi - x_lo);
         }
       while (status == GSL_CONTINUE && iter < max_iter);

       gsl_root_fsolver_free (s);

       return status;
     }

Here are the results of the iterations:

     $ ./a.out
     using brent method
      iter [    lower,     upper]      root        err  err(est)
         1 [1.0000000, 5.0000000] 1.0000000 -1.2360680 4.0000000
         2 [1.0000000, 3.0000000] 3.0000000 +0.7639320 2.0000000
         3 [2.0000000, 3.0000000] 2.0000000 -0.2360680 1.0000000
         4 [2.2000000, 3.0000000] 2.2000000 -0.0360680 0.8000000
         5 [2.2000000, 2.2366300] 2.2366300 +0.0005621 0.0366300
     Converged:
         6 [2.2360634, 2.2366300] 2.2360634 -0.0000046 0.0005666

If the program is modified to use the bisection solver instead of
Brent’s method, by changing *note gsl_root_fsolver_brent: a7e. to *note
gsl_root_fsolver_bisection: a7c. the slower convergence of the Bisection
method can be observed:

     $ ./a.out
     using bisection method
      iter [    lower,     upper]      root        err  err(est)
         1 [0.0000000, 2.5000000] 1.2500000 -0.9860680 2.5000000
         2 [1.2500000, 2.5000000] 1.8750000 -0.3610680 1.2500000
         3 [1.8750000, 2.5000000] 2.1875000 -0.0485680 0.6250000
         4 [2.1875000, 2.5000000] 2.3437500 +0.1076820 0.3125000
         5 [2.1875000, 2.3437500] 2.2656250 +0.0295570 0.1562500
         6 [2.1875000, 2.2656250] 2.2265625 -0.0095055 0.0781250
         7 [2.2265625, 2.2656250] 2.2460938 +0.0100258 0.0390625
         8 [2.2265625, 2.2460938] 2.2363281 +0.0002601 0.0195312
         9 [2.2265625, 2.2363281] 2.2314453 -0.0046227 0.0097656
        10 [2.2314453, 2.2363281] 2.2338867 -0.0021813 0.0048828
        11 [2.2338867, 2.2363281] 2.2351074 -0.0009606 0.0024414
     Converged:
        12 [2.2351074, 2.2363281] 2.2357178 -0.0003502 0.0012207

The next program solves the same function using a derivative solver
instead.

     #include <stdio.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_roots.h>

     #include "demo_fn.h"
     #include "demo_fn.c"

     int
     main (void)
     {
       int status;
       int iter = 0, max_iter = 100;
       const gsl_root_fdfsolver_type *T;
       gsl_root_fdfsolver *s;
       double x0, x = 5.0, r_expected = sqrt (5.0);
       gsl_function_fdf FDF;
       struct quadratic_params params = {1.0, 0.0, -5.0};

       FDF.f = &quadratic;
       FDF.df = &quadratic_deriv;
       FDF.fdf = &quadratic_fdf;
       FDF.params = &params;

       T = gsl_root_fdfsolver_newton;
       s = gsl_root_fdfsolver_alloc (T);
       gsl_root_fdfsolver_set (s, &FDF, x);

       printf ("using %s method\n",
               gsl_root_fdfsolver_name (s));

       printf ("%-5s %10s %10s %10s\n",
               "iter", "root", "err", "err(est)");
       do
         {
           iter++;
           status = gsl_root_fdfsolver_iterate (s);
           x0 = x;
           x = gsl_root_fdfsolver_root (s);
           status = gsl_root_test_delta (x, x0, 0, 1e-3);

           if (status == GSL_SUCCESS)
             printf ("Converged:\n");

           printf ("%5d %10.7f %+10.7f %10.7f\n",
                   iter, x, x - r_expected, x - x0);
         }
       while (status == GSL_CONTINUE && iter < max_iter);

       gsl_root_fdfsolver_free (s);
       return status;
     }

Here are the results for Newton’s method:

     $ ./a.out
     using newton method
     iter        root        err   err(est)
         1  3.0000000 +0.7639320 -2.0000000
         2  2.3333333 +0.0972654 -0.6666667
         3  2.2380952 +0.0020273 -0.0952381
     Converged:
         4  2.2360689 +0.0000009 -0.0020263

Note that the error can be estimated more accurately by taking the
difference between the current iterate and next iterate rather than the
previous iterate.  The other derivative solvers can be investigated by
changing *note gsl_root_fdfsolver_newton: a81. to *note
gsl_root_fdfsolver_secant: a82. or *note gsl_root_fdfsolver_steffenson:
a83.


File: gsl-ref.info,  Node: References and Further Reading<29>,  Prev: Examples<27>,  Up: One Dimensional Root-Finding

36.11 References and Further Reading
====================================

For information on the Brent-Dekker algorithm see the following two
papers,

   * R. P. Brent, “An algorithm with guaranteed convergence for finding
     a zero of a function”, `Computer Journal', 14 (1971) 422–425

   * J. C. P. Bus and T. J. Dekker, “Two Efficient Algorithms with
     Guaranteed Convergence for Finding a Zero of a Function”, `ACM
     Transactions of Mathematical Software', Vol.: 1 No.: 4 (1975)
     330–345


File: gsl-ref.info,  Node: One Dimensional Minimization,  Next: Multidimensional Root-Finding,  Prev: One Dimensional Root-Finding,  Up: Top

37 One Dimensional Minimization
*******************************

This chapter describes routines for finding minima of arbitrary
one-dimensional functions.  The library provides low level components
for a variety of iterative minimizers and convergence tests.  These can
be combined by the user to achieve the desired solution, with full
access to the intermediate steps of the algorithms.  Each class of
methods uses the same framework, so that you can switch between
minimizers at runtime without needing to recompile your program.  Each
instance of a minimizer keeps track of its own state, allowing the
minimizers to be used in multi-threaded programs.

The header file ‘gsl_min.h’ contains prototypes for the minimization
functions and related declarations.  To use the minimization algorithms
to find the maximum of a function simply invert its sign.

* Menu:

* Overview: Overview<2>.
* Caveats: Caveats<2>.
* Initializing the Minimizer::
* Providing the function to minimize::
* Iteration: Iteration<2>.
* Stopping Parameters::
* Minimization Algorithms::
* Examples: Examples<28>.
* References and Further Reading: References and Further Reading<30>.


File: gsl-ref.info,  Node: Overview<2>,  Next: Caveats<2>,  Up: One Dimensional Minimization

37.1 Overview
=============

The minimization algorithms begin with a bounded region known to contain
a minimum.  The region is described by a lower bound a and an upper
bound b, with an estimate of the location of the minimum x, as shown in
Fig.  %s.

[image src="min-interval.png"]

Figure: Function with lower and upper bounds with an estimate of the
minimum.

The value of the function at x must be less than the value of the
function at the ends of the interval,

     f(a) > f(x) < f(b)

This condition guarantees that a minimum is contained somewhere within
the interval.  On each iteration a new point x' is selected using one of
the available algorithms.  If the new point is a better estimate of the
minimum, i.e.: where f(x') < f(x), then the current estimate of the
minimum x is updated.  The new point also allows the size of the bounded
interval to be reduced, by choosing the most compact set of points which
satisfies the constraint f(a) > f(x) < f(b).  The interval is reduced
until it encloses the true minimum to a desired tolerance.  This
provides a best estimate of the location of the minimum and a rigorous
error estimate.

Several bracketing algorithms are available within a single framework.
The user provides a high-level driver for the algorithm, and the library
provides the individual functions necessary for each of the steps.
There are three main phases of the iteration.  The steps are,

   * initialize minimizer state, ‘s’, for algorithm ‘T’

   * update ‘s’ using the iteration ‘T’

   * test ‘s’ for convergence, and repeat iteration if necessary

The state for the minimizers is held in a *note gsl_min_fminimizer: a8a.
struct.  The updating procedure uses only function evaluations (not
derivatives).


File: gsl-ref.info,  Node: Caveats<2>,  Next: Initializing the Minimizer,  Prev: Overview<2>,  Up: One Dimensional Minimization

37.2 Caveats
============

Note that minimization functions can only search for one minimum at a
time.  When there are several minima in the search area, the first
minimum to be found will be returned; however it is difficult to predict
which of the minima this will be.  `In most cases, no error will be
reported if you try to find a minimum in an area where there is more
than one.'

With all minimization algorithms it can be difficult to determine the
location of the minimum to full numerical precision.  The behavior of
the function in the region of the minimum x^* can be approximated by a
Taylor expansion,

     y = f(x^*) + (1/2) f''(x^*) (x - x^*)^2

and the second term of this expansion can be lost when added to the
first term at finite precision.  This magnifies the error in locating
x^*, making it proportional to \sqrt \epsilon (where \epsilon is the
relative accuracy of the floating point numbers).  For functions with
higher order minima, such as x^4, the magnification of the error is
correspondingly worse.  The best that can be achieved is to converge to
the limit of numerical accuracy in the function values, rather than the
location of the minimum itself.


File: gsl-ref.info,  Node: Initializing the Minimizer,  Next: Providing the function to minimize,  Prev: Caveats<2>,  Up: One Dimensional Minimization

37.3 Initializing the Minimizer
===============================

 -- Type: gsl_min_fminimizer

     This is a workspace for minimizing functions.

 -- Function: gsl_min_fminimizer * gsl_min_fminimizer_alloc (const
          gsl_min_fminimizer_type * T)

     This function returns a pointer to a newly allocated instance of a
     minimizer of type ‘T’.  For example, the following code creates an
     instance of a golden section minimizer:

          const gsl_min_fminimizer_type * T = gsl_min_fminimizer_goldensection;
          gsl_min_fminimizer * s = gsl_min_fminimizer_alloc (T);

     If there is insufficient memory to create the minimizer then the
     function returns a null pointer and the error handler is invoked
     with an error code of *note GSL_ENOMEM: 2a.

 -- Function: int gsl_min_fminimizer_set (gsl_min_fminimizer * s,
          gsl_function * f, double x_minimum, double x_lower,
          double x_upper)

     This function sets, or resets, an existing minimizer ‘s’ to use the
     function ‘f’ and the initial search interval [‘x_lower’,
     ‘x_upper’], with a guess for the location of the minimum
     ‘x_minimum’.

     If the interval given does not contain a minimum, then the function
     returns an error code of *note GSL_EINVAL: 2b.

 -- Function: int gsl_min_fminimizer_set_with_values (gsl_min_fminimizer
          * s, gsl_function * f, double x_minimum, double f_minimum,
          double x_lower, double f_lower, double x_upper,
          double f_upper)

     This function is equivalent to *note gsl_min_fminimizer_set(): a8e.
     but uses the values ‘f_minimum’, ‘f_lower’ and ‘f_upper’ instead of
     computing ‘f(x_minimum)’, ‘f(x_lower)’ and ‘f(x_upper)’.

 -- Function: void gsl_min_fminimizer_free (gsl_min_fminimizer * s)

     This function frees all the memory associated with the minimizer
     ‘s’.

 -- Function: const char * gsl_min_fminimizer_name (const
          gsl_min_fminimizer * s)

     This function returns a pointer to the name of the minimizer.  For
     example:

          printf ("s is a '%s' minimizer\n", gsl_min_fminimizer_name (s));

     would print something like ‘s is a 'brent' minimizer’.


File: gsl-ref.info,  Node: Providing the function to minimize,  Next: Iteration<2>,  Prev: Initializing the Minimizer,  Up: One Dimensional Minimization

37.4 Providing the function to minimize
=======================================

You must provide a continuous function of one variable for the
minimizers to operate on.  In order to allow for general parameters the
functions are defined by a *note gsl_function: a6c. data type (*note
Providing the function to solve: a6a.).


File: gsl-ref.info,  Node: Iteration<2>,  Next: Stopping Parameters,  Prev: Providing the function to minimize,  Up: One Dimensional Minimization

37.5 Iteration
==============

The following functions drive the iteration of each algorithm.  Each
function performs one iteration to update the state of any minimizer of
the corresponding type.  The same functions work for all minimizers so
that different methods can be substituted at runtime without
modifications to the code.

 -- Function: int gsl_min_fminimizer_iterate (gsl_min_fminimizer * s)

     This function performs a single iteration of the minimizer ‘s’.  If
     the iteration encounters an unexpected problem then an error code
     will be returned,

     ‘GSL_EBADFUNC’

          the iteration encountered a singular point where the function
          evaluated to ‘Inf’ or ‘NaN’.

     ‘GSL_FAILURE’

          the algorithm could not improve the current best approximation
          or bounding interval.

The minimizer maintains a current best estimate of the position of the
minimum at all times, and the current interval bounding the minimum.
This information can be accessed with the following auxiliary functions,

 -- Function: double gsl_min_fminimizer_x_minimum (const
          gsl_min_fminimizer * s)

     This function returns the current estimate of the position of the
     minimum for the minimizer ‘s’.

 -- Function: double gsl_min_fminimizer_x_upper (const
          gsl_min_fminimizer * s)
 -- Function: double gsl_min_fminimizer_x_lower (const
          gsl_min_fminimizer * s)

     These functions return the current upper and lower bound of the
     interval for the minimizer ‘s’.

 -- Function: double gsl_min_fminimizer_f_minimum (const
          gsl_min_fminimizer * s)
 -- Function: double gsl_min_fminimizer_f_upper (const
          gsl_min_fminimizer * s)
 -- Function: double gsl_min_fminimizer_f_lower (const
          gsl_min_fminimizer * s)

     These functions return the value of the function at the current
     estimate of the minimum and at the upper and lower bounds of the
     interval for the minimizer ‘s’.


File: gsl-ref.info,  Node: Stopping Parameters,  Next: Minimization Algorithms,  Prev: Iteration<2>,  Up: One Dimensional Minimization

37.6 Stopping Parameters
========================

A minimization procedure should stop when one of the following
conditions is true:

   * A minimum has been found to within the user-specified precision.

   * A user-specified maximum number of iterations has been reached.

   * An error has occurred.

The handling of these conditions is under user control.  The function
below allows the user to test the precision of the current result.

 -- Function: int gsl_min_test_interval (double x_lower, double x_upper,
          double epsabs, double epsrel)

     This function tests for the convergence of the interval [‘x_lower’,
     ‘x_upper’] with absolute error ‘epsabs’ and relative error
     ‘epsrel’.  The test returns ‘GSL_SUCCESS’ if the following
     condition is achieved,

          |a - b| < epsabs + epsrel min(|a|,|b|)

     when the interval x = [a,b] does not include the origin.  If the
     interval includes the origin then \min(|a|,|b|) is replaced by zero
     (which is the minimum value of |x| over the interval).  This
     ensures that the relative error is accurately estimated for minima
     close to the origin.

     This condition on the interval also implies that any estimate of
     the minimum x_m in the interval satisfies the same condition with
     respect to the true minimum x_m^*,

          |x_m - x_m^*| < epsabs + epsrel x_m^*

     assuming that the true minimum x_m^* is contained within the
     interval.


File: gsl-ref.info,  Node: Minimization Algorithms,  Next: Examples<28>,  Prev: Stopping Parameters,  Up: One Dimensional Minimization

37.7 Minimization Algorithms
============================

The minimization algorithms described in this section require an initial
interval which is guaranteed to contain a minimum—if a and b are the
endpoints of the interval and x is an estimate of the minimum then f(a)
> f(x) < f(b).  This ensures that the function has at least one minimum
somewhere in the interval.  If a valid initial interval is used then
these algorithm cannot fail, provided the function is well-behaved.

 -- Type: gsl_min_fminimizer_type

      -- Variable: gsl_min_fminimizer_goldensection

          The `golden section algorithm' is the simplest method of
          bracketing the minimum of a function.  It is the slowest
          algorithm provided by the library, with linear convergence.

          On each iteration, the algorithm first compares the
          subintervals from the endpoints to the current minimum.  The
          larger subinterval is divided in a golden section (using the
          famous ratio (3-\sqrt 5)/2 \approx 0.3819660 and the value of
          the function at this new point is calculated.  The new value
          is used with the constraint f(a') > f(x') < f(b') to a select
          new interval containing the minimum, by discarding the least
          useful point.  This procedure can be continued indefinitely
          until the interval is sufficiently small.  Choosing the golden
          section as the bisection ratio can be shown to provide the
          fastest convergence for this type of algorithm.

      -- Variable: gsl_min_fminimizer_brent

          The `Brent minimization algorithm' combines a parabolic
          interpolation with the golden section algorithm.  This
          produces a fast algorithm which is still robust.

          The outline of the algorithm can be summarized as follows: on
          each iteration Brent’s method approximates the function using
          an interpolating parabola through three existing points.  The
          minimum of the parabola is taken as a guess for the minimum.
          If it lies within the bounds of the current interval then the
          interpolating point is accepted, and used to generate a
          smaller interval.  If the interpolating point is not accepted
          then the algorithm falls back to an ordinary golden section
          step.  The full details of Brent’s method include some
          additional checks to improve convergence.

      -- Variable: gsl_min_fminimizer_quad_golden

          This is a variant of Brent’s algorithm which uses the
          safeguarded step-length algorithm of Gill and Murray.


File: gsl-ref.info,  Node: Examples<28>,  Next: References and Further Reading<30>,  Prev: Minimization Algorithms,  Up: One Dimensional Minimization

37.8 Examples
=============

The following program uses the Brent algorithm to find the minimum of
the function f(x) = \cos(x) + 1, which occurs at x = \pi.  The starting
interval is (0,6), with an initial guess for the minimum of 2.

     #include <stdio.h>
     #include <gsl/gsl_errno.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_min.h>

     double fn1 (double x, void * params)
     {
       (void)(params); /* avoid unused parameter warning */
       return cos(x) + 1.0;
     }

     int
     main (void)
     {
       int status;
       int iter = 0, max_iter = 100;
       const gsl_min_fminimizer_type *T;
       gsl_min_fminimizer *s;
       double m = 2.0, m_expected = M_PI;
       double a = 0.0, b = 6.0;
       gsl_function F;

       F.function = &fn1;
       F.params = 0;

       T = gsl_min_fminimizer_brent;
       s = gsl_min_fminimizer_alloc (T);
       gsl_min_fminimizer_set (s, &F, m, a, b);

       printf ("using %s method\n",
               gsl_min_fminimizer_name (s));

       printf ("%5s [%9s, %9s] %9s %10s %9s\n",
               "iter", "lower", "upper", "min",
               "err", "err(est)");

       printf ("%5d [%.7f, %.7f] %.7f %+.7f %.7f\n",
               iter, a, b,
               m, m - m_expected, b - a);

       do
         {
           iter++;
           status = gsl_min_fminimizer_iterate (s);

           m = gsl_min_fminimizer_x_minimum (s);
           a = gsl_min_fminimizer_x_lower (s);
           b = gsl_min_fminimizer_x_upper (s);

           status
             = gsl_min_test_interval (a, b, 0.001, 0.0);

           if (status == GSL_SUCCESS)
             printf ("Converged:\n");

           printf ("%5d [%.7f, %.7f] "
                   "%.7f %+.7f %.7f\n",
                   iter, a, b,
                   m, m - m_expected, b - a);
         }
       while (status == GSL_CONTINUE && iter < max_iter);

       gsl_min_fminimizer_free (s);

       return status;
     }

Here are the results of the minimization procedure.

     using brent method
      iter [    lower,     upper]       min        err  err(est)
         0 [0.0000000, 6.0000000] 2.0000000 -1.1415927 6.0000000
         1 [2.0000000, 6.0000000] 3.5278640 +0.3862713 4.0000000
         2 [2.0000000, 3.5278640] 3.1748217 +0.0332290 1.5278640
         3 [2.0000000, 3.1748217] 3.1264576 -0.0151351 1.1748217
         4 [3.1264576, 3.1748217] 3.1414743 -0.0001183 0.0483641
         5 [3.1414743, 3.1748217] 3.1415930 +0.0000004 0.0333474
     Converged:
         6 [3.1414743, 3.1415930] 3.1415927 +0.0000000 0.0001187


File: gsl-ref.info,  Node: References and Further Reading<30>,  Prev: Examples<28>,  Up: One Dimensional Minimization

37.9 References and Further Reading
===================================

Further information on Brent’s algorithm is available in the following
book,

   * Richard Brent, `Algorithms for minimization without derivatives',
     Prentice-Hall (1973), republished by Dover in paperback (2002),
     ISBN 0-486-41998-3.


File: gsl-ref.info,  Node: Multidimensional Root-Finding,  Next: Multidimensional Minimization,  Prev: One Dimensional Minimization,  Up: Top

38 Multidimensional Root-Finding
********************************

This chapter describes functions for multidimensional root-finding
(solving nonlinear systems with n equations in n unknowns).  The library
provides low level components for a variety of iterative solvers and
convergence tests.  These can be combined by the user to achieve the
desired solution, with full access to the intermediate steps of the
iteration.  Each class of methods uses the same framework, so that you
can switch between solvers at runtime without needing to recompile your
program.  Each instance of a solver keeps track of its own state,
allowing the solvers to be used in multi-threaded programs.  The solvers
are based on the original Fortran library MINPACK.

The header file ‘gsl_multiroots.h’ contains prototypes for the
multidimensional root finding functions and related declarations.

* Menu:

* Overview: Overview<3>.
* Initializing the Solver: Initializing the Solver<2>.
* Providing the function to solve: Providing the function to solve<2>.
* Iteration: Iteration<3>.
* Search Stopping Parameters: Search Stopping Parameters<2>.
* Algorithms using Derivatives::
* Algorithms without Derivatives::
* Examples: Examples<29>.
* References and Further Reading: References and Further Reading<31>.


File: gsl-ref.info,  Node: Overview<3>,  Next: Initializing the Solver<2>,  Up: Multidimensional Root-Finding

38.1 Overview
=============

The problem of multidimensional root finding requires the simultaneous
solution of n equations, f_i, in n variables, x_i,

     f_i (x_1, ..., x_n) = 0    for i = 1 ... n.

In general there are no bracketing methods available for n dimensional
systems, and no way of knowing whether any solutions exist.  All
algorithms proceed from an initial guess using a variant of the Newton
iteration,

     x -> x' = x - J^{-1} f(x)

where x, f are vector quantities and J is the Jacobian matrix J_{ij} =
\partial f_i / \partial x_j.  Additional strategies can be used to
enlarge the region of convergence.  These include requiring a decrease
in the norm |f| on each step proposed by Newton’s method, or taking
steepest-descent steps in the direction of the negative gradient of |f|.

Several root-finding algorithms are available within a single framework.
The user provides a high-level driver for the algorithms, and the
library provides the individual functions necessary for each of the
steps.  There are three main phases of the iteration.  The steps are,

   * initialize solver state, ‘s’, for algorithm ‘T’

   * update ‘s’ using the iteration ‘T’

   * test ‘s’ for convergence, and repeat iteration if necessary

The evaluation of the Jacobian matrix can be problematic, either because
programming the derivatives is intractable or because computation of the
n^2 terms of the matrix becomes too expensive.  For these reasons the
algorithms provided by the library are divided into two classes
according to whether the derivatives are available or not.

The state for solvers with an analytic Jacobian matrix is held in a
*note gsl_multiroot_fdfsolver: aa7. struct.  The updating procedure
requires both the function and its derivatives to be supplied by the
user.

The state for solvers which do not use an analytic Jacobian matrix is
held in a *note gsl_multiroot_fsolver: aa8. struct.  The updating
procedure uses only function evaluations (not derivatives).  The
algorithms estimate the matrix J or J^{-1} by approximate methods.


File: gsl-ref.info,  Node: Initializing the Solver<2>,  Next: Providing the function to solve<2>,  Prev: Overview<3>,  Up: Multidimensional Root-Finding

38.2 Initializing the Solver
============================

The following functions initialize a multidimensional solver, either
with or without derivatives.  The solver itself depends only on the
dimension of the problem and the algorithm and can be reused for
different problems.

 -- Type: gsl_multiroot_fsolver

     This is a workspace for multidimensional root-finding without
     derivatives.

 -- Type: gsl_multiroot_fdfsolver

     This is a workspace for multidimensional root-finding with
     derivatives.

 -- Function: gsl_multiroot_fsolver * gsl_multiroot_fsolver_alloc (const
          gsl_multiroot_fsolver_type * T, size_t n)

     This function returns a pointer to a newly allocated instance of a
     solver of type ‘T’ for a system of ‘n’ dimensions.  For example,
     the following code creates an instance of a hybrid solver, to solve
     a 3-dimensional system of equations:

          const gsl_multiroot_fsolver_type * T = gsl_multiroot_fsolver_hybrid;
          gsl_multiroot_fsolver * s = gsl_multiroot_fsolver_alloc (T, 3);

     If there is insufficient memory to create the solver then the
     function returns a null pointer and the error handler is invoked
     with an error code of *note GSL_ENOMEM: 2a.

 -- Function: gsl_multiroot_fdfsolver * gsl_multiroot_fdfsolver_alloc
          (const gsl_multiroot_fdfsolver_type * T, size_t n)

     This function returns a pointer to a newly allocated instance of a
     derivative solver of type ‘T’ for a system of ‘n’ dimensions.  For
     example, the following code creates an instance of a Newton-Raphson
     solver, for a 2-dimensional system of equations:

          const gsl_multiroot_fdfsolver_type * T = gsl_multiroot_fdfsolver_newton;
          gsl_multiroot_fdfsolver * s = gsl_multiroot_fdfsolver_alloc (T, 2);

     If there is insufficient memory to create the solver then the
     function returns a null pointer and the error handler is invoked
     with an error code of *note GSL_ENOMEM: 2a.

 -- Function: int gsl_multiroot_fsolver_set (gsl_multiroot_fsolver * s,
          gsl_multiroot_function * f, const gsl_vector * x)
 -- Function: int gsl_multiroot_fdfsolver_set (gsl_multiroot_fdfsolver
          * s, gsl_multiroot_function_fdf * fdf, const gsl_vector * x)

     These functions set, or reset, an existing solver ‘s’ to use the
     function ‘f’ or function and derivative ‘fdf’, and the initial
     guess ‘x’.  Note that the initial position is copied from ‘x’, this
     argument is not modified by subsequent iterations.

 -- Function: void gsl_multiroot_fsolver_free (gsl_multiroot_fsolver
          * s)
 -- Function: void gsl_multiroot_fdfsolver_free (gsl_multiroot_fdfsolver
          * s)

     These functions free all the memory associated with the solver ‘s’.

 -- Function: const char * gsl_multiroot_fsolver_name (const
          gsl_multiroot_fsolver * s)
 -- Function: const char * gsl_multiroot_fdfsolver_name (const
          gsl_multiroot_fdfsolver * s)

     These functions return a pointer to the name of the solver.  For
     example:

          printf ("s is a '%s' solver\n", gsl_multiroot_fdfsolver_name (s));

     would print something like ‘s is a 'newton' solver’.


File: gsl-ref.info,  Node: Providing the function to solve<2>,  Next: Iteration<3>,  Prev: Initializing the Solver<2>,  Up: Multidimensional Root-Finding

38.3 Providing the function to solve
====================================

You must provide n functions of n variables for the root finders to
operate on.  In order to allow for general parameters the functions are
defined by the following data types:

 -- Type: gsl_multiroot_function

     This data type defines a general system of functions with
     parameters.

     ‘int (* f) (const gsl_vector * x, void * params, gsl_vector * f)’

          this function should store the vector result f(x,params) in
          ‘f’ for argument ‘x’ and parameters ‘params’, returning an
          appropriate error code if the function cannot be computed.

     ‘size_t n’

          the dimension of the system, i.e.  the number of components of
          the vectors ‘x’ and ‘f’.

     ‘void * params’

          a pointer to the parameters of the function.

Here is an example using Powell’s test function,

     f_1(x) = A x_0 x_1 - 1,
     f_2(x) = exp(-x_0) + exp(-x_1) - (1 + 1/A)

with A = 10^4.  The following code defines a *note
gsl_multiroot_function: ab3. system ‘F’ which you could pass to a
solver:

     struct powell_params { double A; };

     int
     powell (gsl_vector * x, void * p, gsl_vector * f) {
        struct powell_params * params
          = (struct powell_params *)p;
        const double A = (params->A);
        const double x0 = gsl_vector_get(x,0);
        const double x1 = gsl_vector_get(x,1);

        gsl_vector_set (f, 0, A * x0 * x1 - 1);
        gsl_vector_set (f, 1, (exp(-x0) + exp(-x1)
                               - (1.0 + 1.0/A)));
        return GSL_SUCCESS
     }

     gsl_multiroot_function F;
     struct powell_params params = { 10000.0 };

     F.f = &powell;
     F.n = 2;
     F.params = &params;

 -- Type: gsl_multiroot_function_fdf

     This data type defines a general system of functions with
     parameters and the corresponding Jacobian matrix of derivatives,

     ‘int (* f) (const gsl_vector * x, void * params, gsl_vector * f)’

          this function should store the vector result f(x,params) in
          ‘f’ for argument ‘x’ and parameters ‘params’, returning an
          appropriate error code if the function cannot be computed.

     ‘int (* df) (const gsl_vector * x, void * params, gsl_matrix * J)’

          this function should store the ‘n’-by-‘n’ matrix result

               J_ij = d f_i(x,params) / d x_j

          in ‘J’ for argument ‘x’ and parameters ‘params’, returning an
          appropriate error code if the function cannot be computed.

     ‘int (* fdf) (const gsl_vector * x, void * params, gsl_vector * f,
     gsl_matrix * J)’

          This function should set the values of the ‘f’ and ‘J’ as
          above, for arguments ‘x’ and parameters ‘params’.  This
          function provides an optimization of the separate functions
          for f(x) and J(x)—it is always faster to compute the function
          and its derivative at the same time.

     ‘size_t n’

          the dimension of the system, i.e.  the number of components of
          the vectors ‘x’ and ‘f’.

     ‘void * params’

          a pointer to the parameters of the function.

The example of Powell’s test function defined above can be extended to
include analytic derivatives using the following code:

     int
     powell_df (gsl_vector * x, void * p, gsl_matrix * J)
     {
        struct powell_params * params
          = (struct powell_params *)p;
        const double A = (params->A);
        const double x0 = gsl_vector_get(x,0);
        const double x1 = gsl_vector_get(x,1);
        gsl_matrix_set (J, 0, 0, A * x1);
        gsl_matrix_set (J, 0, 1, A * x0);
        gsl_matrix_set (J, 1, 0, -exp(-x0));
        gsl_matrix_set (J, 1, 1, -exp(-x1));
        return GSL_SUCCESS
     }

     int
     powell_fdf (gsl_vector * x, void * p,
                 gsl_matrix * f, gsl_matrix * J) {
        struct powell_params * params
          = (struct powell_params *)p;
        const double A = (params->A);
        const double x0 = gsl_vector_get(x,0);
        const double x1 = gsl_vector_get(x,1);

        const double u0 = exp(-x0);
        const double u1 = exp(-x1);

        gsl_vector_set (f, 0, A * x0 * x1 - 1);
        gsl_vector_set (f, 1, u0 + u1 - (1 + 1/A));

        gsl_matrix_set (J, 0, 0, A * x1);
        gsl_matrix_set (J, 0, 1, A * x0);
        gsl_matrix_set (J, 1, 0, -u0);
        gsl_matrix_set (J, 1, 1, -u1);
        return GSL_SUCCESS
     }

     gsl_multiroot_function_fdf FDF;

     FDF.f = &powell_f;
     FDF.df = &powell_df;
     FDF.fdf = &powell_fdf;
     FDF.n = 2;
     FDF.params = 0;

Note that the function ‘powell_fdf’ is able to reuse existing terms from
the function when calculating the Jacobian, thus saving time.


File: gsl-ref.info,  Node: Iteration<3>,  Next: Search Stopping Parameters<2>,  Prev: Providing the function to solve<2>,  Up: Multidimensional Root-Finding

38.4 Iteration
==============

The following functions drive the iteration of each algorithm.  Each
function performs one iteration to update the state of any solver of the
corresponding type.  The same functions work for all solvers so that
different methods can be substituted at runtime without modifications to
the code.

 -- Function: int gsl_multiroot_fsolver_iterate (gsl_multiroot_fsolver
          * s)
 -- Function: int gsl_multiroot_fdfsolver_iterate
          (gsl_multiroot_fdfsolver * s)

     These functions perform a single iteration of the solver ‘s’.  If
     the iteration encounters an unexpected problem then an error code
     will be returned,

     ‘GSL_EBADFUNC’

          the iteration encountered a singular point where the function
          or its derivative evaluated to ‘Inf’ or ‘NaN’.

     ‘GSL_ENOPROG’

          the iteration is not making any progress, preventing the
          algorithm from continuing.

The solver maintains a current best estimate of the root ‘s->x’ and its
function value ‘s->f’ at all times.  This information can be accessed
with the following auxiliary functions,

 -- Function: gsl_vector * gsl_multiroot_fsolver_root (const
          gsl_multiroot_fsolver * s)
 -- Function: gsl_vector * gsl_multiroot_fdfsolver_root (const
          gsl_multiroot_fdfsolver * s)

     These functions return the current estimate of the root for the
     solver ‘s’, given by ‘s->x’.

 -- Function: gsl_vector * gsl_multiroot_fsolver_f (const
          gsl_multiroot_fsolver * s)
 -- Function: gsl_vector * gsl_multiroot_fdfsolver_f (const
          gsl_multiroot_fdfsolver * s)

     These functions return the function value f(x) at the current
     estimate of the root for the solver ‘s’, given by ‘s->f’.

 -- Function: gsl_vector * gsl_multiroot_fsolver_dx (const
          gsl_multiroot_fsolver * s)
 -- Function: gsl_vector * gsl_multiroot_fdfsolver_dx (const
          gsl_multiroot_fdfsolver * s)

     These functions return the last step dx taken by the solver ‘s’,
     given by ‘s->dx’.


File: gsl-ref.info,  Node: Search Stopping Parameters<2>,  Next: Algorithms using Derivatives,  Prev: Iteration<3>,  Up: Multidimensional Root-Finding

38.5 Search Stopping Parameters
===============================

A root finding procedure should stop when one of the following
conditions is true:

   * A multidimensional root has been found to within the user-specified
     precision.

   * A user-specified maximum number of iterations has been reached.

   * An error has occurred.

The handling of these conditions is under user control.  The functions
below allow the user to test the precision of the current result in
several standard ways.

 -- Function: int gsl_multiroot_test_delta (const gsl_vector * dx, const
          gsl_vector * x, double epsabs, double epsrel)

     This function tests for the convergence of the sequence by
     comparing the last step ‘dx’ with the absolute error ‘epsabs’ and
     relative error ‘epsrel’ to the current position ‘x’.  The test
     returns ‘GSL_SUCCESS’ if the following condition is achieved,

          |dx_i| < epsabs + epsrel |x_i|

     for each component of ‘x’ and returns ‘GSL_CONTINUE’ otherwise.

 -- Function: int gsl_multiroot_test_residual (const gsl_vector * f,
          double epsabs)

     This function tests the residual value ‘f’ against the absolute
     error bound ‘epsabs’.  The test returns ‘GSL_SUCCESS’ if the
     following condition is achieved,

          \sum_i |f_i| < epsabs

     and returns ‘GSL_CONTINUE’ otherwise.  This criterion is suitable
     for situations where the precise location of the root, x, is
     unimportant provided a value can be found where the residual is
     small enough.


File: gsl-ref.info,  Node: Algorithms using Derivatives,  Next: Algorithms without Derivatives,  Prev: Search Stopping Parameters<2>,  Up: Multidimensional Root-Finding

38.6 Algorithms using Derivatives
=================================

The root finding algorithms described in this section make use of both
the function and its derivative.  They require an initial guess for the
location of the root, but there is no absolute guarantee of
convergence—the function must be suitable for this technique and the
initial guess must be sufficiently close to the root for it to work.
When the conditions are satisfied then convergence is quadratic.

 -- Type: gsl_multiroot_fdfsolver_type

     The following are available algorithms for minimizing functions
     using derivatives.

      -- Variable: gsl_multiroot_fdfsolver_hybridsj

          This is a modified version of Powell’s Hybrid method as
          implemented in the HYBRJ algorithm in MINPACK. Minpack was
          written by Jorge J. Moré, Burton S. Garbow and Kenneth E.
          Hillstrom.  The Hybrid algorithm retains the fast convergence
          of Newton’s method but will also reduce the residual when
          Newton’s method is unreliable.

          The algorithm uses a generalized trust region to keep each
          step under control.  In order to be accepted a proposed new
          position x' must satisfy the condition |D (x' - x)| < \delta,
          where D is a diagonal scaling matrix and \delta is the size of
          the trust region.  The components of D are computed
          internally, using the column norms of the Jacobian to estimate
          the sensitivity of the residual to each component of x.  This
          improves the behavior of the algorithm for badly scaled
          functions.

          On each iteration the algorithm first determines the standard
          Newton step by solving the system J dx = - f.  If this step
          falls inside the trust region it is used as a trial step in
          the next stage.  If not, the algorithm uses the linear
          combination of the Newton and gradient directions which is
          predicted to minimize the norm of the function while staying
          inside the trust region,

               dx = - \alpha J^{-1} f(x) - \beta \nabla |f(x)|^2

          This combination of Newton and gradient directions is referred
          to as a `dogleg step'.

          The proposed step is now tested by evaluating the function at
          the resulting point, x'.  If the step reduces the norm of the
          function sufficiently then it is accepted and size of the
          trust region is increased.  If the proposed step fails to
          improve the solution then the size of the trust region is
          decreased and another trial step is computed.

          The speed of the algorithm is increased by computing the
          changes to the Jacobian approximately, using a rank-1 update.
          If two successive attempts fail to reduce the residual then
          the full Jacobian is recomputed.  The algorithm also monitors
          the progress of the solution and returns an error if several
          steps fail to make any improvement,

          ‘GSL_ENOPROG’

               the iteration is not making any progress, preventing the
               algorithm from continuing.

          ‘GSL_ENOPROGJ’

               re-evaluations of the Jacobian indicate that the
               iteration is not making any progress, preventing the
               algorithm from continuing.

      -- Variable: gsl_multiroot_fdfsolver_hybridj

          This algorithm is an unscaled version of HYBRIDSJ. The steps
          are controlled by a spherical trust region |x' - x| < \delta,
          instead of a generalized region.  This can be useful if the
          generalized region estimated by HYBRIDSJ is inappropriate.

      -- Variable: gsl_multiroot_fdfsolver_newton

          Newton’s Method is the standard root-polishing algorithm.  The
          algorithm begins with an initial guess for the location of the
          solution.  On each iteration a linear approximation to the
          function F is used to estimate the step which will zero all
          the components of the residual.  The iteration is defined by
          the following sequence,

               x -> x' = x - J^{-1} f(x)

          where the Jacobian matrix J is computed from the derivative
          functions provided by ‘f’.  The step dx is obtained by solving
          the linear system,

               J dx = - f(x)

          using LU decomposition.  If the Jacobian matrix is singular,
          an error code of *note GSL_EDOM: 28. is returned.

      -- Variable: gsl_multiroot_fdfsolver_gnewton

          This is a modified version of Newton’s method which attempts
          to improve global convergence by requiring every step to
          reduce the Euclidean norm of the residual, |f(x)|.  If the
          Newton step leads to an increase in the norm then a reduced
          step of relative size,

               t = (\sqrt{1 + 6 r} - 1) / (3 r)

          is proposed, with r being the ratio of norms
          |f(x')|^2/|f(x)|^2.  This procedure is repeated until a
          suitable step size is found.


File: gsl-ref.info,  Node: Algorithms without Derivatives,  Next: Examples<29>,  Prev: Algorithms using Derivatives,  Up: Multidimensional Root-Finding

38.7 Algorithms without Derivatives
===================================

The algorithms described in this section do not require any derivative
information to be supplied by the user.  Any derivatives needed are
approximated by finite differences.  Note that if the
finite-differencing step size chosen by these routines is inappropriate,
an explicit user-supplied numerical derivative can always be used with
the algorithms described in the previous section.

 -- Type: gsl_multiroot_fsolver_type

     The following are available algorithms for minimizing functions
     without derivatives.

      -- Variable: gsl_multiroot_fsolver_hybrids

          This is a version of the Hybrid algorithm which replaces calls
          to the Jacobian function by its finite difference
          approximation.  The finite difference approximation is
          computed using ‘gsl_multiroots_fdjac()’ with a relative step
          size of ‘GSL_SQRT_DBL_EPSILON’.  Note that this step size will
          not be suitable for all problems.

      -- Variable: gsl_multiroot_fsolver_hybrid

          This is a finite difference version of the Hybrid algorithm
          without internal scaling.

      -- Variable: gsl_multiroot_fsolver_dnewton

          The `discrete Newton algorithm' is the simplest method of
          solving a multidimensional system.  It uses the Newton
          iteration

               x -> x - J^{-1} f(x)

          where the Jacobian matrix J is approximated by taking finite
          differences of the function ‘f’.  The approximation scheme
          used by this implementation is,

               J_{ij} = (f_i(x + \delta_j) - f_i(x)) / \delta_j

          where \delta_j is a step of size \sqrt\epsilon |x_j| with
          \epsilon being the machine precision (\epsilon \approx 2.22
          \times 10^{-16}).  The order of convergence of Newton’s
          algorithm is quadratic, but the finite differences require n^2
          function evaluations on each iteration.  The algorithm may
          become unstable if the finite differences are not a good
          approximation to the true derivatives.

      -- Variable: gsl_multiroot_fsolver_broyden

          The `Broyden algorithm' is a version of the discrete Newton
          algorithm which attempts to avoids the expensive update of the
          Jacobian matrix on each iteration.  The changes to the
          Jacobian are also approximated, using a rank-1 update,

               J^{-1} \to J^{-1} - (J^{-1} df - dx) dx^T J^{-1} / dx^T J^{-1} df

          where the vectors dx and df are the changes in x and f.  On
          the first iteration the inverse Jacobian is estimated using
          finite differences, as in the discrete Newton algorithm.

          This approximation gives a fast update but is unreliable if
          the changes are not small, and the estimate of the inverse
          Jacobian becomes worse as time passes.  The algorithm has a
          tendency to become unstable unless it starts close to the
          root.  The Jacobian is refreshed if this instability is
          detected (consult the source for details).

          This algorithm is included only for demonstration purposes,
          and is not recommended for serious use.


File: gsl-ref.info,  Node: Examples<29>,  Next: References and Further Reading<31>,  Prev: Algorithms without Derivatives,  Up: Multidimensional Root-Finding

38.8 Examples
=============

The multidimensional solvers are used in a similar way to the
one-dimensional root finding algorithms.  This first example
demonstrates the HYBRIDS scaled-hybrid algorithm, which does not require
derivatives.  The program solves the Rosenbrock system of equations,

     f_1 (x, y) = a (1 - x)
     f_2 (x, y) = b (y - x^2)

with a = 1, b = 10.  The solution of this system lies at (x,y) = (1,1)
in a narrow valley.

The first stage of the program is to define the system of equations:

     #include <stdlib.h>
     #include <stdio.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_multiroots.h>

     struct rparams
       {
         double a;
         double b;
       };

     int
     rosenbrock_f (const gsl_vector * x, void *params,
                   gsl_vector * f)
     {
       double a = ((struct rparams *) params)->a;
       double b = ((struct rparams *) params)->b;

       const double x0 = gsl_vector_get (x, 0);
       const double x1 = gsl_vector_get (x, 1);

       const double y0 = a * (1 - x0);
       const double y1 = b * (x1 - x0 * x0);

       gsl_vector_set (f, 0, y0);
       gsl_vector_set (f, 1, y1);

       return GSL_SUCCESS;
     }

The main program begins by creating the function object ‘f’, with the
arguments ‘(x,y)’ and parameters ‘(a,b)’.  The solver ‘s’ is initialized
to use this function, with the *note gsl_multiroot_fsolver_hybrids: ac9.
method:

     int
     main (void)
     {
       const gsl_multiroot_fsolver_type *T;
       gsl_multiroot_fsolver *s;

       int status;
       size_t i, iter = 0;

       const size_t n = 2;
       struct rparams p = {1.0, 10.0};
       gsl_multiroot_function f = {&rosenbrock_f, n, &p};

       double x_init[2] = {-10.0, -5.0};
       gsl_vector *x = gsl_vector_alloc (n);

       gsl_vector_set (x, 0, x_init[0]);
       gsl_vector_set (x, 1, x_init[1]);

       T = gsl_multiroot_fsolver_hybrids;
       s = gsl_multiroot_fsolver_alloc (T, 2);
       gsl_multiroot_fsolver_set (s, &f, x);

       print_state (iter, s);

       do
         {
           iter++;
           status = gsl_multiroot_fsolver_iterate (s);

           print_state (iter, s);

           if (status)   /* check if solver is stuck */
             break;

           status =
             gsl_multiroot_test_residual (s->f, 1e-7);
         }
       while (status == GSL_CONTINUE && iter < 1000);

       printf ("status = %s\n", gsl_strerror (status));

       gsl_multiroot_fsolver_free (s);
       gsl_vector_free (x);
       return 0;
     }

Note that it is important to check the return status of each solver
step, in case the algorithm becomes stuck.  If an error condition is
detected, indicating that the algorithm cannot proceed, then the error
can be reported to the user, a new starting point chosen or a different
algorithm used.

The intermediate state of the solution is displayed by the following
function.  The solver state contains the vector ‘s->x’ which is the
current position, and the vector ‘s->f’ with corresponding function
values:

     int
     print_state (size_t iter, gsl_multiroot_fsolver * s)
     {
       printf ("iter = %3u x = % .3f % .3f "
               "f(x) = % .3e % .3e\n",
               iter,
               gsl_vector_get (s->x, 0),
               gsl_vector_get (s->x, 1),
               gsl_vector_get (s->f, 0),
               gsl_vector_get (s->f, 1));
     }

Here are the results of running the program.  The algorithm is started
at (-10,-5) far from the solution.  Since the solution is hidden in a
narrow valley the earliest steps follow the gradient of the function
downhill, in an attempt to reduce the large value of the residual.  Once
the root has been approximately located, on iteration 8, the Newton
behavior takes over and convergence is very rapid:

     iter =  0 x = -10.000  -5.000  f(x) = 1.100e+01 -1.050e+03
     iter =  1 x = -10.000  -5.000  f(x) = 1.100e+01 -1.050e+03
     iter =  2 x =  -3.976  24.827  f(x) = 4.976e+00  9.020e+01
     iter =  3 x =  -3.976  24.827  f(x) = 4.976e+00  9.020e+01
     iter =  4 x =  -3.976  24.827  f(x) = 4.976e+00  9.020e+01
     iter =  5 x =  -1.274  -5.680  f(x) = 2.274e+00 -7.302e+01
     iter =  6 x =  -1.274  -5.680  f(x) = 2.274e+00 -7.302e+01
     iter =  7 x =   0.249   0.298  f(x) = 7.511e-01  2.359e+00
     iter =  8 x =   0.249   0.298  f(x) = 7.511e-01  2.359e+00
     iter =  9 x =   1.000   0.878  f(x) = 1.268e-10 -1.218e+00
     iter = 10 x =   1.000   0.989  f(x) = 1.124e-11 -1.080e-01
     iter = 11 x =   1.000   1.000  f(x) = 0.000e+00  0.000e+00
     status = success

Note that the algorithm does not update the location on every iteration.
Some iterations are used to adjust the trust-region parameter, after
trying a step which was found to be divergent, or to recompute the
Jacobian, when poor convergence behavior is detected.

The next example program adds derivative information, in order to
accelerate the solution.  There are two derivative functions
‘rosenbrock_df’ and ‘rosenbrock_fdf’.  The latter computes both the
function and its derivative simultaneously.  This allows the
optimization of any common terms.  For simplicity we substitute calls to
the separate ‘f’ and ‘df’ functions at this point in the code below:

     int
     rosenbrock_df (const gsl_vector * x, void *params,
                    gsl_matrix * J)
     {
       const double a = ((struct rparams *) params)->a;
       const double b = ((struct rparams *) params)->b;

       const double x0 = gsl_vector_get (x, 0);

       const double df00 = -a;
       const double df01 = 0;
       const double df10 = -2 * b  * x0;
       const double df11 = b;

       gsl_matrix_set (J, 0, 0, df00);
       gsl_matrix_set (J, 0, 1, df01);
       gsl_matrix_set (J, 1, 0, df10);
       gsl_matrix_set (J, 1, 1, df11);

       return GSL_SUCCESS;
     }

     int
     rosenbrock_fdf (const gsl_vector * x, void *params,
                     gsl_vector * f, gsl_matrix * J)
     {
       rosenbrock_f (x, params, f);
       rosenbrock_df (x, params, J);

       return GSL_SUCCESS;
     }

The main program now makes calls to the corresponding ‘fdfsolver’
versions of the functions:

     int
     main (void)
     {
       const gsl_multiroot_fdfsolver_type *T;
       gsl_multiroot_fdfsolver *s;

       int status;
       size_t i, iter = 0;

       const size_t n = 2;
       struct rparams p = {1.0, 10.0};
       gsl_multiroot_function_fdf f = {&rosenbrock_f,
                                       &rosenbrock_df,
                                       &rosenbrock_fdf,
                                       n, &p};

       double x_init[2] = {-10.0, -5.0};
       gsl_vector *x = gsl_vector_alloc (n);

       gsl_vector_set (x, 0, x_init[0]);
       gsl_vector_set (x, 1, x_init[1]);

       T = gsl_multiroot_fdfsolver_gnewton;
       s = gsl_multiroot_fdfsolver_alloc (T, n);
       gsl_multiroot_fdfsolver_set (s, &f, x);

       print_state (iter, s);

       do
         {
           iter++;

           status = gsl_multiroot_fdfsolver_iterate (s);

           print_state (iter, s);

           if (status)
             break;

           status = gsl_multiroot_test_residual (s->f, 1e-7);
         }
       while (status == GSL_CONTINUE && iter < 1000);

       printf ("status = %s\n", gsl_strerror (status));

       gsl_multiroot_fdfsolver_free (s);
       gsl_vector_free (x);
       return 0;
     }

The addition of derivative information to the *note
gsl_multiroot_fsolver_hybrids: ac9. solver does not make any significant
difference to its behavior, since it able to approximate the Jacobian
numerically with sufficient accuracy.  To illustrate the behavior of a
different derivative solver we switch to *note
gsl_multiroot_fdfsolver_gnewton: ac6.  This is a traditional Newton
solver with the constraint that it scales back its step if the full step
would lead “uphill”.  Here is the output for the *note
gsl_multiroot_fdfsolver_gnewton: ac6. algorithm:

     iter = 0 x = -10.000  -5.000 f(x) =  1.100e+01 -1.050e+03
     iter = 1 x =  -4.231 -65.317 f(x) =  5.231e+00 -8.321e+02
     iter = 2 x =   1.000 -26.358 f(x) = -8.882e-16 -2.736e+02
     iter = 3 x =   1.000   1.000 f(x) = -2.220e-16 -4.441e-15
     status = success

The convergence is much more rapid, but takes a wide excursion out to
the point (-4.23,-65.3).  This could cause the algorithm to go astray in
a realistic application.  The hybrid algorithm follows the downhill path
to the solution more reliably.


File: gsl-ref.info,  Node: References and Further Reading<31>,  Prev: Examples<29>,  Up: Multidimensional Root-Finding

38.9 References and Further Reading
===================================

The original version of the Hybrid method is described in the following
articles by Powell,

   * M.J.D. Powell, “A Hybrid Method for Nonlinear Equations” (Chap 6, p
     87–114) and “A Fortran Subroutine for Solving systems of Nonlinear
     Algebraic Equations” (Chap 7, p 115–161), in `Numerical Methods for
     Nonlinear Algebraic Equations', P. Rabinowitz, editor.  Gordon and
     Breach, 1970.

The following papers are also relevant to the algorithms described in
this section,

   * J.J. Moré, M.Y. Cosnard, “Numerical Solution of Nonlinear
     Equations”, `ACM Transactions on Mathematical Software', Vol 5, No
     1, (1979), p 64–85

   * C.G. Broyden, “A Class of Methods for Solving Nonlinear
     Simultaneous Equations”, `Mathematics of Computation', Vol 19
     (1965), p 577–593

   * J.J. Moré, B.S. Garbow, K.E. Hillstrom, “Testing Unconstrained
     Optimization Software”, ACM Transactions on Mathematical Software,
     Vol 7, No 1 (1981), p 17–41


File: gsl-ref.info,  Node: Multidimensional Minimization,  Next: Linear Least-Squares Fitting,  Prev: Multidimensional Root-Finding,  Up: Top

39 Multidimensional Minimization
********************************

This chapter describes routines for finding minima of arbitrary
multidimensional functions.  The library provides low level components
for a variety of iterative minimizers and convergence tests.  These can
be combined by the user to achieve the desired solution, while providing
full access to the intermediate steps of the algorithms.  Each class of
methods uses the same framework, so that you can switch between
minimizers at runtime without needing to recompile your program.  Each
instance of a minimizer keeps track of its own state, allowing the
minimizers to be used in multi-threaded programs.  The minimization
algorithms can be used to maximize a function by inverting its sign.

The header file ‘gsl_multimin.h’ contains prototypes for the
minimization functions and related declarations.

* Menu:

* Overview: Overview<4>.
* Caveats: Caveats<3>.
* Initializing the Multidimensional Minimizer::
* Providing a function to minimize::
* Iteration: Iteration<4>.
* Stopping Criteria::
* Algorithms with Derivatives::
* Algorithms without Derivatives: Algorithms without Derivatives<2>.
* Examples: Examples<30>.
* References and Further Reading: References and Further Reading<32>.


File: gsl-ref.info,  Node: Overview<4>,  Next: Caveats<3>,  Up: Multidimensional Minimization

39.1 Overview
=============

The problem of multidimensional minimization requires finding a point x
such that the scalar function,

     f(x_1, \dots, x_n)

takes a value which is lower than at any neighboring point.  For smooth
functions the gradient g = \nabla f vanishes at the minimum.  In general
there are no bracketing methods available for the minimization of
n-dimensional functions.  The algorithms proceed from an initial guess
using a search algorithm which attempts to move in a downhill direction.

Algorithms making use of the gradient of the function perform a
one-dimensional line minimisation along this direction until the lowest
point is found to a suitable tolerance.  The search direction is then
updated with local information from the function and its derivatives,
and the whole process repeated until the true n-dimensional minimum is
found.

Algorithms which do not require the gradient of the function use
different strategies.  For example, the Nelder-Mead Simplex algorithm
maintains n+1 trial parameter vectors as the vertices of a n-dimensional
simplex.  On each iteration it tries to improve the worst vertex of the
simplex by geometrical transformations.  The iterations are continued
until the overall size of the simplex has decreased sufficiently.

Both types of algorithms use a standard framework.  The user provides a
high-level driver for the algorithms, and the library provides the
individual functions necessary for each of the steps.  There are three
main phases of the iteration.  The steps are,

   * initialize minimizer state, ‘s’, for algorithm ‘T’

   * update ‘s’ using the iteration ‘T’

   * test ‘s’ for convergence, and repeat iteration if necessary

Each iteration step consists either of an improvement to the
line-minimisation in the current direction or an update to the search
direction itself.  The state for the minimizers is held in a *note
gsl_multimin_fdfminimizer: ad2. struct or a *note
gsl_multimin_fminimizer: ad3. struct.


File: gsl-ref.info,  Node: Caveats<3>,  Next: Initializing the Multidimensional Minimizer,  Prev: Overview<4>,  Up: Multidimensional Minimization

39.2 Caveats
============

Note that the minimization algorithms can only search for one local
minimum at a time.  When there are several local minima in the search
area, the first minimum to be found will be returned; however it is
difficult to predict which of the minima this will be.  In most cases,
no error will be reported if you try to find a local minimum in an area
where there is more than one.

It is also important to note that the minimization algorithms find local
minima; there is no way to determine whether a minimum is a global
minimum of the function in question.


File: gsl-ref.info,  Node: Initializing the Multidimensional Minimizer,  Next: Providing a function to minimize,  Prev: Caveats<3>,  Up: Multidimensional Minimization

39.3 Initializing the Multidimensional Minimizer
================================================

The following function initializes a multidimensional minimizer.  The
minimizer itself depends only on the dimension of the problem and the
algorithm and can be reused for different problems.

 -- Type: gsl_multimin_fdfminimizer

     This is a workspace for minimizing functions using derivatives.

 -- Type: gsl_multimin_fminimizer

     This is a workspace for minimizing functions without derivatives.

 -- Function: gsl_multimin_fdfminimizer *
          gsl_multimin_fdfminimizer_alloc (const
          gsl_multimin_fdfminimizer_type * T, size_t n)
 -- Function: gsl_multimin_fminimizer * gsl_multimin_fminimizer_alloc
          (const gsl_multimin_fminimizer_type * T, size_t n)

     This function returns a pointer to a newly allocated instance of a
     minimizer of type ‘T’ for an ‘n’-dimension function.  If there is
     insufficient memory to create the minimizer then the function
     returns a null pointer and the error handler is invoked with an
     error code of *note GSL_ENOMEM: 2a.

 -- Function: int gsl_multimin_fdfminimizer_set
          (gsl_multimin_fdfminimizer * s, gsl_multimin_function_fdf
          * fdf, const gsl_vector * x, double step_size, double tol)
 -- Function: int gsl_multimin_fminimizer_set (gsl_multimin_fminimizer
          * s, gsl_multimin_function * f, const gsl_vector * x, const
          gsl_vector * step_size)

     The function *note gsl_multimin_fdfminimizer_set(): ad8.
     initializes the minimizer ‘s’ to minimize the function ‘fdf’
     starting from the initial point ‘x’.  The size of the first trial
     step is given by ‘step_size’.  The accuracy of the line
     minimization is specified by ‘tol’.  The precise meaning of this
     parameter depends on the method used.  Typically the line
     minimization is considered successful if the gradient of the
     function g is orthogonal to the current search direction p to a
     relative accuracy of ‘tol’, where p \cdot g < tol |p| |g|.  A ‘tol’
     value of 0.1 is suitable for most purposes, since line minimization
     only needs to be carried out approximately.  Note that setting
     ‘tol’ to zero will force the use of “exact” line-searches, which
     are extremely expensive.

     The function *note gsl_multimin_fminimizer_set(): ad9. initializes
     the minimizer ‘s’ to minimize the function ‘f’, starting from the
     initial point ‘x’.  The size of the initial trial steps is given in
     vector ‘step_size’.  The precise meaning of this parameter depends
     on the method used.

 -- Function: void gsl_multimin_fdfminimizer_free
          (gsl_multimin_fdfminimizer * s)
 -- Function: void gsl_multimin_fminimizer_free (gsl_multimin_fminimizer
          * s)

     This function frees all the memory associated with the minimizer
     ‘s’.

 -- Function: const char * gsl_multimin_fdfminimizer_name (const
          gsl_multimin_fdfminimizer * s)
 -- Function: const char * gsl_multimin_fminimizer_name (const
          gsl_multimin_fminimizer * s)

     This function returns a pointer to the name of the minimizer.  For
     example:

          printf ("s is a '%s' minimizer\n", gsl_multimin_fdfminimizer_name (s));

     would print something like ‘s is a 'conjugate_pr' minimizer’.


File: gsl-ref.info,  Node: Providing a function to minimize,  Next: Iteration<4>,  Prev: Initializing the Multidimensional Minimizer,  Up: Multidimensional Minimization

39.4 Providing a function to minimize
=====================================

You must provide a parametric function of n variables for the minimizers
to operate on.  You may also need to provide a routine which calculates
the gradient of the function and a third routine which calculates both
the function value and the gradient together.  In order to allow for
general parameters the functions are defined by the following data
types:

 -- Type: gsl_multimin_function_fdf

     This data type defines a general function of n variables with
     parameters and the corresponding gradient vector of derivatives,

     ‘double (* f) (const gsl_vector * x, void * params)’

          this function should return the result f(x,params) for
          argument ‘x’ and parameters ‘params’.  If the function cannot
          be computed, an error value of *note GSL_NAN: 3c. should be
          returned.

     ‘void (* df) (const gsl_vector * x, void * params, gsl_vector * g)’

          this function should store the ‘n’-dimensional gradient

               g_i = d f(x,params) / d x_i

          in the vector ‘g’ for argument ‘x’ and parameters ‘params’,
          returning an appropriate error code if the function cannot be
          computed.

     ‘void (* fdf) (const gsl_vector * x, void * params, double * f,
     gsl_vector * g)’

          This function should set the values of the ‘f’ and ‘g’ as
          above, for arguments ‘x’ and parameters ‘params’.  This
          function provides an optimization of the separate functions
          for f(x) and g(x)—it is always faster to compute the function
          and its derivative at the same time.

     ‘size_t n’

          the dimension of the system, i.e.  the number of components of
          the vectors ‘x’.

     ‘void * params’

          a pointer to the parameters of the function.

 -- Type: gsl_multimin_function

     This data type defines a general function of n variables with
     parameters,

     ‘double (* f) (const gsl_vector * x, void * params)’

          this function should return the result f(x,params) for
          argument ‘x’ and parameters ‘params’.  If the function cannot
          be computed, an error value of *note GSL_NAN: 3c. should be
          returned.

     ‘size_t n’

          the dimension of the system, i.e.  the number of components of
          the vectors ‘x’.

     ‘void * params’

          a pointer to the parameters of the function.
The following example function defines a simple two-dimensional
paraboloid with five parameters,

     /* Paraboloid centered on (p[0],p[1]), with
        scale factors (p[2],p[3]) and minimum p[4] */

     double
     my_f (const gsl_vector *v, void *params)
     {
       double x, y;
       double *p = (double *)params;

       x = gsl_vector_get(v, 0);
       y = gsl_vector_get(v, 1);

       return p[2] * (x - p[0]) * (x - p[0]) +
                p[3] * (y - p[1]) * (y - p[1]) + p[4];
     }

     /* The gradient of f, df = (df/dx, df/dy). */
     void
     my_df (const gsl_vector *v, void *params,
            gsl_vector *df)
     {
       double x, y;
       double *p = (double *)params;

       x = gsl_vector_get(v, 0);
       y = gsl_vector_get(v, 1);

       gsl_vector_set(df, 0, 2.0 * p[2] * (x - p[0]));
       gsl_vector_set(df, 1, 2.0 * p[3] * (y - p[1]));
     }

     /* Compute both f and df together. */
     void
     my_fdf (const gsl_vector *x, void *params,
             double *f, gsl_vector *df)
     {
       *f = my_f(x, params);
       my_df(x, params, df);
     }

The function can be initialized using the following code:

     gsl_multimin_function_fdf my_func;

     /* Paraboloid center at (1,2), scale factors (10, 20),
        minimum value 30 */
     double p[5] = { 1.0, 2.0, 10.0, 20.0, 30.0 };

     my_func.n = 2;  /* number of function components */
     my_func.f = &my_f;
     my_func.df = &my_df;
     my_func.fdf = &my_fdf;
     my_func.params = (void *)p;


File: gsl-ref.info,  Node: Iteration<4>,  Next: Stopping Criteria,  Prev: Providing a function to minimize,  Up: Multidimensional Minimization

39.5 Iteration
==============

The following function drives the iteration of each algorithm.  The
function performs one iteration to update the state of the minimizer.
The same function works for all minimizers so that different methods can
be substituted at runtime without modifications to the code.

 -- Function: int gsl_multimin_fdfminimizer_iterate
          (gsl_multimin_fdfminimizer * s)
 -- Function: int gsl_multimin_fminimizer_iterate
          (gsl_multimin_fminimizer * s)

     These functions perform a single iteration of the minimizer ‘s’.
     If the iteration encounters an unexpected problem then an error
     code will be returned.  The error code ‘GSL_ENOPROG’ signifies that
     the minimizer is unable to improve on its current estimate, either
     due to numerical difficulty or because a genuine local minimum has
     been reached.

The minimizer maintains a current best estimate of the minimum at all
times.  This information can be accessed with the following auxiliary
functions,

 -- Function: gsl_vector * gsl_multimin_fdfminimizer_x (const
          gsl_multimin_fdfminimizer * s)
 -- Function: gsl_vector * gsl_multimin_fminimizer_x (const
          gsl_multimin_fminimizer * s)
 -- Function: double gsl_multimin_fdfminimizer_minimum (const
          gsl_multimin_fdfminimizer * s)
 -- Function: double gsl_multimin_fminimizer_minimum (const
          gsl_multimin_fminimizer * s)
 -- Function: gsl_vector * gsl_multimin_fdfminimizer_gradient (const
          gsl_multimin_fdfminimizer * s)
 -- Function: gsl_vector * gsl_multimin_fdfminimizer_dx (const
          gsl_multimin_fdfminimizer * s)
 -- Function: double gsl_multimin_fminimizer_size (const
          gsl_multimin_fminimizer * s)

     These functions return the current best estimate of the location of
     the minimum, the value of the function at that point, its gradient,
     the last step increment of the estimate, and minimizer specific
     characteristic size for the minimizer ‘s’.

 -- Function: int gsl_multimin_fdfminimizer_restart
          (gsl_multimin_fdfminimizer * s)

     This function resets the minimizer ‘s’ to use the current point as
     a new starting point.


File: gsl-ref.info,  Node: Stopping Criteria,  Next: Algorithms with Derivatives,  Prev: Iteration<4>,  Up: Multidimensional Minimization

39.6 Stopping Criteria
======================

A minimization procedure should stop when one of the following
conditions is true:

   * A minimum has been found to within the user-specified precision.

   * A user-specified maximum number of iterations has been reached.

   * An error has occurred.

The handling of these conditions is under user control.  The functions
below allow the user to test the precision of the current result.

 -- Function: int gsl_multimin_test_gradient (const gsl_vector * g,
          double epsabs)

     This function tests the norm of the gradient ‘g’ against the
     absolute tolerance ‘epsabs’.  The gradient of a multidimensional
     function goes to zero at a minimum.  The test returns ‘GSL_SUCCESS’
     if the following condition is achieved,

          |g| < epsabs

     and returns ‘GSL_CONTINUE’ otherwise.  A suitable choice of
     ‘epsabs’ can be made from the desired accuracy in the function for
     small variations in x.  The relationship between these quantities
     is given by \delta{f} = g\,\delta{x}.

 -- Function: int gsl_multimin_test_size (const double size,
          double epsabs)

     This function tests the minimizer specific characteristic size (if
     applicable to the used minimizer) against absolute tolerance
     ‘epsabs’.  The test returns ‘GSL_SUCCESS’ if the size is smaller
     than tolerance, otherwise ‘GSL_CONTINUE’ is returned.


File: gsl-ref.info,  Node: Algorithms with Derivatives,  Next: Algorithms without Derivatives<2>,  Prev: Stopping Criteria,  Up: Multidimensional Minimization

39.7 Algorithms with Derivatives
================================

There are several minimization methods available.  The best choice of
algorithm depends on the problem.  The algorithms described in this
section use the value of the function and its gradient at each
evaluation point.

 -- Type: gsl_multimin_fdfminimizer_type

     This type specifies a minimization algorithm using gradients.

      -- Variable: gsl_multimin_fdfminimizer_conjugate_fr

          This is the Fletcher-Reeves conjugate gradient algorithm.  The
          conjugate gradient algorithm proceeds as a succession of line
          minimizations.  The sequence of search directions is used to
          build up an approximation to the curvature of the function in
          the neighborhood of the minimum.

          An initial search direction ‘p’ is chosen using the gradient,
          and line minimization is carried out in that direction.  The
          accuracy of the line minimization is specified by the
          parameter ‘tol’.  The minimum along this line occurs when the
          function gradient ‘g’ and the search direction ‘p’ are
          orthogonal.  The line minimization terminates when p\cdot g <
          tol |p| |g|.  The search direction is updated using the
          Fletcher-Reeves formula p' = g' - \beta g where
          \beta=-|g'|^2/|g|^2, and the line minimization is then
          repeated for the new search direction.

      -- Variable: gsl_multimin_fdfminimizer_conjugate_pr

          This is the Polak-Ribiere conjugate gradient algorithm.  It is
          similar to the Fletcher-Reeves method, differing only in the
          choice of the coefficient \beta.  Both methods work well when
          the evaluation point is close enough to the minimum of the
          objective function that it is well approximated by a quadratic
          hypersurface.

      -- Variable: gsl_multimin_fdfminimizer_vector_bfgs2
      -- Variable: gsl_multimin_fdfminimizer_vector_bfgs

          These methods use the vector Broyden-Fletcher-Goldfarb-Shanno
          (BFGS) algorithm.  This is a quasi-Newton method which builds
          up an approximation to the second derivatives of the function
          f using the difference between successive gradient vectors.
          By combining the first and second derivatives the algorithm is
          able to take Newton-type steps towards the function minimum,
          assuming quadratic behavior in that region.

          The ‘bfgs2’ version of this minimizer is the most efficient
          version available, and is a faithful implementation of the
          line minimization scheme described in Fletcher’s `Practical
          Methods of Optimization', Algorithms 2.6.2 and 2.6.4.  It
          supersedes the original ‘bfgs’ routine and requires
          substantially fewer function and gradient evaluations.  The
          user-supplied tolerance ‘tol’ corresponds to the parameter
          \sigma used by Fletcher.  A value of 0.1 is recommended for
          typical use (larger values correspond to less accurate line
          searches).

      -- Variable: gsl_multimin_fdfminimizer_steepest_descent

          The steepest descent algorithm follows the downhill gradient
          of the function at each step.  When a downhill step is
          successful the step-size is increased by a factor of two.  If
          the downhill step leads to a higher function value then the
          algorithm backtracks and the step size is decreased using the
          parameter ‘tol’.  A suitable value of ‘tol’ for most
          applications is 0.1.  The steepest descent method is
          inefficient and is included only for demonstration purposes.


File: gsl-ref.info,  Node: Algorithms without Derivatives<2>,  Next: Examples<30>,  Prev: Algorithms with Derivatives,  Up: Multidimensional Minimization

39.8 Algorithms without Derivatives
===================================

The algorithms described in this section use only the value of the
function at each evaluation point.

 -- Type: gsl_multimin_fminimizer_type

     This type specifies minimization algorithms which do not use
     gradients.

      -- Variable: gsl_multimin_fminimizer_nmsimplex2
      -- Variable: gsl_multimin_fminimizer_nmsimplex

          These methods use the Simplex algorithm of Nelder and Mead.
          Starting from the initial vector x = p_0, the algorithm
          constructs an additional n vectors p_i using the step size
          vector s = step\_size as follows:

               p_0 = (x_0, x_1, ... , x_n)
               p_1 = (x_0 + s_0, x_1, ... , x_n)
               p_2 = (x_0, x_1 + s_1, ... , x_n)
               ... = ...
               p_n = (x_0, x_1, ... , x_n + s_n)

          These vectors form the n+1 vertices of a simplex in n
          dimensions.  On each iteration the algorithm uses simple
          geometrical transformations to update the vector corresponding
          to the highest function value.  The geometric transformations
          are reflection, reflection followed by expansion, contraction
          and multiple contraction.  Using these transformations the
          simplex moves through the space towards the minimum, where it
          contracts itself.

          After each iteration, the best vertex is returned.  Note, that
          due to the nature of the algorithm not every step improves the
          current best parameter vector.  Usually several iterations are
          required.

          The minimizer-specific characteristic size is calculated as
          the average distance from the geometrical center of the
          simplex to all its vertices.  This size can be used as a
          stopping criteria, as the simplex contracts itself near the
          minimum.  The size is returned by the function *note
          gsl_multimin_fminimizer_size(): aeb.

          The *note gsl_multimin_fminimizer_nmsimplex2: af9. version of
          this minimiser is a new O(N) operations implementation of the
          earlier O(N^2) operations *note
          gsl_multimin_fminimizer_nmsimplex: afa. minimiser.  It uses
          the same underlying algorithm, but the simplex updates are
          computed more efficiently for high-dimensional problems.  In
          addition, the size of simplex is calculated as the RMS
          distance of each vertex from the center rather than the mean
          distance, allowing a linear update of this quantity on each
          step.  The memory usage is O(N^2) for both algorithms.

      -- Variable: gsl_multimin_fminimizer_nmsimplex2rand

          This method is a variant of *note
          gsl_multimin_fminimizer_nmsimplex2: af9. which initialises the
          simplex around the starting point ‘x’ using a
          randomly-oriented set of basis vectors instead of the fixed
          coordinate axes.  The final dimensions of the simplex are
          scaled along the coordinate axes by the vector ‘step_size’.
          The randomisation uses a simple deterministic generator so
          that repeated calls to *note gsl_multimin_fminimizer_set():
          ad9. for a given solver object will vary the orientation in a
          well-defined way.


File: gsl-ref.info,  Node: Examples<30>,  Next: References and Further Reading<32>,  Prev: Algorithms without Derivatives<2>,  Up: Multidimensional Minimization

39.9 Examples
=============

This example program finds the minimum of the *note paraboloid function:
ae1. defined earlier.  The location of the minimum is offset from the
origin in x and y, and the function value at the minimum is non-zero.
The main program is given below, it requires the example function given
earlier in this chapter.

     int
     main (void)
     {
       size_t iter = 0;
       int status;

       const gsl_multimin_fdfminimizer_type *T;
       gsl_multimin_fdfminimizer *s;

       /* Position of the minimum (1,2), scale factors
          10,20, height 30. */
       double par[5] = { 1.0, 2.0, 10.0, 20.0, 30.0 };

       gsl_vector *x;
       gsl_multimin_function_fdf my_func;

       my_func.n = 2;
       my_func.f = my_f;
       my_func.df = my_df;
       my_func.fdf = my_fdf;
       my_func.params = par;

       /* Starting point, x = (5,7) */
       x = gsl_vector_alloc (2);
       gsl_vector_set (x, 0, 5.0);
       gsl_vector_set (x, 1, 7.0);

       T = gsl_multimin_fdfminimizer_conjugate_fr;
       s = gsl_multimin_fdfminimizer_alloc (T, 2);

       gsl_multimin_fdfminimizer_set (s, &my_func, x, 0.01, 1e-4);

       do
         {
           iter++;
           status = gsl_multimin_fdfminimizer_iterate (s);

           if (status)
             break;

           status = gsl_multimin_test_gradient (s->gradient, 1e-3);

           if (status == GSL_SUCCESS)
             printf ("Minimum found at:\n");

           printf ("%5d %.5f %.5f %10.5f\n", iter,
                   gsl_vector_get (s->x, 0),
                   gsl_vector_get (s->x, 1),
                   s->f);

         }
       while (status == GSL_CONTINUE && iter < 100);

       gsl_multimin_fdfminimizer_free (s);
       gsl_vector_free (x);

       return 0;
     }

The initial step-size is chosen as 0.01, a conservative estimate in this
case, and the line minimization parameter is set at 0.0001.  The program
terminates when the norm of the gradient has been reduced below 0.001.
The output of the program is shown below,

              x       y         f
         1 4.99629 6.99072  687.84780
         2 4.98886 6.97215  683.55456
         3 4.97400 6.93501  675.01278
         4 4.94429 6.86073  658.10798
         5 4.88487 6.71217  625.01340
         6 4.76602 6.41506  561.68440
         7 4.52833 5.82083  446.46694
         8 4.05295 4.63238  261.79422
         9 3.10219 2.25548   75.49762
        10 2.85185 1.62963   67.03704
        11 2.19088 1.76182   45.31640
        12 0.86892 2.02622   30.18555
     Minimum found at:
        13 1.00000 2.00000   30.00000

Note that the algorithm gradually increases the step size as it
successfully moves downhill, as can be seen by plotting the successive
points in Fig.  %s.

[image src="multimin.png"]

Figure: Function contours with path taken by minimization algorithm

The conjugate gradient algorithm finds the minimum on its second
direction because the function is purely quadratic.  Additional
iterations would be needed for a more complicated function.

Here is another example using the Nelder-Mead Simplex algorithm to
minimize the same example object function, as above.

     int
     main(void)
     {
       double par[5] = {1.0, 2.0, 10.0, 20.0, 30.0};

       const gsl_multimin_fminimizer_type *T =
         gsl_multimin_fminimizer_nmsimplex2;
       gsl_multimin_fminimizer *s = NULL;
       gsl_vector *ss, *x;
       gsl_multimin_function minex_func;

       size_t iter = 0;
       int status;
       double size;

       /* Starting point */
       x = gsl_vector_alloc (2);
       gsl_vector_set (x, 0, 5.0);
       gsl_vector_set (x, 1, 7.0);

       /* Set initial step sizes to 1 */
       ss = gsl_vector_alloc (2);
       gsl_vector_set_all (ss, 1.0);

       /* Initialize method and iterate */
       minex_func.n = 2;
       minex_func.f = my_f;
       minex_func.params = par;

       s = gsl_multimin_fminimizer_alloc (T, 2);
       gsl_multimin_fminimizer_set (s, &minex_func, x, ss);

       do
         {
           iter++;
           status = gsl_multimin_fminimizer_iterate(s);

           if (status)
             break;

           size = gsl_multimin_fminimizer_size (s);
           status = gsl_multimin_test_size (size, 1e-2);

           if (status == GSL_SUCCESS)
             {
               printf ("converged to minimum at\n");
             }

           printf ("%5d %10.3e %10.3e f() = %7.3f size = %.3f\n",
                   iter,
                   gsl_vector_get (s->x, 0),
                   gsl_vector_get (s->x, 1),
                   s->fval, size);
         }
       while (status == GSL_CONTINUE && iter < 100);

       gsl_vector_free(x);
       gsl_vector_free(ss);
       gsl_multimin_fminimizer_free (s);

       return status;
     }

The minimum search stops when the Simplex size drops to 0.01.  The
output is shown below.

         1  6.500e+00  5.000e+00 f() = 512.500 size = 1.130
         2  5.250e+00  4.000e+00 f() = 290.625 size = 1.409
         3  5.250e+00  4.000e+00 f() = 290.625 size = 1.409
         4  5.500e+00  1.000e+00 f() = 252.500 size = 1.409
         5  2.625e+00  3.500e+00 f() = 101.406 size = 1.847
         6  2.625e+00  3.500e+00 f() = 101.406 size = 1.847
         7  0.000e+00  3.000e+00 f() =  60.000 size = 1.847
         8  2.094e+00  1.875e+00 f() =  42.275 size = 1.321
         9  2.578e-01  1.906e+00 f() =  35.684 size = 1.069
        10  5.879e-01  2.445e+00 f() =  35.664 size = 0.841
        11  1.258e+00  2.025e+00 f() =  30.680 size = 0.476
        12  1.258e+00  2.025e+00 f() =  30.680 size = 0.367
        13  1.093e+00  1.849e+00 f() =  30.539 size = 0.300
        14  8.830e-01  2.004e+00 f() =  30.137 size = 0.172
        15  8.830e-01  2.004e+00 f() =  30.137 size = 0.126
        16  9.582e-01  2.060e+00 f() =  30.090 size = 0.106
        17  1.022e+00  2.004e+00 f() =  30.005 size = 0.063
        18  1.022e+00  2.004e+00 f() =  30.005 size = 0.043
        19  1.022e+00  2.004e+00 f() =  30.005 size = 0.043
        20  1.022e+00  2.004e+00 f() =  30.005 size = 0.027
        21  1.022e+00  2.004e+00 f() =  30.005 size = 0.022
        22  9.920e-01  1.997e+00 f() =  30.001 size = 0.016
        23  9.920e-01  1.997e+00 f() =  30.001 size = 0.013
     converged to minimum at
        24  9.920e-01  1.997e+00 f() =  30.001 size = 0.008

The simplex size first increases, while the simplex moves towards the
minimum.  After a while the size begins to decrease as the simplex
contracts around the minimum.


File: gsl-ref.info,  Node: References and Further Reading<32>,  Prev: Examples<30>,  Up: Multidimensional Minimization

39.10 References and Further Reading
====================================

The conjugate gradient and BFGS methods are described in detail in the
following book,

   * R. Fletcher, `Practical Methods of Optimization (Second Edition)'
     Wiley (1987), ISBN 0471915475.

A brief description of multidimensional minimization algorithms and more
recent references can be found in,

   * C.W. Ueberhuber, `Numerical Computation (Volume 2)', Chapter 14,
     Section 4.4 “Minimization Methods”, p.: 325–335, Springer (1997),
     ISBN 3-540-62057-5.

The simplex algorithm is described in the following paper,

   * J.A. Nelder and R. Mead, `A simplex method for function
     minimization', Computer Journal vol.: 7 (1965), 308–313.


File: gsl-ref.info,  Node: Linear Least-Squares Fitting,  Next: Nonlinear Least-Squares Fitting,  Prev: Multidimensional Minimization,  Up: Top

40 Linear Least-Squares Fitting
*******************************

This chapter describes routines for performing least squares fits to
experimental data using linear combinations of functions.  The data may
be weighted or unweighted, i.e.  with known or unknown errors.  For
weighted data the functions compute the best fit parameters and their
associated covariance matrix.  For unweighted data the covariance matrix
is estimated from the scatter of the points, giving a
variance-covariance matrix.

The functions are divided into separate versions for simple one- or
two-parameter regression and multiple-parameter fits.

* Menu:

* Overview: Overview<5>.
* Linear regression::
* Multi-parameter regression::
* Regularized regression::
* Robust linear regression::
* Large dense linear systems::
* Troubleshooting::
* Examples: Examples<31>.
* References and Further Reading: References and Further Reading<33>.


File: gsl-ref.info,  Node: Overview<5>,  Next: Linear regression,  Up: Linear Least-Squares Fitting

40.1 Overview
=============

Least-squares fits are found by minimizing \chi^2 (chi-squared), the
weighted sum of squared residuals over n experimental datapoints (x_i,
y_i) for the model Y(c,x),

     \chi^2 = \sum_i w_i (y_i - Y(c, x_i))^2

The p parameters of the model are c = \{c_0, c_1, \dots\}.  The weight
factors w_i are given by w_i = 1/\sigma_i^2 where \sigma_i is the
experimental error on the data-point y_i.  The errors are assumed to be
Gaussian and uncorrelated.  For unweighted data the chi-squared sum is
computed without any weight factors.

The fitting routines return the best-fit parameters c and their p \times
p covariance matrix.  The covariance matrix measures the statistical
errors on the best-fit parameters resulting from the errors on the data,
\sigma_i, and is defined as

     C_{ab} = <\delta c_a \delta c_b>

where \langle \, \rangle denotes an average over the Gaussian error
distributions of the underlying datapoints.

The covariance matrix is calculated by error propagation from the data
errors \sigma_i.  The change in a fitted parameter \delta c_a caused by
a small change in the data \delta y_i is given by

     \delta c_a = \sum_i (dc_a/dy_i) \delta y_i

allowing the covariance matrix to be written in terms of the errors on
the data,

     C_{ab} = \sum_{i,j} (dc_a/dy_i) (dc_b/dy_j) <\delta y_i \delta y_j>

For uncorrelated data the fluctuations of the underlying datapoints
satisfy

     <\delta y_i \delta y_j> = \sigma_i^2 \delta_{ij}

giving a corresponding parameter covariance matrix of

     C_{ab} = \sum_i (1/w_i) (dc_a/dy_i) (dc_b/dy_i)

When computing the covariance matrix for unweighted data, i.e.  data
with unknown errors, the weight factors w_i in this sum are replaced by
the single estimate w = 1/\sigma^2, where \sigma^2 is the computed
variance of the residuals about the best-fit model, \sigma^2 = \sum (y_i
- Y(c,x_i))^2 / (n-p).  This is referred to as the `variance-covariance
matrix'.

The standard deviations of the best-fit parameters are given by the
square root of the corresponding diagonal elements of the covariance
matrix, \sigma_{c_a} = \sqrt{C_{aa}}.  The correlation coefficient of
the fit parameters c_a and c_b is given by \rho_{ab} = C_{ab} /
\sqrt{C_{aa} C_{bb}}.


File: gsl-ref.info,  Node: Linear regression,  Next: Multi-parameter regression,  Prev: Overview<5>,  Up: Linear Least-Squares Fitting

40.2 Linear regression
======================

The functions in this section are used to fit simple one or two
parameter linear regression models.  The functions are declared in the
header file ‘gsl_fit.h’.

* Menu:

* Linear regression with a constant term::
* Linear regression without a constant term::


File: gsl-ref.info,  Node: Linear regression with a constant term,  Next: Linear regression without a constant term,  Up: Linear regression

40.2.1 Linear regression with a constant term
---------------------------------------------

The functions described in this section can be used to perform
least-squares fits to a straight line model, Y(c,x) = c_0 + c_1 x.

 -- Function: int gsl_fit_linear (const double * x, const
          size_t xstride, const double * y, const size_t ystride,
          size_t n, double * c0, double * c1, double * cov00, double
          * cov01, double * cov11, double * sumsq)

     This function computes the best-fit linear regression coefficients
     (‘c0’, ‘c1’) of the model Y = c_0 + c_1 X for the dataset (‘x’,
     ‘y’), two vectors of length ‘n’ with strides ‘xstride’ and
     ‘ystride’.  The errors on ‘y’ are assumed unknown so the
     variance-covariance matrix for the parameters (‘c0’, ‘c1’) is
     estimated from the scatter of the points around the best-fit line
     and returned via the parameters (‘cov00’, ‘cov01’, ‘cov11’).  The
     sum of squares of the residuals from the best-fit line is returned
     in ‘sumsq’.  Note: the correlation coefficient of the data can be
     computed using *note gsl_stats_correlation(): 7cf, it does not
     depend on the fit.

 -- Function: int gsl_fit_wlinear (const double * x, const
          size_t xstride, const double * w, const size_t wstride, const
          double * y, const size_t ystride, size_t n, double * c0,
          double * c1, double * cov00, double * cov01, double * cov11,
          double * chisq)

     This function computes the best-fit linear regression coefficients
     (‘c0’, ‘c1’) of the model Y = c_0 + c_1 X for the weighted dataset
     (‘x’, ‘y’), two vectors of length ‘n’ with strides ‘xstride’ and
     ‘ystride’.  The vector ‘w’, of length ‘n’ and stride ‘wstride’,
     specifies the weight of each datapoint.  The weight is the
     reciprocal of the variance for each datapoint in ‘y’.

     The covariance matrix for the parameters (‘c0’, ‘c1’) is computed
     using the weights and returned via the parameters (‘cov00’,
     ‘cov01’, ‘cov11’).  The weighted sum of squares of the residuals
     from the best-fit line, \chi^2, is returned in ‘chisq’.

 -- Function: int gsl_fit_linear_est (double x, double c0, double c1,
          double cov00, double cov01, double cov11, double * y, double
          * y_err)

     This function uses the best-fit linear regression coefficients
     ‘c0’, ‘c1’ and their covariance ‘cov00’, ‘cov01’, ‘cov11’ to
     compute the fitted function ‘y’ and its standard deviation ‘y_err’
     for the model Y = c_0 + c_1 X at the point ‘x’.


File: gsl-ref.info,  Node: Linear regression without a constant term,  Prev: Linear regression with a constant term,  Up: Linear regression

40.2.2 Linear regression without a constant term
------------------------------------------------

The functions described in this section can be used to perform
least-squares fits to a straight line model without a constant term, Y =
c_1 X.

 -- Function: int gsl_fit_mul (const double * x, const size_t xstride,
          const double * y, const size_t ystride, size_t n, double * c1,
          double * cov11, double * sumsq)

     This function computes the best-fit linear regression coefficient
     ‘c1’ of the model Y = c_1 X for the datasets (‘x’, ‘y’), two
     vectors of length ‘n’ with strides ‘xstride’ and ‘ystride’.  The
     errors on ‘y’ are assumed unknown so the variance of the parameter
     ‘c1’ is estimated from the scatter of the points around the
     best-fit line and returned via the parameter ‘cov11’.  The sum of
     squares of the residuals from the best-fit line is returned in
     ‘sumsq’.

 -- Function: int gsl_fit_wmul (const double * x, const size_t xstride,
          const double * w, const size_t wstride, const double * y,
          const size_t ystride, size_t n, double * c1, double * cov11,
          double * sumsq)

     This function computes the best-fit linear regression coefficient
     ‘c1’ of the model Y = c_1 X for the weighted datasets (‘x’, ‘y’),
     two vectors of length ‘n’ with strides ‘xstride’ and ‘ystride’.
     The vector ‘w’, of length ‘n’ and stride ‘wstride’, specifies the
     weight of each datapoint.  The weight is the reciprocal of the
     variance for each datapoint in ‘y’.

     The variance of the parameter ‘c1’ is computed using the weights
     and returned via the parameter ‘cov11’.  The weighted sum of
     squares of the residuals from the best-fit line, \chi^2, is
     returned in ‘chisq’.

 -- Function: int gsl_fit_mul_est (double x, double c1, double cov11,
          double * y, double * y_err)

     This function uses the best-fit linear regression coefficient ‘c1’
     and its covariance ‘cov11’ to compute the fitted function ‘y’ and
     its standard deviation ‘y_err’ for the model Y = c_1 X at the point
     ‘x’.


File: gsl-ref.info,  Node: Multi-parameter regression,  Next: Regularized regression,  Prev: Linear regression,  Up: Linear Least-Squares Fitting

40.3 Multi-parameter regression
===============================

This section describes routines which perform least squares fits to a
linear model by minimizing the cost function

     \chi^2 = \sum_i w_i (y_i - \sum_j X_{ij} c_j)^2 = || y - Xc ||_W^2

where y is a vector of n observations, X is an n-by-p matrix of
predictor variables, c is a vector of the p unknown best-fit parameters
to be estimated, and ||r||_W^2 = r^T W r.  The matrix W =
\diag(w_1,w_2,...,w_n) defines the weights or uncertainties of the
observation vector.

This formulation can be used for fits to any number of functions and/or
variables by preparing the n-by-p matrix X appropriately.  For example,
to fit to a p-th order polynomial in ‘x’, use the following matrix,

     X_{ij} = x_i^j

where the index i runs over the observations and the index j runs from 0
to p-1.

To fit to a set of p sinusoidal functions with fixed frequencies
\omega_1, \omega_2, \ldots, \omega_p, use,

     X_{ij} = \sin(\omega_j x_i)

To fit to p independent variables x_1, x_2, \ldots, x_p, use,

     X_{ij} = x_j(i)

where x_j(i) is the i-th value of the predictor variable x_j.

The solution of the general linear least-squares system requires an
additional working space for intermediate results, such as the singular
value decomposition of the matrix X.

These functions are declared in the header file ‘gsl_multifit.h’.

 -- Type: gsl_multifit_linear_workspace

     This workspace contains internal variables for fitting
     multi-parameter models.

 -- Function: gsl_multifit_linear_workspace * gsl_multifit_linear_alloc
          (const size_t n, const size_t p)

     This function allocates a workspace for fitting a model to a
     maximum of ‘n’ observations using a maximum of ‘p’ parameters.  The
     user may later supply a smaller least squares system if desired.
     The size of the workspace is O(np + p^2).

 -- Function: void gsl_multifit_linear_free
          (gsl_multifit_linear_workspace * work)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: int gsl_multifit_linear_svd (const gsl_matrix * X,
          gsl_multifit_linear_workspace * work)

     This function performs a singular value decomposition of the matrix
     ‘X’ and stores the SVD factors internally in ‘work’.

 -- Function: int gsl_multifit_linear_bsvd (const gsl_matrix * X,
          gsl_multifit_linear_workspace * work)

     This function performs a singular value decomposition of the matrix
     ‘X’ and stores the SVD factors internally in ‘work’.  The matrix
     ‘X’ is first balanced by applying column scaling factors to improve
     the accuracy of the singular values.

 -- Function: int gsl_multifit_linear (const gsl_matrix * X, const
          gsl_vector * y, gsl_vector * c, gsl_matrix * cov, double
          * chisq, gsl_multifit_linear_workspace * work)

     This function computes the best-fit parameters ‘c’ of the model y =
     X c for the observations ‘y’ and the matrix of predictor variables
     ‘X’, using the preallocated workspace provided in ‘work’.  The
     p-by-p variance-covariance matrix of the model parameters ‘cov’ is
     set to \sigma^2 (X^T X)^{-1}, where \sigma is the standard
     deviation of the fit residuals.  The sum of squares of the
     residuals from the best-fit, \chi^2, is returned in ‘chisq’.  If
     the coefficient of determination is desired, it can be computed
     from the expression R^2 = 1 - \chi^2 / TSS, where the total sum of
     squares (TSS) of the observations ‘y’ may be computed from *note
     gsl_stats_tss(): 7bc.

     The best-fit is found by singular value decomposition of the matrix
     ‘X’ using the modified Golub-Reinsch SVD algorithm, with column
     scaling to improve the accuracy of the singular values.  Any
     components which have zero singular value (to machine precision)
     are discarded from the fit.

 -- Function: int gsl_multifit_linear_tsvd (const gsl_matrix * X, const
          gsl_vector * y, const double tol, gsl_vector * c, gsl_matrix
          * cov, double * chisq, size_t * rank,
          gsl_multifit_linear_workspace * work)

     This function computes the best-fit parameters ‘c’ of the model y =
     X c for the observations ‘y’ and the matrix of predictor variables
     ‘X’, using a truncated SVD expansion.  Singular values which
     satisfy s_i \le tol \times s_0 are discarded from the fit, where
     s_0 is the largest singular value.  The p-by-p variance-covariance
     matrix of the model parameters ‘cov’ is set to \sigma^2 (X^T
     X)^{-1}, where \sigma is the standard deviation of the fit
     residuals.  The sum of squares of the residuals from the best-fit,
     \chi^2, is returned in ‘chisq’.  The effective rank (number of
     singular values used in solution) is returned in ‘rank’.  If the
     coefficient of determination is desired, it can be computed from
     the expression R^2 = 1 - \chi^2 / TSS, where the total sum of
     squares (TSS) of the observations ‘y’ may be computed from *note
     gsl_stats_tss(): 7bc.

 -- Function: int gsl_multifit_wlinear (const gsl_matrix * X, const
          gsl_vector * w, const gsl_vector * y, gsl_vector * c,
          gsl_matrix * cov, double * chisq,
          gsl_multifit_linear_workspace * work)

     This function computes the best-fit parameters ‘c’ of the weighted
     model y = X c for the observations ‘y’ with weights ‘w’ and the
     matrix of predictor variables ‘X’, using the preallocated workspace
     provided in ‘work’.  The p-by-p covariance matrix of the model
     parameters ‘cov’ is computed as (X^T W X)^{-1}.  The weighted sum
     of squares of the residuals from the best-fit, \chi^2, is returned
     in ‘chisq’.  If the coefficient of determination is desired, it can
     be computed from the expression R^2 = 1 - \chi^2 / WTSS, where the
     weighted total sum of squares (WTSS) of the observations ‘y’ may be
     computed from *note gsl_stats_wtss(): 7d9.

 -- Function: int gsl_multifit_wlinear_tsvd (const gsl_matrix * X, const
          gsl_vector * w, const gsl_vector * y, const double tol,
          gsl_vector * c, gsl_matrix * cov, double * chisq, size_t
          * rank, gsl_multifit_linear_workspace * work)

     This function computes the best-fit parameters ‘c’ of the weighted
     model y = X c for the observations ‘y’ with weights ‘w’ and the
     matrix of predictor variables ‘X’, using a truncated SVD expansion.
     Singular values which satisfy s_i \le tol \times s_0 are discarded
     from the fit, where s_0 is the largest singular value.  The p-by-p
     covariance matrix of the model parameters ‘cov’ is computed as (X^T
     W X)^{-1}.  The weighted sum of squares of the residuals from the
     best-fit, \chi^2, is returned in ‘chisq’.  The effective rank of
     the system (number of singular values used in the solution) is
     returned in ‘rank’.  If the coefficient of determination is
     desired, it can be computed from the expression R^2 = 1 - \chi^2 /
     WTSS, where the weighted total sum of squares (WTSS) of the
     observations ‘y’ may be computed from *note gsl_stats_wtss(): 7d9.

 -- Function: int gsl_multifit_linear_est (const gsl_vector * x, const
          gsl_vector * c, const gsl_matrix * cov, double * y, double
          * y_err)

     This function uses the best-fit multilinear regression coefficients
     ‘c’ and their covariance matrix ‘cov’ to compute the fitted
     function value ‘y’ and its standard deviation ‘y_err’ for the model
     y = x.c at the point ‘x’.

 -- Function: int gsl_multifit_linear_residuals (const gsl_matrix * X,
          const gsl_vector * y, const gsl_vector * c, gsl_vector * r)

     This function computes the vector of residuals r = y - X c for the
     observations ‘y’, coefficients ‘c’ and matrix of predictor
     variables ‘X’.

 -- Function: size_t gsl_multifit_linear_rank (const double tol, const
          gsl_multifit_linear_workspace * work)

     This function returns the rank of the matrix X which must first
     have its singular value decomposition computed.  The rank is
     computed by counting the number of singular values \sigma_j which
     satisfy \sigma_j > tol \times \sigma_0, where \sigma_0 is the
     largest singular value.


File: gsl-ref.info,  Node: Regularized regression,  Next: Robust linear regression,  Prev: Multi-parameter regression,  Up: Linear Least-Squares Fitting

40.4 Regularized regression
===========================

Ordinary weighted least squares models seek a solution vector c which
minimizes the residual

     \chi^2 = || y - Xc ||_W^2

where y is the n-by-1 observation vector, X is the n-by-p design matrix,
c is the p-by-1 solution vector, W = \diag(w_1,...,w_n) is the data
weighting matrix, and ||r||_W^2 = r^T W r.  In cases where the least
squares matrix X is ill-conditioned, small perturbations (ie: noise) in
the observation vector could lead to widely different solution vectors
c.  One way of dealing with ill-conditioned matrices is to use a
“truncated SVD” in which small singular values, below some given
tolerance, are discarded from the solution.  The truncated SVD method is
available using the functions *note gsl_multifit_linear_tsvd(): b13. and
*note gsl_multifit_wlinear_tsvd(): b15.  Another way to help solve
ill-posed problems is to include a regularization term in the least
squares minimization

     \chi^2 = || y - Xc ||_W^2 + \lambda^2 || L c ||^2

for a suitably chosen regularization parameter \lambda and matrix L.
This type of regularization is known as Tikhonov, or ridge, regression.
In some applications, L is chosen as the identity matrix, giving
preference to solution vectors c with smaller norms.  Including this
regularization term leads to the explicit “normal equations” solution

     c = ( X^T W X + \lambda^2 L^T L )^-1 X^T W y

which reduces to the ordinary least squares solution when L = 0.  In
practice, it is often advantageous to transform a regularized least
squares system into the form

     \chi^2 = || y~ - X~ c~ ||^2 + \lambda^2 || c~ ||^2

This is known as the Tikhonov “standard form” and has the normal
equations solution

     \tilde{c} = ( \tilde{X}^T \tilde{X} + \lambda^2 I )^{-1} \tilde{X}^T \tilde{y}

For an m-by-p matrix L which is full rank and has m >= p (ie: L is
square or has more rows than columns), we can calculate the “thin” QR
decomposition of L, and note that ||L c|| = ||R c|| since the Q factor
will not change the norm.  Since R is p-by-p, we can then use the
transformation

     X~ = sqrt(W) X R^-1
     y~ = sqrt(W) y
     c~ = R c

to achieve the standard form.  For a rectangular matrix L with m < p, a
more sophisticated approach is needed (see Hansen 1998, chapter 2.3).
In practice, the normal equations solution above is not desirable due to
numerical instabilities, and so the system is solved using the singular
value decomposition of the matrix \tilde{X}.  The matrix L is often
chosen as the identity matrix, or as a first or second finite difference
operator, to ensure a smoothly varying coefficient vector c, or as a
diagonal matrix to selectively damp each model parameter differently.
If L \ne I, the user must first convert the least squares problem to
standard form using *note gsl_multifit_linear_stdform1(): b1b. or *note
gsl_multifit_linear_stdform2(): b1c, solve the system, and then
backtransform the solution vector to recover the solution of the
original problem (see *note gsl_multifit_linear_genform1(): b1d. and
*note gsl_multifit_linear_genform2(): b1e.).

In many regularization problems, care must be taken when choosing the
regularization parameter \lambda.  Since both the residual norm ||y - X
c|| and solution norm ||L c|| are being minimized, the parameter \lambda
represents a tradeoff between minimizing either the residuals or the
solution vector.  A common tool for visualizing the comprimise between
the minimization of these two quantities is known as the L-curve.  The
L-curve is a log-log plot of the residual norm ||y - X c|| on the
horizontal axis and the solution norm ||L c|| on the vertical axis.
This curve nearly always as an L shaped appearance, with a distinct
corner separating the horizontal and vertical sections of the curve.
The regularization parameter corresponding to this corner is often
chosen as the optimal value.  GSL provides routines to calculate the
L-curve for all relevant regularization parameters as well as locating
the corner.

Another method of choosing the regularization parameter is known as
Generalized Cross Validation (GCV). This method is based on the idea
that if an arbitrary element y_i is left out of the right hand side, the
resulting regularized solution should predict this element accurately.
This leads to choosing the parameter \lambda which minimizes the GCV
function

     G(\lambda) = (||y - X c_{\lambda}||^2) / Tr(I_n - X X^I)^2

where X_{\lambda}^I is the matrix which relates the solution c_{\lambda}
to the right hand side y, ie: c_{\lambda} = X_{\lambda}^I y.  GSL
provides routines to compute the GCV curve and its minimum.

For most applications, the steps required to solve a regularized least
squares problem are as follows:

  1. Construct the least squares system (X, y, W, L)

  2. Transform the system to standard form (\tilde{X}, \tilde{y}).  This
     step can be skipped if L = I_p and W = I_n.

  3. Calculate the SVD of \tilde{X}.

  4. Determine an appropriate regularization parameter \lambda (using
     for example L-curve or GCV analysis).

  5. Solve the standard form system using the chosen \lambda and the SVD
     of \tilde{X}.

  6. Backtransform the standard form solution \tilde{c} to recover the
     original solution vector c.

 -- Function: int gsl_multifit_linear_stdform1 (const gsl_vector * L,
          const gsl_matrix * X, const gsl_vector * y, gsl_matrix * Xs,
          gsl_vector * ys, gsl_multifit_linear_workspace * work)
 -- Function: int gsl_multifit_linear_wstdform1 (const gsl_vector * L,
          const gsl_matrix * X, const gsl_vector * w, const gsl_vector
          * y, gsl_matrix * Xs, gsl_vector * ys,
          gsl_multifit_linear_workspace * work)

     These functions define a regularization matrix L =
     \diag(l_0,l_1,...,l_{p-1}).  The diagonal matrix element l_i is
     provided by the i-th element of the input vector ‘L’.  The n-by-p
     least squares matrix ‘X’ and vector ‘y’ of length n are then
     converted to standard form as described above and the parameters
     (\tilde{X}, \tilde{y}) are stored in ‘Xs’ and ‘ys’ on output.  ‘Xs’
     and ‘ys’ have the same dimensions as ‘X’ and ‘y’.  Optional data
     weights may be supplied in the vector ‘w’ of length n.  In order to
     apply this transformation, L^{-1} must exist and so none of the l_i
     may be zero.  After the standard form system has been solved, use
     *note gsl_multifit_linear_genform1(): b1d. to recover the original
     solution vector.  It is allowed to have ‘X’ = ‘Xs’ and ‘y’ = ‘ys’
     for an in-place transform.  In order to perform a weighted
     regularized fit with L = I, the user may call *note
     gsl_multifit_linear_applyW(): b20. to convert to standard form.

 -- Function: int gsl_multifit_linear_L_decomp (gsl_matrix * L,
          gsl_vector * tau)

     This function factors the m-by-p regularization matrix ‘L’ into a
     form needed for the later transformation to standard form.  ‘L’ may
     have any number of rows m.  If m \ge p the QR decomposition of ‘L’
     is computed and stored in ‘L’ on output.  If m < p, the QR
     decomposition of L^T is computed and stored in ‘L’ on output.  On
     output, the Householder scalars are stored in the vector ‘tau’ of
     size MIN(m,p).  These outputs will be used by *note
     gsl_multifit_linear_wstdform2(): b22. to complete the
     transformation to standard form.

 -- Function: int gsl_multifit_linear_stdform2 (const gsl_matrix * LQR,
          const gsl_vector * Ltau, const gsl_matrix * X, const
          gsl_vector * y, gsl_matrix * Xs, gsl_vector * ys, gsl_matrix
          * M, gsl_multifit_linear_workspace * work)
 -- Function: int gsl_multifit_linear_wstdform2 (const gsl_matrix * LQR,
          const gsl_vector * Ltau, const gsl_matrix * X, const
          gsl_vector * w, const gsl_vector * y, gsl_matrix * Xs,
          gsl_vector * ys, gsl_matrix * M, gsl_multifit_linear_workspace
          * work)

     These functions convert the least squares system (‘X’, ‘y’, ‘W’, L)
     to standard form (\tilde{X}, \tilde{y}) which are stored in ‘Xs’
     and ‘ys’ respectively.  The m-by-p regularization matrix ‘L’ is
     specified by the inputs ‘LQR’ and ‘Ltau’, which are outputs from
     *note gsl_multifit_linear_L_decomp(): b21.  The dimensions of the
     standard form parameters (\tilde{X}, \tilde{y}) depend on whether m
     is larger or less than p.  For m \ge p, ‘Xs’ is n-by-p, ‘ys’ is
     n-by-1, and ‘M’ is not used.  For m < p, ‘Xs’ is (n - p + m)-by-m,
     ‘ys’ is (n - p + m)-by-1, and ‘M’ is additional n-by-p workspace,
     which is required to recover the original solution vector after the
     system has been solved (see *note gsl_multifit_linear_genform2():
     b1e.).  Optional data weights may be supplied in the vector ‘w’ of
     length n, where W = \diag(w).

 -- Function: int gsl_multifit_linear_solve (const double lambda, const
          gsl_matrix * Xs, const gsl_vector * ys, gsl_vector * cs,
          double * rnorm, double * snorm, gsl_multifit_linear_workspace
          * work)

     This function computes the regularized best-fit parameters
     \tilde{c} which minimize the cost function \chi^2 = || \tilde{y} -
     \tilde{X} \tilde{c} ||^2 + \lambda^2 || \tilde{c} ||^2 which is in
     standard form.  The least squares system must therefore be
     converted to standard form prior to calling this function.  The
     observation vector \tilde{y} is provided in ‘ys’ and the matrix of
     predictor variables \tilde{X} in ‘Xs’.  The solution vector
     \tilde{c} is returned in ‘cs’, which has length min(m,p).  The SVD
     of ‘Xs’ must be computed prior to calling this function, using
     *note gsl_multifit_linear_svd(): b10.  The regularization parameter
     \lambda is provided in ‘lambda’.  The residual norm || \tilde{y} -
     \tilde{X} \tilde{c} || = ||y - X c||_W is returned in ‘rnorm’.  The
     solution norm || \tilde{c} || = ||L c|| is returned in ‘snorm’.

 -- Function: int gsl_multifit_linear_genform1 (const gsl_vector * L,
          const gsl_vector * cs, gsl_vector * c,
          gsl_multifit_linear_workspace * work)

     After a regularized system has been solved with L =
     \diag(\l_0,\l_1,...,\l_{p-1}), this function backtransforms the
     standard form solution vector ‘cs’ to recover the solution vector
     of the original problem ‘c’.  The diagonal matrix elements l_i are
     provided in the vector ‘L’.  It is allowed to have ‘c’ = ‘cs’ for
     an in-place transform.

 -- Function: int gsl_multifit_linear_genform2 (const gsl_matrix * LQR,
          const gsl_vector * Ltau, const gsl_matrix * X, const
          gsl_vector * y, const gsl_vector * cs, const gsl_matrix * M,
          gsl_vector * c, gsl_multifit_linear_workspace * work)
 -- Function: int gsl_multifit_linear_wgenform2 (const gsl_matrix * LQR,
          const gsl_vector * Ltau, const gsl_matrix * X, const
          gsl_vector * w, const gsl_vector * y, const gsl_vector * cs,
          const gsl_matrix * M, gsl_vector * c,
          gsl_multifit_linear_workspace * work)

     After a regularized system has been solved with a general
     rectangular matrix L, specified by (‘LQR’, ‘Ltau’), this function
     backtransforms the standard form solution ‘cs’ to recover the
     solution vector of the original problem, which is stored in ‘c’, of
     length p.  The original least squares matrix and observation vector
     are provided in ‘X’ and ‘y’ respectively.  ‘M’ is the matrix
     computed by *note gsl_multifit_linear_stdform2(): b1c.  For
     weighted fits, the weight vector ‘w’ must also be supplied.

 -- Function: int gsl_multifit_linear_applyW (const gsl_matrix * X,
          const gsl_vector * w, const gsl_vector * y, gsl_matrix * WX,
          gsl_vector * Wy)

     For weighted least squares systems with L = I, this function may be
     used to convert the system to standard form by applying the weight
     matrix W = \diag(w) to the least squares matrix ‘X’ and observation
     vector ‘y’.  On output, ‘WX’ is equal to W^{1/2} X and ‘Wy’ is
     equal to W^{1/2} y.  It is allowed for ‘WX’ = ‘X’ and ‘Wy’ = ‘y’
     for an in-place transform.

 -- Function: int gsl_multifit_linear_lcurve (const gsl_vector * y,
          gsl_vector * reg_param, gsl_vector * rho, gsl_vector * eta,
          gsl_multifit_linear_workspace * work)

     This function computes the L-curve for a least squares system using
     the right hand side vector ‘y’ and the SVD decomposition of the
     least squares matrix ‘X’, which must be provided to *note
     gsl_multifit_linear_svd(): b10. prior to calling this function.
     The output vectors ‘reg_param’, ‘rho’, and ‘eta’ must all be the
     same size, and will contain the regularization parameters
     \lambda_i, residual norms ||y - X c_i||, and solution norms || L
     c_i || which compose the L-curve, where c_i is the regularized
     solution vector corresponding to \lambda_i.  The user may determine
     the number of points on the L-curve by adjusting the size of these
     input arrays.  The regularization parameters \lambda_i are
     estimated from the singular values of ‘X’, and chosen to represent
     the most relevant portion of the L-curve.

 -- Function: int gsl_multifit_linear_lcorner (const gsl_vector * rho,
          const gsl_vector * eta, size_t * idx)

     This function attempts to locate the corner of the L-curve (||y - X
     c||, ||L c||) defined by the ‘rho’ and ‘eta’ input arrays
     respectively.  The corner is defined as the point of maximum
     curvature of the L-curve in log-log scale.  The ‘rho’ and ‘eta’
     arrays can be outputs of *note gsl_multifit_linear_lcurve(): b25.
     The algorithm used simply fits a circle to 3 consecutive points on
     the L-curve and uses the circle’s radius to determine the curvature
     at the middle point.  Therefore, the input array sizes must be \ge
     3.  With more points provided for the L-curve, a better estimate of
     the curvature can be obtained.  The array index corresponding to
     maximum curvature (ie: the corner) is returned in ‘idx’.  If the
     input arrays contain colinear points, this function could fail and
     return *note GSL_EINVAL: 2b.

 -- Function: int gsl_multifit_linear_lcorner2 (const gsl_vector
          * reg_param, const gsl_vector * eta, size_t * idx)

     This function attempts to locate the corner of an alternate L-curve
     (\lambda^2, ||L c||^2) studied by Rezghi and Hosseini, 2009.  This
     alternate L-curve can provide better estimates of the
     regularization parameter for smooth solution vectors.  The
     regularization parameters \lambda and solution norms ||L c|| are
     provided in the ‘reg_param’ and ‘eta’ input arrays respectively.
     The corner is defined as the point of maximum curvature of this
     alternate L-curve in linear scale.  The ‘reg_param’ and ‘eta’
     arrays can be outputs of *note gsl_multifit_linear_lcurve(): b25.
     The algorithm used simply fits a circle to 3 consecutive points on
     the L-curve and uses the circle’s radius to determine the curvature
     at the middle point.  Therefore, the input array sizes must be \ge
     3.  With more points provided for the L-curve, a better estimate of
     the curvature can be obtained.  The array index corresponding to
     maximum curvature (ie: the corner) is returned in ‘idx’.  If the
     input arrays contain colinear points, this function could fail and
     return *note GSL_EINVAL: 2b.

 -- Function: int gsl_multifit_linear_gcv_init (const gsl_vector * y,
          gsl_vector * reg_param, gsl_vector * UTy, double * delta0,
          gsl_multifit_linear_workspace * work)

     This function performs some initialization in preparation for
     computing the GCV curve and its minimum.  The right hand side
     vector is provided in ‘y’.  On output, ‘reg_param’ is set to a
     vector of regularization parameters in decreasing order and may be
     of any size.  The vector ‘UTy’ of size p is set to U^T y.  The
     parameter ‘delta0’ is needed for subsequent steps of the GCV
     calculation.

 -- Function: int gsl_multifit_linear_gcv_curve (const gsl_vector
          * reg_param, const gsl_vector * UTy, const double delta0,
          gsl_vector * G, gsl_multifit_linear_workspace * work)

     This funtion calculates the GCV curve G(\lambda) and stores it in
     ‘G’ on output, which must be the same size as ‘reg_param’.  The
     inputs ‘reg_param’, ‘UTy’ and ‘delta0’ are computed in *note
     gsl_multifit_linear_gcv_init(): b28.

 -- Function: int gsl_multifit_linear_gcv_min (const gsl_vector
          * reg_param, const gsl_vector * UTy, const gsl_vector * G,
          const double delta0, double * lambda,
          gsl_multifit_linear_workspace * work)

     This function computes the value of the regularization parameter
     which minimizes the GCV curve G(\lambda) and stores it in ‘lambda’.
     The input ‘G’ is calculated by *note
     gsl_multifit_linear_gcv_curve(): b29. and the inputs ‘reg_param’,
     ‘UTy’ and ‘delta0’ are computed by *note
     gsl_multifit_linear_gcv_init(): b28.

 -- Function: double gsl_multifit_linear_gcv_calc (const double lambda,
          const gsl_vector * UTy, const double delta0,
          gsl_multifit_linear_workspace * work)

     This function returns the value of the GCV curve G(\lambda)
     corresponding to the input ‘lambda’.

 -- Function: int gsl_multifit_linear_gcv (const gsl_vector * y,
          gsl_vector * reg_param, gsl_vector * G, double * lambda,
          double * G_lambda, gsl_multifit_linear_workspace * work)

     This function combines the steps ‘gcv_init’, ‘gcv_curve’, and
     ‘gcv_min’ defined above into a single function.  The input ‘y’ is
     the right hand side vector.  On output, ‘reg_param’ and ‘G’, which
     must be the same size, are set to vectors of \lambda and G(\lambda)
     values respectively.  The output ‘lambda’ is set to the optimal
     value of \lambda which minimizes the GCV curve.  The minimum value
     of the GCV curve is returned in ‘G_lambda’.

 -- Function: int gsl_multifit_linear_Lk (const size_t p, const
          size_t k, gsl_matrix * L)

     This function computes the discrete approximation to the derivative
     operator L_k of order ‘k’ on a regular grid of ‘p’ points and
     stores it in ‘L’.  The dimensions of ‘L’ are (p-k)-by-p.

 -- Function: int gsl_multifit_linear_Lsobolev (const size_t p, const
          size_t kmax, const gsl_vector * alpha, gsl_matrix * L,
          gsl_multifit_linear_workspace * work)

     This function computes the regularization matrix ‘L’ corresponding
     to the weighted Sobolov norm ||L c||^2 = \sum_k \alpha_k^2 ||L_k
     c||^2 where L_k approximates the derivative operator of order k.
     This regularization norm can be useful in applications where it is
     necessary to smooth several derivatives of the solution.  ‘p’ is
     the number of model parameters, ‘kmax’ is the highest derivative to
     include in the summation above, and *note alpha: 49a. is the vector
     of weights of size ‘kmax’ + 1, where ‘alpha[k]’ = \alpha_k is the
     weight assigned to the derivative of order k.  The output matrix
     ‘L’ is size ‘p’-by-‘p’ and upper triangular.

 -- Function: double gsl_multifit_linear_rcond (const
          gsl_multifit_linear_workspace * work)

     This function returns the reciprocal condition number of the least
     squares matrix X, defined as the ratio of the smallest and largest
     singular values, rcond = \sigma_{min}/\sigma_{max}.  The routine
     *note gsl_multifit_linear_svd(): b10. must first be called to
     compute the SVD of X.


File: gsl-ref.info,  Node: Robust linear regression,  Next: Large dense linear systems,  Prev: Regularized regression,  Up: Linear Least-Squares Fitting

40.5 Robust linear regression
=============================

Ordinary least squares (OLS) models are often heavily influenced by the
presence of outliers.  Outliers are data points which do not follow the
general trend of the other observations, although there is strictly no
precise definition of an outlier.  Robust linear regression refers to
regression algorithms which are robust to outliers.  The most common
type of robust regression is M-estimation.  The general M-estimator
minimizes the objective function

     \sum_i \rho(e_i) = \sum_i \rho (y_i - Y(c, x_i))

where e_i = y_i - Y(c, x_i) is the residual of the ith data point, and
\rho(e_i) is a function which should have the following properties:

   * \rho(e) \ge 0

   * \rho(0) = 0

   * \rho(-e) = \rho(e)

   * \rho(e_1) > \rho(e_2) for |e_1| > |e_2|

The special case of ordinary least squares is given by \rho(e_i) =
e_i^2.  Letting \psi = \rho' be the derivative of \rho, differentiating
the objective function with respect to the coefficients c and setting
the partial derivatives to zero produces the system of equations

     \sum_i \psi(e_i) X_i = 0

where X_i is a vector containing row i of the design matrix X. Next, we
define a weight function w(e) = \psi(e)/e, and let w_i = w(e_i):

     \sum_i w_i e_i X_i = 0

This system of equations is equivalent to solving a weighted ordinary
least squares problem, minimizing \chi^2 = \sum_i w_i e_i^2.  The
weights however, depend on the residuals e_i, which depend on the
coefficients c, which depend on the weights.  Therefore, an iterative
solution is used, called Iteratively Reweighted Least Squares (IRLS).

  1. Compute initial estimates of the coefficients c^{(0)} using
     ordinary least squares

  2. For iteration k, form the residuals e_i^{(k)} = (y_i - X_i
     c^{(k-1)})/(t \sigma^{(k)} \sqrt{1 - h_i}), where t is a tuning
     constant depending on the choice of \psi, and h_i are the
     statistical leverages (diagonal elements of the matrix X (X^T
     X)^{-1} X^T). Including t and h_i in the residual calculation has
     been shown to improve the convergence of the method.  The residual
     standard deviation is approximated as \sigma^{(k)} = MAD / 0.6745,
     where MAD is the Median-Absolute-Deviation of the n-p largest
     residuals from the previous iteration.

  3. Compute new weights w_i^{(k)} = \psi(e_i^{(k)})/e_i^{(k)}.

  4. Compute new coefficients c^{(k)} by solving the weighted least
     squares problem with weights w_i^{(k)}.

  5. Steps 2 through 4 are iterated until the coefficients converge or
     until some maximum iteration limit is reached.  Coefficients are
     tested for convergence using the critera:

          |c_i^(k) - c_i^(k-1)| <= \epsilon * max(|c_i^(k)|, |c_i^(k-1)|)

     for all 0 \le i < p where \epsilon is a small tolerance factor.

The key to this method lies in selecting the function \psi(e_i) to
assign smaller weights to large residuals, and larger weights to smaller
residuals.  As the iteration proceeds, outliers are assigned smaller and
smaller weights, eventually having very little or no effect on the
fitted model.

 -- Type: gsl_multifit_robust_workspace

     This workspace is used for robust least squares fitting.

 -- Function: gsl_multifit_robust_workspace * gsl_multifit_robust_alloc
          (const gsl_multifit_robust_type * T, const size_t n, const
          size_t p)

     This function allocates a workspace for fitting a model to ‘n’
     observations using ‘p’ parameters.  The size of the workspace is
     O(np + p^2).  The type ‘T’ specifies the function \psi and can be
     selected from the following choices.

      -- Type: gsl_multifit_robust_type

           -- Variable: gsl_multifit_robust_default

               This specifies the *note gsl_multifit_robust_bisquare:
               b35. type (see below) and is a good general purpose
               choice for robust regression.

           -- Variable: gsl_multifit_robust_bisquare

               This is Tukey’s biweight (bisquare) function and is a
               good general purpose choice for robust regression.  The
               weight function is given by

                    w(e) = { (1 - e^2)^2, |e| <= 1
                           {     0,       |e| > 1

               and the default tuning constant is t = 4.685.

           -- Variable: gsl_multifit_robust_cauchy

               This is Cauchy’s function, also known as the Lorentzian
               function.  This function does not guarantee a unique
               solution, meaning different choices of the coefficient
               vector ‘c’ could minimize the objective function.
               Therefore this option should be used with care.  The
               weight function is given by

                    w(e) = 1 / (1 + e^2)

               and the default tuning constant is t = 2.385.

           -- Variable: gsl_multifit_robust_fair

               This is the fair \rho function, which guarantees a unique
               solution and has continuous derivatives to three orders.
               The weight function is given by

                    w(e) = 1 / (1 + |e|)

               and the default tuning constant is t = 1.400.

           -- Variable: gsl_multifit_robust_huber

               This specifies Huber’s \rho function, which is a parabola
               in the vicinity of zero and increases linearly for a
               given threshold |e| > t.  This function is also
               considered an excellent general purpose robust estimator,
               however, occasional difficulties can be encountered due
               to the discontinuous first derivative of the \psi
               function.  The weight function is given by

                    w(e) = 1/max(1,|e|)

               and the default tuning constant is t = 1.345.

           -- Variable: gsl_multifit_robust_ols

               This specifies the ordinary least squares solution, which
               can be useful for quickly checking the difference between
               the various robust and OLS solutions.  The weight
               function is given by

                    w(e) = 1

               and the default tuning constant is t = 1.

           -- Variable: gsl_multifit_robust_welsch

               This specifies the Welsch function which can perform well
               in cases where the residuals have an exponential
               distribution.  The weight function is given by

                    w(e) = \exp{(-e^2)}

               and the default tuning constant is t = 2.985.

 -- Function: void gsl_multifit_robust_free
          (gsl_multifit_robust_workspace * w)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: const char * gsl_multifit_robust_name (const
          gsl_multifit_robust_workspace * w)

     This function returns the name of the robust type ‘T’ specified to
     *note gsl_multifit_robust_alloc(): b32.

 -- Function: int gsl_multifit_robust_tune (const double tune,
          gsl_multifit_robust_workspace * w)

     This function sets the tuning constant t used to adjust the
     residuals at each iteration to ‘tune’.  Decreasing the tuning
     constant increases the downweight assigned to large residuals,
     while increasing the tuning constant decreases the downweight
     assigned to large residuals.

 -- Function: int gsl_multifit_robust_maxiter (const size_t maxiter,
          gsl_multifit_robust_workspace * w)

     This function sets the maximum number of iterations in the
     iteratively reweighted least squares algorithm to ‘maxiter’.  By
     default, this value is set to 100 by *note
     gsl_multifit_robust_alloc(): b32.

 -- Function: int gsl_multifit_robust_weights (const gsl_vector * r,
          gsl_vector * wts, gsl_multifit_robust_workspace * w)

     This function assigns weights to the vector ‘wts’ using the
     residual vector ‘r’ and previously specified weighting function.
     The output weights are given by wts_i = w(r_i / (t \sigma)), where
     the weighting functions w are detailed in *note
     gsl_multifit_robust_alloc(): b32.  \sigma is an estimate of the
     residual standard deviation based on the Median-Absolute-Deviation
     and t is the tuning constant.  This function is useful if the user
     wishes to implement their own robust regression rather than using
     the supplied *note gsl_multifit_robust(): b40. routine below.

 -- Function: int gsl_multifit_robust (const gsl_matrix * X, const
          gsl_vector * y, gsl_vector * c, gsl_matrix * cov,
          gsl_multifit_robust_workspace * w)

     This function computes the best-fit parameters ‘c’ of the model y =
     X c for the observations ‘y’ and the matrix of predictor variables
     ‘X’, attemping to reduce the influence of outliers using the
     algorithm outlined above.  The p-by-p variance-covariance matrix of
     the model parameters ‘cov’ is estimated as \sigma^2 (X^T X)^{-1},
     where \sigma is an approximation of the residual standard deviation
     using the theory of robust regression.  Special care must be taken
     when estimating \sigma and other statistics such as R^2, and so
     these are computed internally and are available by calling the
     function *note gsl_multifit_robust_statistics(): b41.

     If the coefficients do not converge within the maximum iteration
     limit, the function returns ‘GSL_EMAXITER’.  In this case, the
     current estimates of the coefficients and covariance matrix are
     returned in ‘c’ and ‘cov’ and the internal fit statistics are
     computed with these estimates.

 -- Function: int gsl_multifit_robust_est (const gsl_vector * x, const
          gsl_vector * c, const gsl_matrix * cov, double * y, double
          * y_err)

     This function uses the best-fit robust regression coefficients ‘c’
     and their covariance matrix ‘cov’ to compute the fitted function
     value ‘y’ and its standard deviation ‘y_err’ for the model y = x
     \cdot c at the point ‘x’.

 -- Function: int gsl_multifit_robust_residuals (const gsl_matrix * X,
          const gsl_vector * y, const gsl_vector * c, gsl_vector * r,
          gsl_multifit_robust_workspace * w)

     This function computes the vector of studentized residuals r_i =
     {y_i - (X c)_i \over \sigma \sqrt{1 - h_i}} for the observations
     ‘y’, coefficients ‘c’ and matrix of predictor variables ‘X’.  The
     routine *note gsl_multifit_robust(): b40. must first be called to
     compute the statisical leverages h_i of the matrix ‘X’ and residual
     standard deviation estimate \sigma.

 -- Function: gsl_multifit_robust_stats gsl_multifit_robust_statistics
          (const gsl_multifit_robust_workspace * w)

     This function returns a structure containing relevant statistics
     from a robust regression.  The function *note
     gsl_multifit_robust(): b40. must be called first to perform the
     regression and calculate these statistics.  The returned *note
     gsl_multifit_robust_stats: b44. structure contains the following
     fields.

      -- Type: gsl_multifit_robust_stats

          ‘double sigma_ols’

               This contains the standard deviation of the residuals as
               computed from ordinary least squares (OLS).

          ‘double sigma_mad’

               This contains an estimate of the standard deviation of
               the final residuals using the Median-Absolute-Deviation
               statistic

          ‘double sigma_rob’

               This contains an estimate of the standard deviation of
               the final residuals from the theory of robust regression
               (see Street et al, 1988).

          ‘double sigma’

               This contains an estimate of the standard deviation of
               the final residuals by attemping to reconcile ‘sigma_rob’
               and ‘sigma_ols’ in a reasonable way.

          ‘double Rsq’

               This contains the R^2 coefficient of determination
               statistic using the estimate ‘sigma’.

          ‘double adj_Rsq’

               This contains the adjusted R^2 coefficient of
               determination statistic using the estimate ‘sigma’.

          ‘double rmse’

               This contains the root mean squared error of the final
               residuals

          ‘double sse’

               This contains the residual sum of squares taking into
               account the robust covariance matrix.

          ‘size_t dof’

               This contains the number of degrees of freedom n - p

          ‘size_t numit’

               Upon successful convergence, this contains the number of
               iterations performed

          ‘gsl_vector * weights’

               This contains the final weight vector of length ‘n’

          ‘gsl_vector * r’

               This contains the final residual vector of length ‘n’, r
               = y - X c


File: gsl-ref.info,  Node: Large dense linear systems,  Next: Troubleshooting,  Prev: Robust linear regression,  Up: Linear Least-Squares Fitting

40.6 Large dense linear systems
===============================

This module is concerned with solving large dense least squares systems
X c = y where the n-by-p matrix X has n >> p (ie: many more rows than
columns).  This type of matrix is called a “tall skinny” matrix, and for
some applications, it may not be possible to fit the entire matrix in
memory at once to use the standard SVD approach.  Therefore, the
algorithms in this module are designed to allow the user to construct
smaller blocks of the matrix X and accumulate those blocks into the
larger system one at a time.  The algorithms in this module never need
to store the entire matrix X in memory.  The large linear least squares
routines support data weights and Tikhonov regularization, and are
designed to minimize the residual

     \chi^2 = || y - Xc ||_W^2 + \lambda^2 || L c ||^2

where y is the n-by-1 observation vector, X is the n-by-p design matrix,
c is the p-by-1 solution vector, W = \diag(w_1,...,w_n) is the data
weighting matrix, L is an m-by-p regularization matrix, \lambda is a
regularization parameter, and ||r||_W^2 = r^T W r.  In the discussion
which follows, we will assume that the system has been converted into
Tikhonov standard form,

     \chi^2 = || y~ - X~ c~ ||^2 + \lambda^2 || c~ ||^2

and we will drop the tilde characters from the various parameters.  For
a discussion of the transformation to standard form, see *note
Regularized regression: b1a.

The basic idea is to partition the matrix X and observation vector y as

     [ X_1 ] c = [ y_1 ]
     [ X_2 ]     [ y_2 ]
     [ X_3 ]     [ y_3 ]
     [ ... ]     [ ... ]
     [ X_k ]     [ y_k ]

into k blocks, where each block (X_i,y_i) may have any number of rows,
but each X_i has p columns.  The sections below describe the methods
available for solving this partitioned system.  The functions are
declared in the header file ‘gsl_multilarge.h’.

* Menu:

* Normal Equations Approach::
* Tall Skinny QR (TSQR) Approach: Tall Skinny QR TSQR Approach.
* Large Dense Linear Systems Solution Steps::
* Large Dense Linear Least Squares Routines::


File: gsl-ref.info,  Node: Normal Equations Approach,  Next: Tall Skinny QR TSQR Approach,  Up: Large dense linear systems

40.6.1 Normal Equations Approach
--------------------------------

The normal equations approach to the large linear least squares problem
described above is popular due to its speed and simplicity.  Since the
normal equations solution to the problem is given by

     c = ( X^T X + \lambda^2 I )^-1 X^T y

only the p-by-p matrix X^T X and p-by-1 vector X^T y need to be stored.
Using the partition scheme described above, these are given by

     X^T X = \sum_i X_i^T X_i
     X^T y = \sum_i X_i^T y_i

Since the matrix X^T X is symmetric, only half of it needs to be
calculated.  Once all of the blocks (X_i,y_i) have been accumulated into
the final X^T X and X^T y, the system can be solved with a Cholesky
factorization of the X^T X matrix.  The X^T X matrix is first
transformed via a diagonal scaling transformation to attempt to reduce
its condition number as much as possible to recover a more accurate
solution vector.  The normal equations approach is the fastest method
for solving the large least squares problem, and is accurate for
well-conditioned matrices X. However, for ill-conditioned matrices, as
is often the case for large systems, this method can suffer from
numerical instabilities (see Trefethen and Bau, 1997).  The number of
operations for this method is O(np^2 + {1 \over 3}p^3).


File: gsl-ref.info,  Node: Tall Skinny QR TSQR Approach,  Next: Large Dense Linear Systems Solution Steps,  Prev: Normal Equations Approach,  Up: Large dense linear systems

40.6.2 Tall Skinny QR (TSQR) Approach
-------------------------------------

An algorithm which has better numerical stability for ill-conditioned
problems is known as the Tall Skinny QR (TSQR) method.  This method is
based on computing the thin QR decomposition of the least squares matrix
X = Q R, where Q is an n-by-p matrix with orthogonal columns, and R is a
p-by-p upper triangular matrix.  Once these factors are calculated, the
residual becomes

     \chi^2 = || Q^T y - R c ||^2 + \lambda^2 || c ||^2

which can be written as the matrix equation

     [ R ; \lambda I ] c = [ Q^T b ; 0 ]

The matrix on the left hand side is now a much smaller 2p-by-p matrix
which can be solved with a standard SVD approach.  The Q matrix is just
as large as the original matrix X, however it does not need to be
explicitly constructed.  The TSQR algorithm computes only the p-by-p
matrix R and the p-by-1 vector Q^T y, and updates these quantities as
new blocks are added to the system.  Each time a new block of rows
(X_i,y_i) is added, the algorithm performs a QR decomposition of the
matrix

     [ R_(i-1) ; X_i ]

where R_{i-1} is the upper triangular R factor for the matrix

     [ X_1 ; ... ; X_(i-1) ]

This QR decomposition is done efficiently taking into account the sparse
structure of R_{i-1}.  See Demmel et al, 2008 for more details on how
this is accomplished.  The number of operations for this method is
O(2np^2 - {2 \over 3}p^3).


File: gsl-ref.info,  Node: Large Dense Linear Systems Solution Steps,  Next: Large Dense Linear Least Squares Routines,  Prev: Tall Skinny QR TSQR Approach,  Up: Large dense linear systems

40.6.3 Large Dense Linear Systems Solution Steps
------------------------------------------------

The typical steps required to solve large regularized linear least
squares problems are as follows:

  1. Choose the regularization matrix L.

  2. Construct a block of rows of the least squares matrix, right hand
     side vector, and weight vector (X_i, y_i, w_i).

  3. Transform the block to standard form (\tilde{X_i}, \tilde{y_i}).
     This step can be skipped if L = I and W = I.

  4. Accumulate the standard form block (\tilde{X_i}, \tilde{y_i}) into
     the system.

  5. Repeat steps 2-4 until the entire matrix and right hand side vector
     have been accumulated.

  6. Determine an appropriate regularization parameter \lambda (using
     for example L-curve analysis).

  7. Solve the standard form system using the chosen \lambda.

  8. Backtransform the standard form solution \tilde{c} to recover the
     original solution vector c.


File: gsl-ref.info,  Node: Large Dense Linear Least Squares Routines,  Prev: Large Dense Linear Systems Solution Steps,  Up: Large dense linear systems

40.6.4 Large Dense Linear Least Squares Routines
------------------------------------------------

 -- Type: gsl_multilarge_linear_workspace

     This workspace contains parameters for solving large linear least
     squares problems.

 -- Function: gsl_multilarge_linear_workspace *
          gsl_multilarge_linear_alloc (const gsl_multilarge_linear_type
          * T, const size_t p)

     This function allocates a workspace for solving large linear least
     squares systems.  The least squares matrix X has ‘p’ columns, but
     may have any number of rows.

      -- Type: gsl_multilarge_linear_type

          The parameter ‘T’ specifies the method to be used for solving
          the large least squares system and may be selected from the
          following choices

           -- Variable: gsl_multilarge_linear_normal

               This specifies the normal equations approach for solving
               the least squares system.  This method is suitable in
               cases where performance is critical and it is known that
               the least squares matrix X is well conditioned.  The size
               of this workspace is O(p^2).

           -- Variable: gsl_multilarge_linear_tsqr

               This specifies the sequential Tall Skinny QR (TSQR)
               approach for solving the least squares system.  This
               method is a good general purpose choice for large
               systems, but requires about twice as many operations as
               the normal equations method for n >> p.  The size of this
               workspace is O(p^2).

 -- Function: void gsl_multilarge_linear_free
          (gsl_multilarge_linear_workspace * w)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: const char * gsl_multilarge_linear_name
          (gsl_multilarge_linear_workspace * w)

     This function returns a string pointer to the name of the
     multilarge solver.

 -- Function: int gsl_multilarge_linear_reset
          (gsl_multilarge_linear_workspace * w)

     This function resets the workspace ‘w’ so it can begin to
     accumulate a new least squares system.

 -- Function: int gsl_multilarge_linear_stdform1 (const gsl_vector * L,
          const gsl_matrix * X, const gsl_vector * y, gsl_matrix * Xs,
          gsl_vector * ys, gsl_multilarge_linear_workspace * work)
 -- Function: int gsl_multilarge_linear_wstdform1 (const gsl_vector * L,
          const gsl_matrix * X, const gsl_vector * w, const gsl_vector
          * y, gsl_matrix * Xs, gsl_vector * ys,
          gsl_multilarge_linear_workspace * work)

     These functions define a regularization matrix L =
     \diag(l_0,l_1,...,l_{p-1}).  The diagonal matrix element l_i is
     provided by the i-th element of the input vector ‘L’.  The block
     (‘X’, ‘y’) is converted to standard form and the parameters
     (\tilde{X}, \tilde{y}) are stored in ‘Xs’ and ‘ys’ on output.  ‘Xs’
     and ‘ys’ have the same dimensions as ‘X’ and ‘y’.  Optional data
     weights may be supplied in the vector ‘w’.  In order to apply this
     transformation, L^{-1} must exist and so none of the l_i may be
     zero.  After the standard form system has been solved, use *note
     gsl_multilarge_linear_genform1(): b54. to recover the original
     solution vector.  It is allowed to have ‘X’ = ‘Xs’ and ‘y’ = ‘ys’
     for an in-place transform.

 -- Function: int gsl_multilarge_linear_L_decomp (gsl_matrix * L,
          gsl_vector * tau)

     This function calculates the QR decomposition of the m-by-p
     regularization matrix ‘L’.  ‘L’ must have m \ge p.  On output, the
     Householder scalars are stored in the vector ‘tau’ of size p.
     These outputs will be used by *note
     gsl_multilarge_linear_wstdform2(): b56. to complete the
     transformation to standard form.

 -- Function: int gsl_multilarge_linear_stdform2 (const gsl_matrix
          * LQR, const gsl_vector * Ltau, const gsl_matrix * X, const
          gsl_vector * y, gsl_matrix * Xs, gsl_vector * ys,
          gsl_multilarge_linear_workspace * work)
 -- Function: int gsl_multilarge_linear_wstdform2 (const gsl_matrix
          * LQR, const gsl_vector * Ltau, const gsl_matrix * X, const
          gsl_vector * w, const gsl_vector * y, gsl_matrix * Xs,
          gsl_vector * ys, gsl_multilarge_linear_workspace * work)

     These functions convert a block of rows (‘X’, ‘y’, ‘w’) to standard
     form (\tilde{X}, \tilde{y}) which are stored in ‘Xs’ and ‘ys’
     respectively.  ‘X’, ‘y’, and ‘w’ must all have the same number of
     rows.  The m-by-p regularization matrix ‘L’ is specified by the
     inputs ‘LQR’ and ‘Ltau’, which are outputs from *note
     gsl_multilarge_linear_L_decomp(): b55.  ‘Xs’ and ‘ys’ have the same
     dimensions as ‘X’ and ‘y’.  After the standard form system has been
     solved, use *note gsl_multilarge_linear_genform2(): b58. to recover
     the original solution vector.  Optional data weights may be
     supplied in the vector ‘w’, where W = \diag(w).

 -- Function: int gsl_multilarge_linear_accumulate (gsl_matrix * X,
          gsl_vector * y, gsl_multilarge_linear_workspace * w)

     This function accumulates the standard form block (X,y) into the
     current least squares system.  ‘X’ and ‘y’ have the same number of
     rows, which can be arbitrary.  ‘X’ must have p columns.  For the
     TSQR method, ‘X’ and ‘y’ are destroyed on output.  For the normal
     equations method, they are both unchanged.

 -- Function: int gsl_multilarge_linear_solve (const double lambda,
          gsl_vector * c, double * rnorm, double * snorm,
          gsl_multilarge_linear_workspace * w)

     After all blocks (X_i,y_i) have been accumulated into the large
     least squares system, this function will compute the solution
     vector which is stored in ‘c’ on output.  The regularization
     parameter \lambda is provided in ‘lambda’.  On output, ‘rnorm’
     contains the residual norm ||y - X c||_W and ‘snorm’ contains the
     solution norm ||L c||.

 -- Function: int gsl_multilarge_linear_genform1 (const gsl_vector * L,
          const gsl_vector * cs, gsl_vector * c,
          gsl_multilarge_linear_workspace * work)

     After a regularized system has been solved with L =
     \diag(\l_0,\l_1,...,\l_{p-1}), this function backtransforms the
     standard form solution vector ‘cs’ to recover the solution vector
     of the original problem ‘c’.  The diagonal matrix elements l_i are
     provided in the vector ‘L’.  It is allowed to have ‘c’ = ‘cs’ for
     an in-place transform.

 -- Function: int gsl_multilarge_linear_genform2 (const gsl_matrix
          * LQR, const gsl_vector * Ltau, const gsl_vector * cs,
          gsl_vector * c, gsl_multilarge_linear_workspace * work)

     After a regularized system has been solved with a regularization
     matrix L, specified by (‘LQR’, ‘Ltau’), this function
     backtransforms the standard form solution ‘cs’ to recover the
     solution vector of the original problem, which is stored in ‘c’, of
     length p.

 -- Function: int gsl_multilarge_linear_lcurve (gsl_vector * reg_param,
          gsl_vector * rho, gsl_vector * eta,
          gsl_multilarge_linear_workspace * work)

     This function computes the L-curve for a large least squares system
     after it has been fully accumulated into the workspace ‘work’.  The
     output vectors ‘reg_param’, ‘rho’, and ‘eta’ must all be the same
     size, and will contain the regularization parameters \lambda_i,
     residual norms ||y - X c_i||, and solution norms || L c_i || which
     compose the L-curve, where c_i is the regularized solution vector
     corresponding to \lambda_i.  The user may determine the number of
     points on the L-curve by adjusting the size of these input arrays.
     For the TSQR method, the regularization parameters \lambda_i are
     estimated from the singular values of the triangular R factor.  For
     the normal equations method, they are estimated from the
     eigenvalues of the X^T X matrix.

 -- Function: int gsl_multilarge_linear_rcond (double * rcond,
          gsl_multilarge_linear_workspace * work)

     This function computes the reciprocal condition number, stored in
     ‘rcond’, of the least squares matrix after it has been accumulated
     into the workspace ‘work’.  For the TSQR algorithm, this is
     accomplished by calculating the SVD of the R factor, which has the
     same singular values as the matrix X. For the normal equations
     method, this is done by computing the eigenvalues of X^T X, which
     could be inaccurate for ill-conditioned matrices X.


File: gsl-ref.info,  Node: Troubleshooting,  Next: Examples<31>,  Prev: Large dense linear systems,  Up: Linear Least-Squares Fitting

40.7 Troubleshooting
====================

When using models based on polynomials, care should be taken when
constructing the design matrix X. If the x values are large, then the
matrix X could be ill-conditioned since its columns are powers of x,
leading to unstable least-squares solutions.  In this case it can often
help to center and scale the x values using the mean and standard
deviation:

     x' = (x - mu)/sigma

and then construct the X matrix using the transformed values x'.


File: gsl-ref.info,  Node: Examples<31>,  Next: References and Further Reading<33>,  Prev: Troubleshooting,  Up: Linear Least-Squares Fitting

40.8 Examples
=============

The example programs in this section demonstrate the various linear
regression methods.

* Menu:

* Simple Linear Regression Example::
* Multi-parameter Linear Regression Example::
* Regularized Linear Regression Example 1::
* Regularized Linear Regression Example 2::
* Robust Linear Regression Example::
* Large Dense Linear Regression Example::


File: gsl-ref.info,  Node: Simple Linear Regression Example,  Next: Multi-parameter Linear Regression Example,  Up: Examples<31>

40.8.1 Simple Linear Regression Example
---------------------------------------

The following program computes a least squares straight-line fit to a
simple dataset, and outputs the best-fit line and its associated one
standard-deviation error bars.

     #include <stdio.h>
     #include <gsl/gsl_fit.h>

     int
     main (void)
     {
       int i, n = 4;
       double x[4] = { 1970, 1980, 1990, 2000 };
       double y[4] = {   12,   11,   14,   13 };
       double w[4] = {  0.1,  0.2,  0.3,  0.4 };

       double c0, c1, cov00, cov01, cov11, chisq;

       gsl_fit_wlinear (x, 1, w, 1, y, 1, n,
                        &c0, &c1, &cov00, &cov01, &cov11,
                        &chisq);

       printf ("# best fit: Y = %g + %g X\n", c0, c1);
       printf ("# covariance matrix:\n");
       printf ("# [ %g, %g\n#   %g, %g]\n",
               cov00, cov01, cov01, cov11);
       printf ("# chisq = %g\n", chisq);

       for (i = 0; i < n; i++)
         printf ("data: %g %g %g\n",
                        x[i], y[i], 1/sqrt(w[i]));

       printf ("\n");

       for (i = -30; i < 130; i++)
         {
           double xf = x[0] + (i/100.0) * (x[n-1] - x[0]);
           double yf, yf_err;

           gsl_fit_linear_est (xf,
                               c0, c1,
                               cov00, cov01, cov11,
                               &yf, &yf_err);

           printf ("fit: %g %g\n", xf, yf);
           printf ("hi : %g %g\n", xf, yf + yf_err);
           printf ("lo : %g %g\n", xf, yf - yf_err);
         }
       return 0;
     }

The following commands extract the data from the output of the program
and display it using the GNU plotutils “graph” utility:

     $ ./demo > tmp
     $ more tmp
     # best fit: Y = -106.6 + 0.06 X
     # covariance matrix:
     # [ 39602, -19.9
     #   -19.9, 0.01]
     # chisq = 0.8

     $ for n in data fit hi lo ;
        do
          grep "^$n" tmp | cut -d: -f2 > $n ;
        done
     $ graph -T X -X x -Y y -y 0 20 -m 0 -S 2 -Ie data
          -S 0 -I a -m 1 fit -m 2 hi -m 2 lo

The result is shown in Fig.  %s.

[image src="fit-wlinear.png"]

Figure: Straight line fit with 1-\sigma error bars


File: gsl-ref.info,  Node: Multi-parameter Linear Regression Example,  Next: Regularized Linear Regression Example 1,  Prev: Simple Linear Regression Example,  Up: Examples<31>

40.8.2 Multi-parameter Linear Regression Example
------------------------------------------------

The following program performs a quadratic fit y = c_0 + c_1 x + c_2 x^2
to a weighted dataset using the generalised linear fitting function
*note gsl_multifit_wlinear(): b14.  The model matrix X for a quadratic
fit is given by,

     X = [ 1   , x_0  , x_0^2 ;
           1   , x_1  , x_1^2 ;
           1   , x_2  , x_2^2 ;
           ... , ...  , ...   ]

where the column of ones corresponds to the constant term c_0.  The two
remaining columns corresponds to the terms c_1 x and c_2 x^2.

The program reads ‘n’ lines of data in the format (‘x’, ‘y’, ‘err’)
where ‘err’ is the error (standard deviation) in the value ‘y’.

     #include <stdio.h>
     #include <gsl/gsl_multifit.h>

     int
     main (int argc, char **argv)
     {
       int i, n;
       double xi, yi, ei, chisq;
       gsl_matrix *X, *cov;
       gsl_vector *y, *w, *c;

       if (argc != 2)
         {
           fprintf (stderr,"usage: fit n < data\n");
           exit (-1);
         }

       n = atoi (argv[1]);

       X = gsl_matrix_alloc (n, 3);
       y = gsl_vector_alloc (n);
       w = gsl_vector_alloc (n);

       c = gsl_vector_alloc (3);
       cov = gsl_matrix_alloc (3, 3);

       for (i = 0; i < n; i++)
         {
           int count = fscanf (stdin, "%lg %lg %lg",
                               &xi, &yi, &ei);

           if (count != 3)
             {
               fprintf (stderr, "error reading file\n");
               exit (-1);
             }

           printf ("%g %g +/- %g\n", xi, yi, ei);

           gsl_matrix_set (X, i, 0, 1.0);
           gsl_matrix_set (X, i, 1, xi);
           gsl_matrix_set (X, i, 2, xi*xi);

           gsl_vector_set (y, i, yi);
           gsl_vector_set (w, i, 1.0/(ei*ei));
         }

       {
         gsl_multifit_linear_workspace * work
           = gsl_multifit_linear_alloc (n, 3);
         gsl_multifit_wlinear (X, w, y, c, cov,
                               &chisq, work);
         gsl_multifit_linear_free (work);
       }

     #define C(i) (gsl_vector_get(c,(i)))
     #define COV(i,j) (gsl_matrix_get(cov,(i),(j)))

       {
         printf ("# best fit: Y = %g + %g X + %g X^2\n",
                 C(0), C(1), C(2));

         printf ("# covariance matrix:\n");
         printf ("[ %+.5e, %+.5e, %+.5e  \n",
                    COV(0,0), COV(0,1), COV(0,2));
         printf ("  %+.5e, %+.5e, %+.5e  \n",
                    COV(1,0), COV(1,1), COV(1,2));
         printf ("  %+.5e, %+.5e, %+.5e ]\n",
                    COV(2,0), COV(2,1), COV(2,2));
         printf ("# chisq = %g\n", chisq);
       }

       gsl_matrix_free (X);
       gsl_vector_free (y);
       gsl_vector_free (w);
       gsl_vector_free (c);
       gsl_matrix_free (cov);

       return 0;
     }

A suitable set of data for fitting can be generated using the following
program.  It outputs a set of points with gaussian errors from the curve
y = e^x in the region 0 < x < 2.

     #include <stdio.h>
     #include <math.h>
     #include <gsl/gsl_randist.h>

     int
     main (void)
     {
       double x;
       const gsl_rng_type * T;
       gsl_rng * r;

       gsl_rng_env_setup ();

       T = gsl_rng_default;
       r = gsl_rng_alloc (T);

       for (x = 0.1; x < 2; x+= 0.1)
         {
           double y0 = exp (x);
           double sigma = 0.1 * y0;
           double dy = gsl_ran_gaussian (r, sigma);

           printf ("%g %g %g\n", x, y0 + dy, sigma);
         }

       gsl_rng_free(r);

       return 0;
     }

The data can be prepared by running the resulting executable program:

     $ GSL_RNG_TYPE=mt19937_1999 ./generate > exp.dat
     $ more exp.dat
     0.1 0.97935 0.110517
     0.2 1.3359 0.12214
     0.3 1.52573 0.134986
     0.4 1.60318 0.149182
     0.5 1.81731 0.164872
     0.6 1.92475 0.182212
     ....

To fit the data use the previous program, with the number of data points
given as the first argument.  In this case there are 19 data points:

     $ ./fit 19 < exp.dat
     0.1 0.97935 +/- 0.110517
     0.2 1.3359 +/- 0.12214
     ...
     # best fit: Y = 1.02318 + 0.956201 X + 0.876796 X^2
     # covariance matrix:
     [ +1.25612e-02, -3.64387e-02, +1.94389e-02
       -3.64387e-02, +1.42339e-01, -8.48761e-02
       +1.94389e-02, -8.48761e-02, +5.60243e-02 ]
     # chisq = 23.0987

The parameters of the quadratic fit match the coefficients of the
expansion of e^x, taking into account the errors on the parameters and
the O(x^3) difference between the exponential and quadratic functions
for the larger values of x.  The errors on the parameters are given by
the square-root of the corresponding diagonal elements of the covariance
matrix.  The chi-squared per degree of freedom is 1.4, indicating a
reasonable fit to the data.

Fig.  %s shows the resulting fit.

[image src="fit-wlinear2.png"]

Figure: Weighted fit example with error bars


File: gsl-ref.info,  Node: Regularized Linear Regression Example 1,  Next: Regularized Linear Regression Example 2,  Prev: Multi-parameter Linear Regression Example,  Up: Examples<31>

40.8.3 Regularized Linear Regression Example 1
----------------------------------------------

The next program demonstrates the difference between ordinary and
regularized least squares when the design matrix is near-singular.  In
this program, we generate two random normally distributed variables u
and v, with v = u + noise so that u and v are nearly colinear.  We then
set a third dependent variable y = u + v + noise and solve for the
coefficients c_1,c_2 of the model Y(c_1,c_2) = c_1 u + c_2 v.  Since u
\approx v, the design matrix X is nearly singular, leading to unstable
ordinary least squares solutions.

Here is the program output:

     matrix condition number = 1.025113e+04

     === Unregularized fit ===
     best fit: y = -43.6588 u + 45.6636 v
     residual norm = 31.6248
     solution norm = 63.1764
     chisq/dof = 1.00213

     === Regularized fit (L-curve) ===
     optimal lambda: 4.51103
     best fit: y = 1.00113 u + 1.0032 v
     residual norm = 31.6547
     solution norm = 1.41728
     chisq/dof = 1.04499

     === Regularized fit (GCV) ===
     optimal lambda: 0.0232029
     best fit: y = -19.8367 u + 21.8417 v
     residual norm = 31.6332
     solution norm = 29.5051
     chisq/dof = 1.00314

We see that the ordinary least squares solution is completely wrong,
while the L-curve regularized method with the optimal \lambda = 4.51103
finds the correct solution c_1 \approx c_2 \approx 1.  The GCV
regularized method finds a regularization parameter \lambda = 0.0232029
which is too small to give an accurate solution, although it performs
better than OLS. The L-curve and its computed corner, as well as the GCV
curve and its minimum are plotted in Fig.  %s.

[image src="regularized.png"]

Figure: L-curve and GCV curve for example program.

The program is given below.

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_multifit.h>

     int
     main()
     {
       const size_t n = 1000; /* number of observations */
       const size_t p = 2;    /* number of model parameters */
       size_t i;
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       gsl_matrix *X = gsl_matrix_alloc(n, p);
       gsl_vector *y = gsl_vector_alloc(n);

       for (i = 0; i < n; ++i)
         {
           /* generate first random variable u */
           double ui = 5.0 * gsl_ran_gaussian(r, 1.0);

           /* set v = u + noise */
           double vi = ui + gsl_ran_gaussian(r, 0.001);

           /* set y = u + v + noise */
           double yi = ui + vi + gsl_ran_gaussian(r, 1.0);

           /* since u =~ v, the matrix X is ill-conditioned */
           gsl_matrix_set(X, i, 0, ui);
           gsl_matrix_set(X, i, 1, vi);

           /* rhs vector */
           gsl_vector_set(y, i, yi);
         }

       {
         const size_t npoints = 200;                   /* number of points on L-curve and GCV curve */
         gsl_multifit_linear_workspace *w =
           gsl_multifit_linear_alloc(n, p);
         gsl_vector *c = gsl_vector_alloc(p);          /* OLS solution */
         gsl_vector *c_lcurve = gsl_vector_alloc(p);   /* regularized solution (L-curve) */
         gsl_vector *c_gcv = gsl_vector_alloc(p);      /* regularized solution (GCV) */
         gsl_vector *reg_param = gsl_vector_alloc(npoints);
         gsl_vector *rho = gsl_vector_alloc(npoints);  /* residual norms */
         gsl_vector *eta = gsl_vector_alloc(npoints);  /* solution norms */
         gsl_vector *G = gsl_vector_alloc(npoints);    /* GCV function values */
         double lambda_l;                              /* optimal regularization parameter (L-curve) */
         double lambda_gcv;                            /* optimal regularization parameter (GCV) */
         double G_gcv;                                 /* G(lambda_gcv) */
         size_t reg_idx;                               /* index of optimal lambda */
         double rcond;                                 /* reciprocal condition number of X */
         double chisq, rnorm, snorm;

         /* compute SVD of X */
         gsl_multifit_linear_svd(X, w);

         rcond = gsl_multifit_linear_rcond(w);
         fprintf(stderr, "matrix condition number = %e\n\n", 1.0 / rcond);

         /* unregularized (standard) least squares fit, lambda = 0 */
         gsl_multifit_linear_solve(0.0, X, y, c, &rnorm, &snorm, w);
         chisq = pow(rnorm, 2.0);

         fprintf(stderr, "=== Unregularized fit ===\n");
         fprintf(stderr, "best fit: y = %g u + %g v\n",
           gsl_vector_get(c, 0), gsl_vector_get(c, 1));
         fprintf(stderr, "residual norm = %g\n", rnorm);
         fprintf(stderr, "solution norm = %g\n", snorm);
         fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));

         /* calculate L-curve and find its corner */
         gsl_multifit_linear_lcurve(y, reg_param, rho, eta, w);
         gsl_multifit_linear_lcorner(rho, eta, &reg_idx);

         /* store optimal regularization parameter */
         lambda_l = gsl_vector_get(reg_param, reg_idx);

         /* regularize with lambda_l */
         gsl_multifit_linear_solve(lambda_l, X, y, c_lcurve, &rnorm, &snorm, w);
         chisq = pow(rnorm, 2.0) + pow(lambda_l * snorm, 2.0);

         fprintf(stderr, "\n=== Regularized fit (L-curve) ===\n");
         fprintf(stderr, "optimal lambda: %g\n", lambda_l);
         fprintf(stderr, "best fit: y = %g u + %g v\n",
                 gsl_vector_get(c_lcurve, 0), gsl_vector_get(c_lcurve, 1));
         fprintf(stderr, "residual norm = %g\n", rnorm);
         fprintf(stderr, "solution norm = %g\n", snorm);
         fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));

         /* calculate GCV curve and find its minimum */
         gsl_multifit_linear_gcv(y, reg_param, G, &lambda_gcv, &G_gcv, w);

         /* regularize with lambda_gcv */
         gsl_multifit_linear_solve(lambda_gcv, X, y, c_gcv, &rnorm, &snorm, w);
         chisq = pow(rnorm, 2.0) + pow(lambda_gcv * snorm, 2.0);

         fprintf(stderr, "\n=== Regularized fit (GCV) ===\n");
         fprintf(stderr, "optimal lambda: %g\n", lambda_gcv);
         fprintf(stderr, "best fit: y = %g u + %g v\n",
                 gsl_vector_get(c_gcv, 0), gsl_vector_get(c_gcv, 1));
         fprintf(stderr, "residual norm = %g\n", rnorm);
         fprintf(stderr, "solution norm = %g\n", snorm);
         fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));

         /* output L-curve and GCV curve */
         for (i = 0; i < npoints; ++i)
           {
             printf("%e %e %e %e\n",
                    gsl_vector_get(reg_param, i),
                    gsl_vector_get(rho, i),
                    gsl_vector_get(eta, i),
                    gsl_vector_get(G, i));
           }

         /* output L-curve corner point */
         printf("\n\n%f %f\n",
                gsl_vector_get(rho, reg_idx),
                gsl_vector_get(eta, reg_idx));

         /* output GCV curve corner minimum */
         printf("\n\n%e %e\n",
                lambda_gcv,
                G_gcv);

         gsl_multifit_linear_free(w);
         gsl_vector_free(c);
         gsl_vector_free(c_lcurve);
         gsl_vector_free(reg_param);
         gsl_vector_free(rho);
         gsl_vector_free(eta);
         gsl_vector_free(G);
       }

       gsl_rng_free(r);
       gsl_matrix_free(X);
       gsl_vector_free(y);

       return 0;
     }


File: gsl-ref.info,  Node: Regularized Linear Regression Example 2,  Next: Robust Linear Regression Example,  Prev: Regularized Linear Regression Example 1,  Up: Examples<31>

40.8.4 Regularized Linear Regression Example 2
----------------------------------------------

The following example program minimizes the cost function

     ||y - X c||^2 + \lambda^2 ||x||^2

where X is the 10-by-8 Hilbert matrix whose entries are given by

     X_{ij} = 1 / (i + j - 1)

and the right hand side vector is given by y =
[1,-1,1,-1,1,-1,1,-1,1,-1]^T. Solutions are computed for \lambda = 0
(unregularized) as well as for optimal parameters \lambda chosen by
analyzing the L-curve and GCV curve.

Here is the program output:

     matrix condition number = 3.565872e+09

     === Unregularized fit ===
     residual norm = 2.15376
     solution norm = 2.92217e+09
     chisq/dof = 2.31934

     === Regularized fit (L-curve) ===
     optimal lambda: 7.11407e-07
     residual norm = 2.60386
     solution norm = 424507
     chisq/dof = 3.43565

     === Regularized fit (GCV) ===
     optimal lambda: 1.72278
     residual norm = 3.1375
     solution norm = 0.139357
     chisq/dof = 4.95076

Here we see the unregularized solution results in a large solution norm
due to the ill-conditioned matrix.  The L-curve solution finds a small
value of \lambda = 7.11e-7 which still results in a badly conditioned
system and a large solution norm.  The GCV method finds a parameter
\lambda = 1.72 which results in a well-conditioned system and small
solution norm.

The L-curve and its computed corner, as well as the GCV curve and its
minimum are plotted in Fig.  %s.

[image src="regularized2.png"]

Figure: L-curve and GCV curve for example program.

The program is given below.

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_multifit.h>
     #include <gsl/gsl_blas.h>

     static int
     hilbert_matrix(gsl_matrix * m)
     {
       const size_t N = m->size1;
       const size_t M = m->size2;
       size_t i, j;

       for (i = 0; i < N; i++)
         {
           for (j = 0; j < M; j++)
             {
               gsl_matrix_set(m, i, j, 1.0/(i+j+1.0));
             }
         }

       return GSL_SUCCESS;
     }

     int
     main()
     {
       const size_t n = 10; /* number of observations */
       const size_t p = 8;  /* number of model parameters */
       size_t i;
       gsl_matrix *X = gsl_matrix_alloc(n, p);
       gsl_vector *y = gsl_vector_alloc(n);

       /* construct Hilbert matrix and rhs vector */
       hilbert_matrix(X);

       {
         double val = 1.0;
         for (i = 0; i < n; ++i)
           {
             gsl_vector_set(y, i, val);
             val *= -1.0;
           }
       }

       {
         const size_t npoints = 200;                   /* number of points on L-curve and GCV curve */
         gsl_multifit_linear_workspace *w =
           gsl_multifit_linear_alloc(n, p);
         gsl_vector *c = gsl_vector_alloc(p);          /* OLS solution */
         gsl_vector *c_lcurve = gsl_vector_alloc(p);   /* regularized solution (L-curve) */
         gsl_vector *c_gcv = gsl_vector_alloc(p);      /* regularized solution (GCV) */
         gsl_vector *reg_param = gsl_vector_alloc(npoints);
         gsl_vector *rho = gsl_vector_alloc(npoints);  /* residual norms */
         gsl_vector *eta = gsl_vector_alloc(npoints);  /* solution norms */
         gsl_vector *G = gsl_vector_alloc(npoints);    /* GCV function values */
         double lambda_l;                              /* optimal regularization parameter (L-curve) */
         double lambda_gcv;                            /* optimal regularization parameter (GCV) */
         double G_gcv;                                 /* G(lambda_gcv) */
         size_t reg_idx;                               /* index of optimal lambda */
         double rcond;                                 /* reciprocal condition number of X */
         double chisq, rnorm, snorm;

         /* compute SVD of X */
         gsl_multifit_linear_svd(X, w);

         rcond = gsl_multifit_linear_rcond(w);
         fprintf(stderr, "matrix condition number = %e\n", 1.0 / rcond);

         /* unregularized (standard) least squares fit, lambda = 0 */
         gsl_multifit_linear_solve(0.0, X, y, c, &rnorm, &snorm, w);
         chisq = pow(rnorm, 2.0);

         fprintf(stderr, "\n=== Unregularized fit ===\n");
         fprintf(stderr, "residual norm = %g\n", rnorm);
         fprintf(stderr, "solution norm = %g\n", snorm);
         fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));

         /* calculate L-curve and find its corner */
         gsl_multifit_linear_lcurve(y, reg_param, rho, eta, w);
         gsl_multifit_linear_lcorner(rho, eta, &reg_idx);

         /* store optimal regularization parameter */
         lambda_l = gsl_vector_get(reg_param, reg_idx);

         /* regularize with lambda_l */
         gsl_multifit_linear_solve(lambda_l, X, y, c_lcurve, &rnorm, &snorm, w);
         chisq = pow(rnorm, 2.0) + pow(lambda_l * snorm, 2.0);

         fprintf(stderr, "\n=== Regularized fit (L-curve) ===\n");
         fprintf(stderr, "optimal lambda: %g\n", lambda_l);
         fprintf(stderr, "residual norm = %g\n", rnorm);
         fprintf(stderr, "solution norm = %g\n", snorm);
         fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));

         /* calculate GCV curve and find its minimum */
         gsl_multifit_linear_gcv(y, reg_param, G, &lambda_gcv, &G_gcv, w);

         /* regularize with lambda_gcv */
         gsl_multifit_linear_solve(lambda_gcv, X, y, c_gcv, &rnorm, &snorm, w);
         chisq = pow(rnorm, 2.0) + pow(lambda_gcv * snorm, 2.0);

         fprintf(stderr, "\n=== Regularized fit (GCV) ===\n");
         fprintf(stderr, "optimal lambda: %g\n", lambda_gcv);
         fprintf(stderr, "residual norm = %g\n", rnorm);
         fprintf(stderr, "solution norm = %g\n", snorm);
         fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));

         /* output L-curve and GCV curve */
         for (i = 0; i < npoints; ++i)
           {
             printf("%e %e %e %e\n",
                    gsl_vector_get(reg_param, i),
                    gsl_vector_get(rho, i),
                    gsl_vector_get(eta, i),
                    gsl_vector_get(G, i));
           }

         /* output L-curve corner point */
         printf("\n\n%f %f\n",
                gsl_vector_get(rho, reg_idx),
                gsl_vector_get(eta, reg_idx));

         /* output GCV curve corner minimum */
         printf("\n\n%e %e\n",
                lambda_gcv,
                G_gcv);

         gsl_multifit_linear_free(w);
         gsl_vector_free(c);
         gsl_vector_free(c_lcurve);
         gsl_vector_free(reg_param);
         gsl_vector_free(rho);
         gsl_vector_free(eta);
         gsl_vector_free(G);
       }

       gsl_matrix_free(X);
       gsl_vector_free(y);

       return 0;
     }


File: gsl-ref.info,  Node: Robust Linear Regression Example,  Next: Large Dense Linear Regression Example,  Prev: Regularized Linear Regression Example 2,  Up: Examples<31>

40.8.5 Robust Linear Regression Example
---------------------------------------

The next program demonstrates the advantage of robust least squares on a
dataset with outliers.  The program generates linear (x,y) data pairs on
the line y = 1.45 x + 3.88, adds some random noise, and inserts 3
outliers into the dataset.  Both the robust and ordinary least squares
(OLS) coefficients are computed for comparison.

     #include <stdio.h>
     #include <gsl/gsl_multifit.h>
     #include <gsl/gsl_randist.h>

     int
     dofit(const gsl_multifit_robust_type *T,
           const gsl_matrix *X, const gsl_vector *y,
           gsl_vector *c, gsl_matrix *cov)
     {
       int s;
       gsl_multifit_robust_workspace * work
         = gsl_multifit_robust_alloc (T, X->size1, X->size2);

       s = gsl_multifit_robust (X, y, c, cov, work);
       gsl_multifit_robust_free (work);

       return s;
     }

     int
     main (int argc, char **argv)
     {
       size_t i;
       size_t n;
       const size_t p = 2; /* linear fit */
       gsl_matrix *X, *cov;
       gsl_vector *x, *y, *c, *c_ols;
       const double a = 1.45; /* slope */
       const double b = 3.88; /* intercept */
       gsl_rng *r;

       if (argc != 2)
         {
           fprintf (stderr,"usage: robfit n\n");
           exit (-1);
         }

       n = atoi (argv[1]);

       X = gsl_matrix_alloc (n, p);
       x = gsl_vector_alloc (n);
       y = gsl_vector_alloc (n);

       c = gsl_vector_alloc (p);
       c_ols = gsl_vector_alloc (p);
       cov = gsl_matrix_alloc (p, p);

       r = gsl_rng_alloc(gsl_rng_default);

       /* generate linear dataset */
       for (i = 0; i < n - 3; i++)
         {
           double dx = 10.0 / (n - 1.0);
           double ei = gsl_rng_uniform(r);
           double xi = -5.0 + i * dx;
           double yi = a * xi + b;

           gsl_vector_set (x, i, xi);
           gsl_vector_set (y, i, yi + ei);
         }

       /* add a few outliers */
       gsl_vector_set(x, n - 3, 4.7);
       gsl_vector_set(y, n - 3, -8.3);

       gsl_vector_set(x, n - 2, 3.5);
       gsl_vector_set(y, n - 2, -6.7);

       gsl_vector_set(x, n - 1, 4.1);
       gsl_vector_set(y, n - 1, -6.0);

       /* construct design matrix X for linear fit */
       for (i = 0; i < n; ++i)
         {
           double xi = gsl_vector_get(x, i);

           gsl_matrix_set (X, i, 0, 1.0);
           gsl_matrix_set (X, i, 1, xi);
         }

       /* perform robust and OLS fit */
       dofit(gsl_multifit_robust_ols, X, y, c_ols, cov);
       dofit(gsl_multifit_robust_bisquare, X, y, c, cov);

       /* output data and model */
       for (i = 0; i < n; ++i)
         {
           double xi = gsl_vector_get(x, i);
           double yi = gsl_vector_get(y, i);
           gsl_vector_view v = gsl_matrix_row(X, i);
           double y_ols, y_rob, y_err;

           gsl_multifit_robust_est(&v.vector, c, cov, &y_rob, &y_err);
           gsl_multifit_robust_est(&v.vector, c_ols, cov, &y_ols, &y_err);

           printf("%g %g %g %g\n", xi, yi, y_rob, y_ols);
         }

     #define C(i) (gsl_vector_get(c,(i)))
     #define COV(i,j) (gsl_matrix_get(cov,(i),(j)))

       {
         printf ("# best fit: Y = %g + %g X\n",
                 C(0), C(1));

         printf ("# covariance matrix:\n");
         printf ("# [ %+.5e, %+.5e\n",
                    COV(0,0), COV(0,1));
         printf ("#   %+.5e, %+.5e\n",
                    COV(1,0), COV(1,1));
       }

       gsl_matrix_free (X);
       gsl_vector_free (x);
       gsl_vector_free (y);
       gsl_vector_free (c);
       gsl_vector_free (c_ols);
       gsl_matrix_free (cov);
       gsl_rng_free(r);

       return 0;
     }

The output from the program is shown in Fig.  %s.

[image src="robust.png"]

Figure: Linear fit to dataset with outliers.


File: gsl-ref.info,  Node: Large Dense Linear Regression Example,  Prev: Robust Linear Regression Example,  Up: Examples<31>

40.8.6 Large Dense Linear Regression Example
--------------------------------------------

The following program demostrates the large dense linear least squares
solvers.  This example is adapted from Trefethen and Bau, and fits the
function f(t) = \exp{(\sin^3{(10t)}}) on the interval [0,1] with a
degree 15 polynomial.  The program generates n = 50000 equally spaced
points t_i on this interval, calculates the function value and adds
random noise to determine the observation value y_i.  The entries of the
least squares matrix are X_{ij} = t_i^j, representing a polynomial fit.
The matrix is highly ill-conditioned, with a condition number of about
2.4 \cdot 10^{11}.  The program accumulates the matrix into the least
squares system in 5 blocks, each with 10000 rows.  This way the full
matrix X is never stored in memory.  We solve the system with both the
normal equations and TSQR methods.  The results are shown in Fig.  %s.
In the top left plot, the TSQR solution provides a reasonable agreement
to the exact solution, while the normal equations method fails
completely since the Cholesky factorization fails due to the
ill-conditioning of the matrix.  In the bottom left plot, we show the
L-curve calculated from TSQR, which exhibits multiple corners.  In the
top right panel, we plot a regularized solution using \lambda = 10^{-5}.
The TSQR and normal solutions now agree, however they are unable to
provide a good fit due to the damping.  This indicates that for some
ill-conditioned problems, regularizing the normal equations does not
improve the solution.  This is further illustrated in the bottom right
panel, where we plot the L-curve calculated from the normal equations.
The curve agrees with the TSQR curve for larger damping parameters, but
for small \lambda, the normal equations approach cannot provide accurate
solution vectors leading to numerical inaccuracies in the left portion
of the curve.

[image src="multilarge.png"]

Figure

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_multifit.h>
     #include <gsl/gsl_multilarge.h>
     #include <gsl/gsl_blas.h>
     #include <gsl/gsl_errno.h>

     /* function to be fitted */
     double
     func(const double t)
     {
       double x = sin(10.0 * t);
       return exp(x*x*x);
     }

     /* construct a row of the least squares matrix */
     int
     build_row(const double t, gsl_vector *row)
     {
       const size_t p = row->size;
       double Xj = 1.0;
       size_t j;

       for (j = 0; j < p; ++j)
         {
           gsl_vector_set(row, j, Xj);
           Xj *= t;
         }

       return 0;
     }

     int
     solve_system(const int print_data, const gsl_multilarge_linear_type * T,
                  const double lambda, const size_t n, const size_t p,
                  gsl_vector * c)
     {
       const size_t nblock = 5;         /* number of blocks to accumulate */
       const size_t nrows = n / nblock; /* number of rows per block */
       gsl_multilarge_linear_workspace * w =
         gsl_multilarge_linear_alloc(T, p);
       gsl_matrix *X = gsl_matrix_alloc(nrows, p);
       gsl_vector *y = gsl_vector_alloc(nrows);
       gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
       const size_t nlcurve = 200;
       gsl_vector *reg_param = gsl_vector_alloc(nlcurve);
       gsl_vector *rho = gsl_vector_calloc(nlcurve);
       gsl_vector *eta = gsl_vector_calloc(nlcurve);
       size_t rowidx = 0;
       double rnorm, snorm, rcond;
       double t = 0.0;
       double dt = 1.0 / (n - 1.0);

       while (rowidx < n)
         {
           size_t nleft = n - rowidx;         /* number of rows left to accumulate */
           size_t nr = GSL_MIN(nrows, nleft); /* number of rows in this block */
           gsl_matrix_view Xv = gsl_matrix_submatrix(X, 0, 0, nr, p);
           gsl_vector_view yv = gsl_vector_subvector(y, 0, nr);
           size_t i;

           /* build (X,y) block with 'nr' rows */
           for (i = 0; i < nr; ++i)
             {
               gsl_vector_view row = gsl_matrix_row(&Xv.matrix, i);
               double fi = func(t);
               double ei = gsl_ran_gaussian (r, 0.1 * fi); /* noise */
               double yi = fi + ei;

               /* construct this row of LS matrix */
               build_row(t, &row.vector);

               /* set right hand side value with added noise */
               gsl_vector_set(&yv.vector, i, yi);

               if (print_data && (i % 100 == 0))
                 printf("%f %f\n", t, yi);

               t += dt;
             }

           /* accumulate (X,y) block into LS system */
           gsl_multilarge_linear_accumulate(&Xv.matrix, &yv.vector, w);

           rowidx += nr;
         }

       if (print_data)
         printf("\n\n");

       /* compute L-curve */
       gsl_multilarge_linear_lcurve(reg_param, rho, eta, w);

       /* solve large LS system and store solution in c */
       gsl_multilarge_linear_solve(lambda, c, &rnorm, &snorm, w);

       /* compute reciprocal condition number */
       gsl_multilarge_linear_rcond(&rcond, w);

       fprintf(stderr, "=== Method %s ===\n", gsl_multilarge_linear_name(w));
       fprintf(stderr, "condition number = %e\n", 1.0 / rcond);
       fprintf(stderr, "residual norm    = %e\n", rnorm);
       fprintf(stderr, "solution norm    = %e\n", snorm);

       /* output L-curve */
       {
         size_t i;
         for (i = 0; i < nlcurve; ++i)
           {
             printf("%.12e %.12e %.12e\n",
                    gsl_vector_get(reg_param, i),
                    gsl_vector_get(rho, i),
                    gsl_vector_get(eta, i));
           }
         printf("\n\n");
       }

       gsl_matrix_free(X);
       gsl_vector_free(y);
       gsl_multilarge_linear_free(w);
       gsl_rng_free(r);
       gsl_vector_free(reg_param);
       gsl_vector_free(rho);
       gsl_vector_free(eta);

       return 0;
     }

     int
     main(int argc, char *argv[])
     {
       const size_t n = 50000;   /* number of observations */
       const size_t p = 16;      /* polynomial order + 1 */
       double lambda = 0.0;      /* regularization parameter */
       gsl_vector *c_tsqr = gsl_vector_calloc(p);
       gsl_vector *c_normal = gsl_vector_calloc(p);

       if (argc > 1)
         lambda = atof(argv[1]);

       /* turn off error handler so normal equations method won't abort */
       gsl_set_error_handler_off();

       /* solve system with TSQR method */
       solve_system(1, gsl_multilarge_linear_tsqr, lambda, n, p, c_tsqr);

       /* solve system with Normal equations method */
       solve_system(0, gsl_multilarge_linear_normal, lambda, n, p, c_normal);

       /* output solutions */
       {
         gsl_vector *v = gsl_vector_alloc(p);
         double t;

         for (t = 0.0; t <= 1.0; t += 0.01)
           {
             double f_exact = func(t);
             double f_tsqr, f_normal;

             build_row(t, v);
             gsl_blas_ddot(v, c_tsqr, &f_tsqr);
             gsl_blas_ddot(v, c_normal, &f_normal);

             printf("%f %e %e %e\n", t, f_exact, f_tsqr, f_normal);
           }

         gsl_vector_free(v);
       }

       gsl_vector_free(c_tsqr);
       gsl_vector_free(c_normal);

       return 0;
     }


File: gsl-ref.info,  Node: References and Further Reading<33>,  Prev: Examples<31>,  Up: Linear Least-Squares Fitting

40.9 References and Further Reading
===================================

A summary of formulas and techniques for least squares fitting can be
found in the “Statistics” chapter of the Annual Review of Particle
Physics prepared by the Particle Data Group,

   * `Review of Particle Properties', R.M. Barnett et al., Physical
     Review D54, 1 (1996) ‘http://pdg.lbl.gov’

The Review of Particle Physics is available online at the website given
above.

The tests used to prepare these routines are based on the NIST
Statistical Reference Datasets.  The datasets and their documentation
are available from NIST at the following website,

‘http://www.nist.gov/itl/div898/strd/index.html’

More information on Tikhonov regularization can be found in

   * Hansen, P. C. (1998), Rank-Deficient and Discrete Ill-Posed
     Problems: Numerical Aspects of Linear Inversion.  SIAM Monogr.  on
     Mathematical Modeling and Computation, Society for Industrial and
     Applied Mathematics

   * M. Rezghi and S. M. Hosseini (2009), A new variant of L-curve for
     Tikhonov regularization, Journal of Computational and Applied
     Mathematics, Volume 231, Issue 2, pages 914-924.

The GSL implementation of robust linear regression closely follows the
publications

   * DuMouchel, W. and F. O’Brien (1989), “Integrating a robust option
     into a multiple regression computing environment,” Computer Science
     and Statistics: Proceedings of the 21st Symposium on the Interface,
     American Statistical Association

   * Street, J.O., R.J. Carroll, and D. Ruppert (1988), “A note on
     computing robust regression estimates via iteratively reweighted
     least squares,” The American Statistician, v.  42, pp.  152-154.

More information about the normal equations and TSQR approach for
solving large linear least squares systems can be found in the
publications

   * Trefethen, L. N. and Bau, D. (1997), “Numerical Linear Algebra”,
     SIAM.

   * Demmel, J., Grigori, L., Hoemmen, M. F., and Langou, J.
     “Communication-optimal parallel and sequential QR and LU
     factorizations”, UCB Technical Report No.  UCB/EECS-2008-89, 2008.


File: gsl-ref.info,  Node: Nonlinear Least-Squares Fitting,  Next: Basis Splines,  Prev: Linear Least-Squares Fitting,  Up: Top

41 Nonlinear Least-Squares Fitting
**********************************

This chapter describes functions for multidimensional nonlinear
least-squares fitting.  There are generally two classes of algorithms
for solving nonlinear least squares problems, which fall under line
search methods and trust region methods.  GSL currently implements only
trust region methods and provides the user with full access to
intermediate steps of the iteration.  The user also has the ability to
tune a number of parameters which affect low-level aspects of the
algorithm which can help to accelerate convergence for the specific
problem at hand.  GSL provides two separate interfaces for nonlinear
least squares fitting.  The first is designed for small to moderate
sized problems, and the second is designed for very large problems,
which may or may not have significant sparse structure.

The header file ‘gsl_multifit_nlinear.h’ contains prototypes for the
multidimensional nonlinear fitting functions and related declarations
relating to the small to moderate sized systems.

The header file ‘gsl_multilarge_nlinear.h’ contains prototypes for the
multidimensional nonlinear fitting functions and related declarations
relating to large systems.

* Menu:

* Overview: Overview<6>.
* Solving the Trust Region Subproblem (TRS): Solving the Trust Region Subproblem TRS.
* Weighted Nonlinear Least-Squares::
* Tunable Parameters::
* Initializing the Solver: Initializing the Solver<3>.
* Providing the Function to be Minimized::
* Iteration: Iteration<5>.
* Testing for Convergence::
* High Level Driver::
* Covariance matrix of best fit parameters::
* Troubleshooting: Troubleshooting<2>.
* Examples: Examples<32>.
* References and Further Reading: References and Further Reading<34>.


File: gsl-ref.info,  Node: Overview<6>,  Next: Solving the Trust Region Subproblem TRS,  Up: Nonlinear Least-Squares Fitting

41.1 Overview
=============

The problem of multidimensional nonlinear least-squares fitting requires
the minimization of the squared residuals of n functions, f_i, in p
parameters, x_i,

     \Phi(x) = (1/2) || f(x) ||^2
             = (1/2) \sum_{i=1}^{n} f_i(x_1, ..., x_p)^2

In trust region methods, the objective (or cost) function \Phi(x) is
approximated by a model function m_k(\delta) in the vicinity of some
point x_k.  The model function is often simply a second order Taylor
series expansion around the point x_k, ie:

     \Phi(x_k + \delta) ~=~ m_k(\delta) = \Phi(x_k) + g_k^T \delta + 1/2 \delta^T B_k \delta

where g_k = \nabla \Phi(x_k) = J^T f is the gradient vector at the point
x_k, B_k = \nabla^2 \Phi(x_k) is the Hessian matrix at x_k, or some
approximation to it, and J is the n-by-p Jacobian matrix

     J_{ij} = d f_i / d x_j

In order to find the next step \delta, we minimize the model function
m_k(\delta), but search for solutions only within a region where we
trust that m_k(\delta) is a good approximation to the objective function
\Phi(x_k + \delta).  In other words, we seek a solution of the trust
region subproblem (TRS)

     \min_(\delta \in R^p) m_k(\delta), s.t. || D_k \delta || <= \Delta_k

where \Delta_k > 0 is the trust region radius and D_k is a scaling
matrix.  If D_k = I, then the trust region is a ball of radius \Delta_k
centered at x_k.  In some applications, the parameter vector x may have
widely different scales.  For example, one parameter might be a
temperature on the order of 10^3 K, while another might be a length on
the order of 10^{-6} m.  In such cases, a spherical trust region may not
be the best choice, since if \Phi changes rapidly along directions with
one scale, and more slowly along directions with a different scale, the
model function m_k may be a poor approximation to \Phi along the rapidly
changing directions.  In such problems, it may be best to use an
elliptical trust region, by setting D_k to a diagonal matrix whose
entries are designed so that the scaled step D_k \delta has entries of
approximately the same order of magnitude.

The trust region subproblem above normally amounts to solving a linear
least squares system (or multiple systems) for the step \delta.  Once
\delta is computed, it is checked whether or not it reduces the
objective function \Phi(x).  A useful statistic for this is to look at
the ratio

     \rho_k = ( \Phi(x_k) - \Phi(x_k + \delta_k) / ( m_k(0) - m_k(\delta_k) )

where the numerator is the actual reduction of the objective function
due to the step \delta_k, and the denominator is the predicted reduction
due to the model m_k.  If \rho_k is negative, it means that the step
\delta_k increased the objective function and so it is rejected.  If
\rho_k is positive, then we have found a step which reduced the
objective function and it is accepted.  Furthermore, if \rho_k is close
to 1, then this indicates that the model function is a good
approximation to the objective function in the trust region, and so on
the next iteration the trust region is enlarged in order to take more
ambitious steps.  When a step is rejected, the trust region is made
smaller and the TRS is solved again.  An outline for the general trust
region method used by GSL can now be given.

`Trust Region Algorithm'

  1. Initialize: given x_0, construct m_0(\delta), D_0 and \Delta_0 > 0

  2. For k = 0, 1, 2, …

       a. If converged, then stop

       b. Solve TRS for trial step \delta_k

       c. Evaluate trial step by computing \rho_k

          1). if step is accepted, set x_{k+1} = x_k + \delta_k and increase radius,

               \Delta_{k+1} = \alpha \Delta_k

          2). if step is rejected, set x_{k+1} = x_k and decrease radius,

               \Delta_{k+1} = {\Delta_k \over \beta}; goto 2(b)

       d. Construct m_{k+1}(\delta) and D_{k+1}

GSL offers the user a number of different algorithms for solving the
trust region subproblem in 2(b), as well as different choices of scaling
matrices D_k and different methods of updating the trust region radius
\Delta_k.  Therefore, while reasonable default methods are provided, the
user has a lot of control to fine-tune the various steps of the
algorithm for their specific problem.


File: gsl-ref.info,  Node: Solving the Trust Region Subproblem TRS,  Next: Weighted Nonlinear Least-Squares,  Prev: Overview<6>,  Up: Nonlinear Least-Squares Fitting

41.2 Solving the Trust Region Subproblem (TRS)
==============================================

Below we describe the methods available for solving the trust region
subproblem.  The methods available provide either exact or approximate
solutions to the trust region subproblem.  In all algorithms below, the
Hessian matrix B_k is approximated as B_k \approx J_k^T J_k, where J_k =
J(x_k).  In all methods, the solution of the TRS involves solving a
linear least squares system involving the Jacobian matrix.  For small to
moderate sized problems (‘gsl_multifit_nlinear’ interface), this is
accomplished by factoring the full Jacobian matrix, which is provided by
the user, with the Cholesky, QR, or SVD decompositions.  For large
systems (‘gsl_multilarge_nlinear’ interface), the user has two choices.
One is to solve the system iteratively, without needing to store the
full Jacobian matrix in memory.  With this method, the user must provide
a routine to calculate the matrix-vector products J u or J^T u for a
given vector u.  This iterative method is particularly useful for
systems where the Jacobian has sparse structure, since forming
matrix-vector products can be done cheaply.  The second option for large
systems involves forming the normal equations matrix J^T J and then
factoring it using a Cholesky decomposition.  The normal equations
matrix is p-by-p, typically much smaller than the full n-by-p Jacobian,
and can usually be stored in memory even if the full Jacobian matrix
cannot.  This option is useful for large, dense systems, or if the
iterative method has difficulty converging.

* Menu:

* Levenberg-Marquardt::
* Levenberg-Marquardt with Geodesic Acceleration::
* Dogleg::
* Double Dogleg::
* Two Dimensional Subspace::
* Steihaug-Toint Conjugate Gradient::


File: gsl-ref.info,  Node: Levenberg-Marquardt,  Next: Levenberg-Marquardt with Geodesic Acceleration,  Up: Solving the Trust Region Subproblem TRS

41.2.1 Levenberg-Marquardt
--------------------------

There is a theorem which states that if \delta_k is a solution to the
trust region subproblem given above, then there exists \mu_k \ge 0 such
that

     ( B_k + \mu_k D_k^T D_k ) \delta_k = -g_k

with \mu_k (\Delta_k - ||D_k \delta_k||) = 0.  This forms the basis of
the Levenberg-Marquardt algorithm, which controls the trust region size
by adjusting the parameter \mu_k rather than the radius \Delta_k
directly.  For each radius \Delta_k, there is a unique parameter \mu_k
which solves the TRS, and they have an inverse relationship, so that
large values of \mu_k correspond to smaller trust regions, while small
values of \mu_k correspond to larger trust regions.

With the approximation B_k \approx J_k^T J_k, on each iteration, in
order to calculate the step \delta_k, the following linear least squares
problem is solved:

     [J_k; sqrt(mu_k) D_k] \delta_k = - [f_k; 0]

If the step \delta_k is accepted, then \mu_k is decreased on the next
iteration in order to take a larger step, otherwise it is increased to
take a smaller step.  The Levenberg-Marquardt algorithm provides an
exact solution of the trust region subproblem, but typically has a
higher computational cost per iteration than the approximate methods
discussed below, since it may need to solve the least squares system
above several times for different values of \mu_k.


File: gsl-ref.info,  Node: Levenberg-Marquardt with Geodesic Acceleration,  Next: Dogleg,  Prev: Levenberg-Marquardt,  Up: Solving the Trust Region Subproblem TRS

41.2.2 Levenberg-Marquardt with Geodesic Acceleration
-----------------------------------------------------

This method applies a so-called geodesic acceleration correction to the
standard Levenberg-Marquardt step \delta_k (Transtrum et al, 2011).  By
interpreting \delta_k as a first order step along a geodesic in the
model parameter space (ie: a velocity \delta_k = v_k), the geodesic
acceleration a_k is a second order correction along the geodesic which
is determined by solving the linear least squares system

     [J_k; sqrt(mu_k) D_k] a_k = - [f_vv(x_k); 0]

where f_{vv} is the second directional derivative of the residual vector
in the velocity direction v, f_{vv}(x) = D_v^2 f = \sum_{\alpha\beta}
v_{\alpha} v_{\beta} \partial_{\alpha} \partial_{\beta} f(x), where
\alpha and \beta are summed over the p parameters.  The new total step
is then \delta_k' = v_k + {1 \over 2}a_k.  The second order correction
a_k can be calculated with a modest additional cost, and has been shown
to dramatically reduce the number of iterations (and expensive Jacobian
evaluations) required to reach convergence on a variety of different
problems.  In order to utilize the geodesic acceleration, the user must
supply a function which provides the second directional derivative
vector f_{vv}(x), or alternatively the library can use a finite
difference method to estimate this vector with one additional function
evaluation of f(x + h v) where h is a tunable step size (see the ‘h_fvv’
parameter description).


File: gsl-ref.info,  Node: Dogleg,  Next: Double Dogleg,  Prev: Levenberg-Marquardt with Geodesic Acceleration,  Up: Solving the Trust Region Subproblem TRS

41.2.3 Dogleg
-------------

This is Powell’s dogleg method, which finds an approximate solution to
the trust region subproblem, by restricting its search to a piecewise
linear “dogleg” path, composed of the origin, the Cauchy point which
represents the model minimizer along the steepest descent direction, and
the Gauss-Newton point, which is the overall minimizer of the
unconstrained model.  The Gauss-Newton step is calculated by solving

     J_k \delta_{gn} = -f_k

which is the main computational task for each iteration, but only needs
to be performed once per iteration.  If the Gauss-Newton point is inside
the trust region, it is selected as the step.  If it is outside, the
method then calculates the Cauchy point, which is located along the
gradient direction.  If the Cauchy point is also outside the trust
region, the method assumes that it is still far from the minimum and so
proceeds along the gradient direction, truncating the step at the trust
region boundary.  If the Cauchy point is inside the trust region, with
the Gauss-Newton point outside, the method uses a dogleg step, which is
a linear combination of the gradient direction and the Gauss-Newton
direction, stopping at the trust region boundary.


File: gsl-ref.info,  Node: Double Dogleg,  Next: Two Dimensional Subspace,  Prev: Dogleg,  Up: Solving the Trust Region Subproblem TRS

41.2.4 Double Dogleg
--------------------

This method is an improvement over the classical dogleg algorithm, which
attempts to include information about the Gauss-Newton step while the
iteration is still far from the minimum.  When the Cauchy point is
inside the trust region and the Gauss-Newton point is outside, the
method computes a scaled Gauss-Newton point and then takes a dogleg step
between the Cauchy point and the scaled Gauss-Newton point.  The scaling
is calculated to ensure that the reduction in the model m_k is about the
same as the reduction provided by the Cauchy point.


File: gsl-ref.info,  Node: Two Dimensional Subspace,  Next: Steihaug-Toint Conjugate Gradient,  Prev: Double Dogleg,  Up: Solving the Trust Region Subproblem TRS

41.2.5 Two Dimensional Subspace
-------------------------------

The dogleg methods restrict the search for the TRS solution to a 1D
curve defined by the Cauchy and Gauss-Newton points.  An improvement to
this is to search for a solution using the full two dimensional subspace
spanned by the Cauchy and Gauss-Newton directions.  The dogleg path is
of course inside this subspace, and so this method solves the TRS at
least as accurately as the dogleg methods.  Since this method searches a
larger subspace for a solution, it can converge more quickly than dogleg
on some problems.  Because the subspace is only two dimensional, this
method is very efficient and the main computation per iteration is to
determine the Gauss-Newton point.


File: gsl-ref.info,  Node: Steihaug-Toint Conjugate Gradient,  Prev: Two Dimensional Subspace,  Up: Solving the Trust Region Subproblem TRS

41.2.6 Steihaug-Toint Conjugate Gradient
----------------------------------------

One difficulty of the dogleg methods is calculating the Gauss-Newton
step when the Jacobian matrix is singular.  The Steihaug-Toint method
also computes a generalized dogleg step, but avoids solving for the
Gauss-Newton step directly, instead using an iterative conjugate
gradient algorithm.  This method performs well at points where the
Jacobian is singular, and is also suitable for large-scale problems
where factoring the Jacobian matrix could be prohibitively expensive.


File: gsl-ref.info,  Node: Weighted Nonlinear Least-Squares,  Next: Tunable Parameters,  Prev: Solving the Trust Region Subproblem TRS,  Up: Nonlinear Least-Squares Fitting

41.3 Weighted Nonlinear Least-Squares
=====================================

Weighted nonlinear least-squares fitting minimizes the function

     \Phi(x) = (1/2) || f(x) ||_W^2
             = (1/2) \sum_{i=1}^{n} f_i(x_1, ..., x_p)^2

where W = \diag(w_1,w_2,...,w_n) is the weighting matrix, and ||f||_W^2
= f^T W f.  The weights w_i are commonly defined as w_i = 1/\sigma_i^2,
where \sigma_i is the error in the i-th measurement.  A simple change of
variables \tilde{f} = W^{1 \over 2} f yields \Phi(x) = {1 \over 2}
||\tilde{f}||^2, which is in the same form as the unweighted case.  The
user can either perform this transform directly on their function
residuals and Jacobian, or use the *note gsl_multifit_nlinear_winit():
b77. interface which automatically performs the correct scaling.  To
manually perform this transformation, the residuals and Jacobian should
be modified according to

     f~_i = f_i / \sigma_i
     J~_ij = 1 / \sigma_i df_i/dx_j

For large systems, the user must perform their own weighting.


File: gsl-ref.info,  Node: Tunable Parameters,  Next: Initializing the Solver<3>,  Prev: Weighted Nonlinear Least-Squares,  Up: Nonlinear Least-Squares Fitting

41.4 Tunable Parameters
=======================

The user can tune nearly all aspects of the iteration at allocation
time.  For the ‘gsl_multifit_nlinear’ interface, the user may modify the
*note gsl_multifit_nlinear_parameters: b7a. structure, which is defined
as follows:

 -- Type: gsl_multifit_nlinear_parameters

          typedef struct
          {
            const gsl_multifit_nlinear_trs *trs;        /* trust region subproblem method */
            const gsl_multifit_nlinear_scale *scale;    /* scaling method */
            const gsl_multifit_nlinear_solver *solver;  /* solver method */
            gsl_multifit_nlinear_fdtype fdtype;         /* finite difference method */
            double factor_up;                           /* factor for increasing trust radius */
            double factor_down;                         /* factor for decreasing trust radius */
            double avmax;                               /* max allowed |a|/|v| */
            double h_df;                                /* step size for finite difference Jacobian */
            double h_fvv;                               /* step size for finite difference fvv */
          } gsl_multifit_nlinear_parameters;

For the ‘gsl_multilarge_nlinear’ interface, the user may modify the
*note gsl_multilarge_nlinear_parameters: b7b. structure, which is
defined as follows:

 -- Type: gsl_multilarge_nlinear_parameters

          typedef struct
          {
            const gsl_multilarge_nlinear_trs *trs;       /* trust region subproblem method */
            const gsl_multilarge_nlinear_scale *scale;   /* scaling method */
            const gsl_multilarge_nlinear_solver *solver; /* solver method */
            gsl_multilarge_nlinear_fdtype fdtype;        /* finite difference method */
            double factor_up;                            /* factor for increasing trust radius */
            double factor_down;                          /* factor for decreasing trust radius */
            double avmax;                                /* max allowed |a|/|v| */
            double h_df;                                 /* step size for finite difference Jacobian */
            double h_fvv;                                /* step size for finite difference fvv */
            size_t max_iter;                             /* maximum iterations for trs method */
            double tol;                                  /* tolerance for solving trs */
          } gsl_multilarge_nlinear_parameters;

Each of these parameters is discussed in further detail below.

 -- Type: gsl_multifit_nlinear_trs
 -- Type: gsl_multilarge_nlinear_trs

     The parameter ‘trs’ determines the method used to solve the trust
     region subproblem, and may be selected from the following choices,

      -- Variable: gsl_multifit_nlinear_trs_lm
      -- Variable: gsl_multilarge_nlinear_trs_lm

          This selects the Levenberg-Marquardt algorithm.

      -- Variable: gsl_multifit_nlinear_trs_lmaccel
      -- Variable: gsl_multilarge_nlinear_trs_lmaccel

          This selects the Levenberg-Marquardt algorithm with geodesic
          acceleration.

      -- Variable: gsl_multifit_nlinear_trs_dogleg
      -- Variable: gsl_multilarge_nlinear_trs_dogleg

          This selects the dogleg algorithm.

      -- Variable: gsl_multifit_nlinear_trs_ddogleg
      -- Variable: gsl_multilarge_nlinear_trs_ddogleg

          This selects the double dogleg algorithm.

      -- Variable: gsl_multifit_nlinear_trs_subspace2D
      -- Variable: gsl_multilarge_nlinear_trs_subspace2D

          This selects the 2D subspace algorithm.

      -- Variable: gsl_multilarge_nlinear_trs_cgst

          This selects the Steihaug-Toint conjugate gradient algorithm.
          This method is available only for large systems.

 -- Type: gsl_multifit_nlinear_scale
 -- Type: gsl_multilarge_nlinear_scale

     The parameter ‘scale’ determines the diagonal scaling matrix D and
     may be selected from the following choices,

      -- Variable: gsl_multifit_nlinear_scale_more
      -- Variable: gsl_multilarge_nlinear_scale_more

          This damping strategy was suggested by Moré, and corresponds
          to D^T D = \max(\diag(J^T J)), in other words the maximum
          elements of \diag(J^T J) encountered thus far in the
          iteration.  This choice of D makes the problem
          scale-invariant, so that if the model parameters x_i are each
          scaled by an arbitrary constant, \tilde{x}_i = a_i x_i, then
          the sequence of iterates produced by the algorithm would be
          unchanged.  This method can work very well in cases where the
          model parameters have widely different scales (ie: if some
          parameters are measured in nanometers, while others are
          measured in degrees Kelvin).  This strategy has been proven
          effective on a large class of problems and so it is the
          library default, but it may not be the best choice for all
          problems.

      -- Variable: gsl_multifit_nlinear_scale_levenberg
      -- Variable: gsl_multilarge_nlinear_scale_levenberg

          This damping strategy was originally suggested by Levenberg,
          and corresponds to D^T D = I. This method has also proven
          effective on a large class of problems, but is not
          scale-invariant.  However, some authors (e.g.  Transtrum and
          Sethna 2012) argue that this choice is better for problems
          which are susceptible to parameter evaporation (ie: parameters
          go to infinity)

      -- Variable: gsl_multifit_nlinear_scale_marquardt
      -- Variable: gsl_multilarge_nlinear_scale_marquardt

          This damping strategy was suggested by Marquardt, and
          corresponds to D^T D = \diag(J^T J). This method is
          scale-invariant, but it is generally considered inferior to
          both the Levenberg and Moré strategies, though may work well
          on certain classes of problems.

 -- Type: gsl_multifit_nlinear_solver
 -- Type: gsl_multilarge_nlinear_solver

     Solving the trust region subproblem on each iteration almost always
     requires the solution of the following linear least squares system

          [J; sqrt(mu) D] \delta = - [f; 0]

     The ‘solver’ parameter determines how the system is solved and can
     be selected from the following choices:

      -- Variable: gsl_multifit_nlinear_solver_qr

          This method solves the system using a rank revealing QR
          decomposition of the Jacobian J. This method will produce
          reliable solutions in cases where the Jacobian is rank
          deficient or near-singular but does require about twice as
          many operations as the Cholesky method discussed below.

      -- Variable: gsl_multifit_nlinear_solver_cholesky
      -- Variable: gsl_multilarge_nlinear_solver_cholesky

          This method solves the alternate normal equations problem

               ( J^T J + \mu D^T D ) \delta = -J^T f

          by using a Cholesky decomposition of the matrix J^T J + \mu
          D^T D. This method is faster than the QR approach, however it
          is susceptible to numerical instabilities if the Jacobian
          matrix is rank deficient or near-singular.  In these cases, an
          attempt is made to reduce the condition number of the matrix
          using Jacobi preconditioning, but for highly ill-conditioned
          problems the QR approach is better.  If it is known that the
          Jacobian matrix is well conditioned, this method is accurate
          and will perform faster than the QR approach.

      -- Variable: gsl_multifit_nlinear_solver_mcholesky
      -- Variable: gsl_multilarge_nlinear_solver_mcholesky

          This method solves the alternate normal equations problem

               ( J^T J + \mu D^T D ) \delta = -J^T f

          by using a modified Cholesky decomposition of the matrix J^T J
          + \mu D^T D. This is more suitable for the dogleg methods
          where the parameter \mu = 0, and the matrix J^T J may be
          ill-conditioned or indefinite causing the standard Cholesky
          decomposition to fail.  This method is based on Level 2 BLAS
          and is thus slower than the standard Cholesky decomposition,
          which is based on Level 3 BLAS.

      -- Variable: gsl_multifit_nlinear_solver_svd

          This method solves the system using a singular value
          decomposition of the Jacobian J. This method will produce the
          most reliable solutions for ill-conditioned Jacobians but is
          also the slowest solver method.

 -- Type: gsl_multifit_nlinear_fdtype

     The parameter ‘fdtype’ specifies whether to use forward or centered
     differences when approximating the Jacobian.  This is only used
     when an analytic Jacobian is not provided to the solver.  This
     parameter may be set to one of the following choices.

      -- Macro: GSL_MULTIFIT_NLINEAR_FWDIFF

          This specifies a forward finite difference to approximate the
          Jacobian matrix.  The Jacobian matrix will be calculated as

               J_ij = 1 / \Delta_j ( f_i(x + \Delta_j e_j) - f_i(x) )

          where \Delta_j = h |x_j| and e_j is the standard j-th
          Cartesian unit basis vector so that x + \Delta_j e_j
          represents a small (forward) perturbation of the j-th
          parameter by an amount \Delta_j.  The perturbation \Delta_j is
          proportional to the current value |x_j| which helps to
          calculate an accurate Jacobian when the various parameters
          have different scale sizes.  The value of h is specified by
          the ‘h_df’ parameter.  The accuracy of this method is O(h),
          and evaluating this matrix requires an additional p function
          evaluations.

      -- Macro: GSL_MULTIFIT_NLINEAR_CTRDIFF

          This specifies a centered finite difference to approximate the
          Jacobian matrix.  The Jacobian matrix will be calculated as

               J_ij = 1 / \Delta_j ( f_i(x + 1/2 \Delta_j e_j) - f_i(x - 1/2 \Delta_j e_j) )

          See above for a description of \Delta_j.  The accuracy of this
          method is O(h^2), but evaluating this matrix requires an
          additional 2p function evaluations.

‘double factor_up’

When a step is accepted, the trust region radius will be increased by
this factor.  The default value is 3.

‘double factor_down’

When a step is rejected, the trust region radius will be decreased by
this factor.  The default value is 2.

‘double avmax’

When using geodesic acceleration to solve a nonlinear least squares
problem, an important parameter to monitor is the ratio of the
acceleration term to the velocity term,

     |a| / |v|

If this ratio is small, it means the acceleration correction is
contributing very little to the step.  This could be because the problem
is not “nonlinear” enough to benefit from the acceleration.  If the
ratio is large (> 1) it means that the acceleration is larger than the
velocity, which shouldn’t happen since the step represents a truncated
series and so the second order term a should be smaller than the first
order term v to guarantee convergence.  Therefore any steps with a ratio
larger than the parameter ‘avmax’ are rejected.  ‘avmax’ is set to 0.75
by default.  For problems which experience difficulty converging, this
threshold could be lowered.

‘double h_df’

This parameter specifies the step size for approximating the Jacobian
matrix with finite differences.  It is set to \sqrt{\epsilon} by
default, where \epsilon is ‘GSL_DBL_EPSILON’.

‘double h_fvv’

When using geodesic acceleration, the user must either supply a function
to calculate f_{vv}(x) or the library can estimate this second
directional derivative using a finite difference method.  When using
finite differences, the library must calculate f(x + h v) where h
represents a small step in the velocity direction.  The parameter
‘h_fvv’ defines this step size and is set to 0.02 by default.


File: gsl-ref.info,  Node: Initializing the Solver<3>,  Next: Providing the Function to be Minimized,  Prev: Tunable Parameters,  Up: Nonlinear Least-Squares Fitting

41.5 Initializing the Solver
============================

 -- Type: gsl_multifit_nlinear_type

     This structure specifies the type of algorithm which will be used
     to solve a nonlinear least squares problem.  It may be selected
     from the following choices,

      -- Variable: gsl_multifit_nlinear_trust

          This specifies a trust region method.  It is currently the
          only implemented nonlinear least squares method.

 -- Function: gsl_multifit_nlinear_workspace *
          gsl_multifit_nlinear_alloc (const gsl_multifit_nlinear_type
          * T, const gsl_multifit_nlinear_parameters * params, const
          size_t n, const size_t p)
 -- Function: gsl_multilarge_nlinear_workspace *
          gsl_multilarge_nlinear_alloc (const
          gsl_multilarge_nlinear_type * T, const
          gsl_multilarge_nlinear_parameters * params, const size_t n,
          const size_t p)

     These functions return a pointer to a newly allocated instance of a
     derivative solver of type ‘T’ for ‘n’ observations and ‘p’
     parameters.  The ‘params’ input specifies a tunable set of
     parameters which will affect important details in each iteration of
     the trust region subproblem algorithm.  It is recommended to start
     with the suggested default parameters (see *note
     gsl_multifit_nlinear_default_parameters(): ba1. and *note
     gsl_multilarge_nlinear_default_parameters(): ba2.) and then tune
     the parameters once the code is working correctly.  See *note
     Tunable Parameters: b78.  for descriptions of the various
     parameters.  For example, the following code creates an instance of
     a Levenberg-Marquardt solver for 100 data points and 3 parameters,
     using suggested defaults:

          const gsl_multifit_nlinear_type * T = gsl_multifit_nlinear_trust;
          gsl_multifit_nlinear_parameters params = gsl_multifit_nlinear_default_parameters();
          gsl_multifit_nlinear_workspace * w = gsl_multifit_nlinear_alloc (T, &params, 100, 3);

     The number of observations ‘n’ must be greater than or equal to
     parameters ‘p’.

     If there is insufficient memory to create the solver then the
     function returns a null pointer and the error handler is invoked
     with an error code of *note GSL_ENOMEM: 2a.

 -- Function: gsl_multifit_nlinear_parameters
          gsl_multifit_nlinear_default_parameters (void)
 -- Function: gsl_multilarge_nlinear_parameters
          gsl_multilarge_nlinear_default_parameters (void)

     These functions return a set of recommended default parameters for
     use in solving nonlinear least squares problems.  The user can tune
     each parameter to improve the performance on their particular
     problem, see *note Tunable Parameters: b78.

 -- Function: int gsl_multifit_nlinear_init (const gsl_vector * x,
          gsl_multifit_nlinear_fdf * fdf, gsl_multifit_nlinear_workspace
          * w)
 -- Function: int gsl_multifit_nlinear_winit (const gsl_vector * x,
          const gsl_vector * wts, gsl_multifit_nlinear_fdf * fdf,
          gsl_multifit_nlinear_workspace * w)
 -- Function: int gsl_multilarge_nlinear_init (const gsl_vector * x,
          gsl_multilarge_nlinear_fdf * fdf,
          gsl_multilarge_nlinear_workspace * w)

     These functions initialize, or reinitialize, an existing workspace
     ‘w’ to use the system ‘fdf’ and the initial guess ‘x’.  See *note
     Providing the Function to be Minimized: ba5. for a description of
     the ‘fdf’ structure.

     Optionally, a weight vector ‘wts’ can be given to perform a
     weighted nonlinear regression.  Here, the weighting matrix is W =
     \diag(w_1,w_2,...,w_n).

 -- Function: void gsl_multifit_nlinear_free
          (gsl_multifit_nlinear_workspace * w)
 -- Function: void gsl_multilarge_nlinear_free
          (gsl_multilarge_nlinear_workspace * w)

     These functions free all the memory associated with the workspace
     ‘w’.

 -- Function: const char * gsl_multifit_nlinear_name (const
          gsl_multifit_nlinear_workspace * w)
 -- Function: const char * gsl_multilarge_nlinear_name (const
          gsl_multilarge_nlinear_workspace * w)

     These functions return a pointer to the name of the solver.  For
     example:

          printf ("w is a '%s' solver\n", gsl_multifit_nlinear_name (w));

     would print something like ‘w is a 'trust-region' solver’.

 -- Function: const char * gsl_multifit_nlinear_trs_name (const
          gsl_multifit_nlinear_workspace * w)
 -- Function: const char * gsl_multilarge_nlinear_trs_name (const
          gsl_multilarge_nlinear_workspace * w)

     These functions return a pointer to the name of the trust region
     subproblem method.  For example:

          printf ("w is a '%s' solver\n", gsl_multifit_nlinear_trs_name (w));

     would print something like ‘w is a 'levenberg-marquardt' solver’.


File: gsl-ref.info,  Node: Providing the Function to be Minimized,  Next: Iteration<5>,  Prev: Initializing the Solver<3>,  Up: Nonlinear Least-Squares Fitting

41.6 Providing the Function to be Minimized
===========================================

The user must provide n functions of p variables for the minimization
algorithm to operate on.  In order to allow for arbitrary parameters the
functions are defined by the following data types:

 -- Type: gsl_multifit_nlinear_fdf

     This data type defines a general system of functions with arbitrary
     parameters, the corresponding Jacobian matrix of derivatives, and
     optionally the second directional derivative of the functions for
     geodesic acceleration.

     ‘int (* f) (const gsl_vector * x, void * params, gsl_vector * f)’

          This function should store the n components of the vector f(x)
          in ‘f’ for argument ‘x’ and arbitrary parameters ‘params’,
          returning an appropriate error code if the function cannot be
          computed.

     ‘int (* df) (const gsl_vector * x, void * params, gsl_matrix * J)’

          This function should store the ‘n’-by-‘p’ matrix result

               J_ij = d f_i(x) / d x_j

          in ‘J’ for argument ‘x’ and arbitrary parameters ‘params’,
          returning an appropriate error code if the matrix cannot be
          computed.  If an analytic Jacobian is unavailable, or too
          expensive to compute, this function pointer may be set to
          ‘NULL’, in which case the Jacobian will be internally computed
          using finite difference approximations of the function ‘f’.

     ‘int (* fvv) (const gsl_vector * x, const gsl_vector * v, void *
     params, gsl_vector * fvv)’

          When geodesic acceleration is enabled, this function should
          store the n components of the vector f_{vv}(x) =
          \sum_{\alpha\beta} v_{\alpha} v_{\beta} {\partial \over
          \partial x_{\alpha}} {\partial \over \partial x_{\beta}} f(x),
          representing second directional derivatives of the function to
          be minimized, into the output ‘fvv’.  The parameter vector is
          provided in ‘x’ and the velocity vector is provided in ‘v’,
          both of which have p components.  The arbitrary parameters are
          given in ‘params’.  If analytic expressions for f_{vv}(x) are
          unavailable or too difficult to compute, this function pointer
          may be set to ‘NULL’, in which case f_{vv}(x) will be computed
          internally using a finite difference approximation.

     ‘size_t n’

          the number of functions, i.e.  the number of components of the
          vector ‘f’.

     ‘size_t p’

          the number of independent variables, i.e.  the number of
          components of the vector ‘x’.

     ‘void * params’

          a pointer to the arbitrary parameters of the function.

     ‘size_t nevalf’

          This does not need to be set by the user.  It counts the
          number of function evaluations and is initialized by the
          ‘_init’ function.

     ‘size_t nevaldf’

          This does not need to be set by the user.  It counts the
          number of Jacobian evaluations and is initialized by the
          ‘_init’ function.

     ‘size_t nevalfvv’

          This does not need to be set by the user.  It counts the
          number of f_{vv}(x) evaluations and is initialized by the
          ‘_init’ function.

 -- Type: gsl_multilarge_nlinear_fdf

     This data type defines a general system of functions with arbitrary
     parameters, a function to compute J u or J^T u for a given vector
     u, the normal equations matrix J^T J, and optionally the second
     directional derivative of the functions for geodesic acceleration.

     ‘int (* f) (const gsl_vector * x, void * params, gsl_vector * f)’

          This function should store the n components of the vector f(x)
          in ‘f’ for argument ‘x’ and arbitrary parameters ‘params’,
          returning an appropriate error code if the function cannot be
          computed.

     ‘int (* df) (CBLAS_TRANSPOSE_t TransJ, const gsl_vector * x, const
     gsl_vector * u, void * params, gsl_vector * v, gsl_matrix * JTJ)’

          If ‘TransJ’ is equal to ‘CblasNoTrans’, then this function
          should compute the matrix-vector product J u and store the
          result in ‘v’.  If ‘TransJ’ is equal to ‘CblasTrans’, then
          this function should compute the matrix-vector product J^T u
          and store the result in ‘v’.  Additionally, the normal
          equations matrix J^T J should be stored in the lower half of
          ‘JTJ’.  The input matrix ‘JTJ’ could be set to ‘NULL’, for
          example by iterative methods which do not require this matrix,
          so the user should check for this prior to constructing the
          matrix.  The input ‘params’ contains the arbitrary parameters.

     ‘int (* fvv) (const gsl_vector * x, const gsl_vector * v, void *
     params, gsl_vector * fvv)’

          When geodesic acceleration is enabled, this function should
          store the n components of the vector f_{vv}(x) =
          \sum_{\alpha\beta} v_{\alpha} v_{\beta} {\partial \over
          \partial x_{\alpha}} {\partial \over \partial x_{\beta}} f(x),
          representing second directional derivatives of the function to
          be minimized, into the output ‘fvv’.  The parameter vector is
          provided in ‘x’ and the velocity vector is provided in ‘v’,
          both of which have p components.  The arbitrary parameters are
          given in ‘params’.  If analytic expressions for f_{vv}(x) are
          unavailable or too difficult to compute, this function pointer
          may be set to ‘NULL’, in which case f_{vv}(x) will be computed
          internally using a finite difference approximation.

     ‘size_t n’

          the number of functions, i.e.  the number of components of the
          vector ‘f’.

     ‘size_t p’

          the number of independent variables, i.e.  the number of
          components of the vector ‘x’.

     ‘void * params’

          a pointer to the arbitrary parameters of the function.

     ‘size_t nevalf’

          This does not need to be set by the user.  It counts the
          number of function evaluations and is initialized by the
          ‘_init’ function.

     ‘size_t nevaldfu’

          This does not need to be set by the user.  It counts the
          number of Jacobian matrix-vector evaluations (J u or J^T u)
          and is initialized by the ‘_init’ function.

     ‘size_t nevaldf2’

          This does not need to be set by the user.  It counts the
          number of J^T J evaluations and is initialized by the ‘_init’
          function.

     ‘size_t nevalfvv’

          This does not need to be set by the user.  It counts the
          number of f_{vv}(x) evaluations and is initialized by the
          ‘_init’ function.

Note that when fitting a non-linear model against experimental data, the
data is passed to the functions above using the ‘params’ argument and
the trial best-fit parameters through the ‘x’ argument.


File: gsl-ref.info,  Node: Iteration<5>,  Next: Testing for Convergence,  Prev: Providing the Function to be Minimized,  Up: Nonlinear Least-Squares Fitting

41.7 Iteration
==============

The following functions drive the iteration of each algorithm.  Each
function performs one iteration of the trust region method and updates
the state of the solver.

 -- Function: int gsl_multifit_nlinear_iterate
          (gsl_multifit_nlinear_workspace * w)
 -- Function: int gsl_multilarge_nlinear_iterate
          (gsl_multilarge_nlinear_workspace * w)

     These functions perform a single iteration of the solver ‘w’.  If
     the iteration encounters an unexpected problem then an error code
     will be returned.  The solver workspace maintains a current
     estimate of the best-fit parameters at all times.

The solver workspace ‘w’ contains the following entries, which can be
used to track the progress of the solution:

‘gsl_vector * x’

     The current position, length p.

‘gsl_vector * f’

     The function residual vector at the current position f(x), length
     n.

‘gsl_matrix * J’

     The Jacobian matrix at the current position J(x), size n-by-p (only
     for ‘gsl_multifit_nlinear’ interface).

‘gsl_vector * dx’

     The difference between the current position and the previous
     position, i.e.  the last step \delta, taken as a vector, length p.

These quantities can be accessed with the following functions,

 -- Function: gsl_vector * gsl_multifit_nlinear_position (const
          gsl_multifit_nlinear_workspace * w)
 -- Function: gsl_vector * gsl_multilarge_nlinear_position (const
          gsl_multilarge_nlinear_workspace * w)

     These functions return the current position x (i.e.  best-fit
     parameters) of the solver ‘w’.

 -- Function: gsl_vector * gsl_multifit_nlinear_residual (const
          gsl_multifit_nlinear_workspace * w)
 -- Function: gsl_vector * gsl_multilarge_nlinear_residual (const
          gsl_multilarge_nlinear_workspace * w)

     These functions return the current residual vector f(x) of the
     solver ‘w’.  For weighted systems, the residual vector includes the
     weighting factor \sqrt{W}.

 -- Function: gsl_matrix * gsl_multifit_nlinear_jac (const
          gsl_multifit_nlinear_workspace * w)

     This function returns a pointer to the n-by-p Jacobian matrix for
     the current iteration of the solver ‘w’.  This function is
     available only for the ‘gsl_multifit_nlinear’ interface.

 -- Function: size_t gsl_multifit_nlinear_niter (const
          gsl_multifit_nlinear_workspace * w)
 -- Function: size_t gsl_multilarge_nlinear_niter (const
          gsl_multilarge_nlinear_workspace * w)

     These functions return the number of iterations performed so far.
     The iteration counter is updated on each call to the ‘_iterate’
     functions above, and reset to 0 in the ‘_init’ functions.

 -- Function: int gsl_multifit_nlinear_rcond (double * rcond, const
          gsl_multifit_nlinear_workspace * w)
 -- Function: int gsl_multilarge_nlinear_rcond (double * rcond, const
          gsl_multilarge_nlinear_workspace * w)

     This function estimates the reciprocal condition number of the
     Jacobian matrix at the current position x and stores it in ‘rcond’.
     The computed value is only an estimate to give the user a guideline
     as to the conditioning of their particular problem.  Its
     calculation is based on which factorization method is used
     (Cholesky, QR, or SVD).

        * For the Cholesky solver, the matrix J^T J is factored at each
          iteration.  Therefore this function will estimate the 1-norm
          condition number rcond^2 = 1/(||J^T J||_1 \cdot ||(J^T
          J)^{-1}||_1)

        * For the QR solver, J is factored as J = Q R at each iteration.
          For simplicity, this function calculates the 1-norm
          conditioning of only the R factor, rcond = 1 / (||R||_1 \cdot
          ||R^{-1}||_1).  This can be computed efficiently since R is
          upper triangular.

        * For the SVD solver, in order to efficiently solve the trust
          region subproblem, the matrix which is factored is J D^{-1},
          instead of J itself.  The resulting singular values are used
          to provide the 2-norm reciprocal condition number, as rcond =
          \sigma_{min} / \sigma_{max}.  Note that when using Moré
          scaling, D \ne I and the resulting ‘rcond’ estimate may be
          significantly different from the true ‘rcond’ of J itself.

 -- Function: double gsl_multifit_nlinear_avratio (const
          gsl_multifit_nlinear_workspace * w)
 -- Function: double gsl_multilarge_nlinear_avratio (const
          gsl_multilarge_nlinear_workspace * w)

     This function returns the current ratio |a| / |v| of the
     acceleration correction term to the velocity step term.  The
     acceleration term is computed only by the *note
     gsl_multifit_nlinear_trs_lmaccel: b80. and *note
     gsl_multilarge_nlinear_trs_lmaccel: b81. methods, so this ratio
     will be zero for other TRS methods.


File: gsl-ref.info,  Node: Testing for Convergence,  Next: High Level Driver,  Prev: Iteration<5>,  Up: Nonlinear Least-Squares Fitting

41.8 Testing for Convergence
============================

A minimization procedure should stop when one of the following
conditions is true:

   * A minimum has been found to within the user-specified precision.

   * A user-specified maximum number of iterations has been reached.

   * An error has occurred.

The handling of these conditions is under user control.  The functions
below allow the user to test the current estimate of the best-fit
parameters in several standard ways.

 -- Function: int gsl_multifit_nlinear_test (const double xtol, const
          double gtol, const double ftol, int * info, const
          gsl_multifit_nlinear_workspace * w)
 -- Function: int gsl_multilarge_nlinear_test (const double xtol, const
          double gtol, const double ftol, int * info, const
          gsl_multilarge_nlinear_workspace * w)

     These functions test for convergence of the minimization method
     using the following criteria:

        * Testing for a small step size relative to the current
          parameter vector

               |\delta_i| \le xtol (|x_i| + xtol)

          for each 0 <= i < p.  Each element of the step vector \delta
          is tested individually in case the different parameters have
          widely different scales.  Adding ‘xtol’ to |x_i| helps the
          test avoid breaking down in situations where the true solution
          value x_i = 0.  If this test succeeds, ‘info’ is set to 1 and
          the function returns ‘GSL_SUCCESS’.

          A general guideline for selecting the step tolerance is to
          choose xtol = 10^{-d} where d is the number of accurate
          decimal digits desired in the solution x.  See Dennis and
          Schnabel for more information.

        * Testing for a small gradient (g = \nabla \Phi(x) = J^T f)
          indicating a local function minimum:

               ||g||_inf <= gtol

          This expression tests whether the ratio (\nabla \Phi)_i x_i /
          \Phi is small.  Testing this scaled gradient is a better than
          \nabla \Phi alone since it is a dimensionless quantity and so
          independent of the scale of the problem.  The ‘max’ arguments
          help ensure the test doesn’t break down in regions where x_i
          or \Phi(x) are close to 0.  If this test succeeds, ‘info’ is
          set to 2 and the function returns ‘GSL_SUCCESS’.

          A general guideline for choosing the gradient tolerance is to
          set ‘gtol = GSL_DBL_EPSILON^(1/3)’.  See Dennis and Schnabel
          for more information.

     If none of the tests succeed, ‘info’ is set to 0 and the function
     returns ‘GSL_CONTINUE’, indicating further iterations are required.


File: gsl-ref.info,  Node: High Level Driver,  Next: Covariance matrix of best fit parameters,  Prev: Testing for Convergence,  Up: Nonlinear Least-Squares Fitting

41.9 High Level Driver
======================

These routines provide a high level wrapper that combines the iteration
and convergence testing for easy use.

 -- Function: int gsl_multifit_nlinear_driver (const size_t maxiter,
          const double xtol, const double gtol, const double ftol, void
          (* callback)(const size_t iter, void * params, const
          gsl_multifit_linear_workspace * w), void * callback_params,
          int * info, gsl_multifit_nlinear_workspace * w)
 -- Function: int gsl_multilarge_nlinear_driver (const size_t maxiter,
          const double xtol, const double gtol, const double ftol, void
          (* callback)(const size_t iter, void * params, const
          gsl_multilarge_linear_workspace * w), void * callback_params,
          int * info, gsl_multilarge_nlinear_workspace * w)

     These functions iterate the nonlinear least squares solver ‘w’ for
     a maximum of ‘maxiter’ iterations.  After each iteration, the
     system is tested for convergence with the error tolerances ‘xtol’,
     ‘gtol’ and ‘ftol’.  Additionally, the user may supply a callback
     function ‘callback’ which is called after each iteration, so that
     the user may save or print relevant quantities for each iteration.
     The parameter ‘callback_params’ is passed to the ‘callback’
     function.  The parameters ‘callback’ and ‘callback_params’ may be
     set to ‘NULL’ to disable this feature.  Upon successful
     convergence, the function returns ‘GSL_SUCCESS’ and sets ‘info’ to
     the reason for convergence (see *note gsl_multifit_nlinear_test():
     bbe.).  If the function has not converged after ‘maxiter’
     iterations, ‘GSL_EMAXITER’ is returned.  In rare cases, during an
     iteration the algorithm may be unable to find a new acceptable step
     \delta to take.  In this case, ‘GSL_ENOPROG’ is returned indicating
     no further progress can be made.  If your problem is having
     difficulty converging, see *note Troubleshooting: bc3. for further
     guidance.


File: gsl-ref.info,  Node: Covariance matrix of best fit parameters,  Next: Troubleshooting<2>,  Prev: High Level Driver,  Up: Nonlinear Least-Squares Fitting

41.10 Covariance matrix of best fit parameters
==============================================

 -- Function: int gsl_multifit_nlinear_covar (const gsl_matrix * J,
          const double epsrel, gsl_matrix * covar)
 -- Function: int gsl_multilarge_nlinear_covar (gsl_matrix * covar,
          gsl_multilarge_nlinear_workspace * w)

     This function computes the covariance matrix of best-fit parameters
     using the Jacobian matrix ‘J’ and stores it in ‘covar’.  The
     parameter ‘epsrel’ is used to remove linear-dependent columns when
     ‘J’ is rank deficient.

     The covariance matrix is given by,

          C = (J^T J)^{-1}

     or in the weighted case,

          C = (J^T W J)^{-1}

     and is computed using the factored form of the Jacobian (Cholesky,
     QR, or SVD). Any columns of R which satisfy

          |R_{kk}| \leq epsrel |R_{11}|

     are considered linearly-dependent and are excluded from the
     covariance matrix (the corresponding rows and columns of the
     covariance matrix are set to zero).

     If the minimisation uses the weighted least-squares function f_i =
     (Y(x, t_i) - y_i) / \sigma_i then the covariance matrix above gives
     the statistical error on the best-fit parameters resulting from the
     Gaussian errors \sigma_i on the underlying data y_i.  This can be
     verified from the relation \delta f = J \delta c and the fact that
     the fluctuations in f from the data y_i are normalised by \sigma_i
     and so satisfy

          <\delta f \delta f^T> = I

     For an unweighted least-squares function f_i = (Y(x, t_i) - y_i)
     the covariance matrix above should be multiplied by the variance of
     the residuals about the best-fit \sigma^2 = \sum (y_i - Y(x,t_i))^2
     / (n-p) to give the variance-covariance matrix \sigma^2 C. This
     estimates the statistical error on the best-fit parameters from the
     scatter of the underlying data.

     For more information about covariance matrices see *note Linear
     Least-Squares Overview: b02.


File: gsl-ref.info,  Node: Troubleshooting<2>,  Next: Examples<32>,  Prev: Covariance matrix of best fit parameters,  Up: Nonlinear Least-Squares Fitting

41.11 Troubleshooting
=====================

When developing a code to solve a nonlinear least squares problem, here
are a few considerations to keep in mind.

  1. The most common difficulty is the accurate implementation of the
     Jacobian matrix.  If the analytic Jacobian is not properly provided
     to the solver, this can hinder and many times prevent convergence
     of the method.  When developing a new nonlinear least squares code,
     it often helps to compare the program output with the internally
     computed finite difference Jacobian and the user supplied analytic
     Jacobian.  If there is a large difference in coefficients, it is
     likely the analytic Jacobian is incorrectly implemented.

  2. If your code is having difficulty converging, the next thing to
     check is the starting point provided to the solver.  The methods of
     this chapter are local methods, meaning if you provide a starting
     point far away from the true minimum, the method may converge to a
     local minimum or not converge at all.  Sometimes it is possible to
     solve a linearized approximation to the nonlinear problem, and use
     the linear solution as the starting point to the nonlinear problem.

  3. If the various parameters of the coefficient vector x vary widely
     in magnitude, then the problem is said to be badly scaled.  The
     methods of this chapter do attempt to automatically rescale the
     elements of x to have roughly the same order of magnitude, but in
     extreme cases this could still cause problems for convergence.  In
     these cases it is recommended for the user to scale their parameter
     vector x so that each parameter spans roughly the same range, say
     [-1,1].  The solution vector can be backscaled to recover the
     original units of the problem.


File: gsl-ref.info,  Node: Examples<32>,  Next: References and Further Reading<34>,  Prev: Troubleshooting<2>,  Up: Nonlinear Least-Squares Fitting

41.12 Examples
==============

The following example programs demonstrate the nonlinear least squares
fitting capabilities.

* Menu:

* Exponential Fitting Example::
* Geodesic Acceleration Example 1::
* Geodesic Acceleration Example 2::
* Comparing TRS Methods Example::
* Large Nonlinear Least Squares Example::


File: gsl-ref.info,  Node: Exponential Fitting Example,  Next: Geodesic Acceleration Example 1,  Up: Examples<32>

41.12.1 Exponential Fitting Example
-----------------------------------

The following example program fits a weighted exponential model with
background to experimental data, Y = A \exp(-\lambda t) + b.  The first
part of the program sets up the functions ‘expb_f()’ and ‘expb_df()’ to
calculate the model and its Jacobian.  The appropriate fitting function
is given by,

     f_i = (A \exp(-\lambda t_i) + b) - y_i

where we have chosen t_i = i T / (N - 1), where N is the number of data
points fitted, so that t_i \in [0, T]. The Jacobian matrix J is the
derivative of these functions with respect to the three parameters (A,
\lambda, b).  It is given by,

     J_{ij} = d f_i / d x_j

where x_0 = A, x_1 = \lambda and x_2 = b.  The i-th row of the Jacobian
is therefore

     J(i,:) = [ \exp(-\lambda t_i) ; -t_i A \exp(-\lambda t_i) ; 1 ]

The main part of the program sets up a Levenberg-Marquardt solver and
some simulated random data.  The data uses the known parameters
(5.0,0.1,1.0) combined with Gaussian noise (standard deviation = 0.1)
with a maximum time T = 40 and N = 100 timesteps.  The initial guess for
the parameters is chosen as (1.0, 1.0, 0.0).  The iteration terminates
when the relative change in x is smaller than 10^{-8}, or when the
magnitude of the gradient falls below 10^{-8}.  Here are the results of
running the program:

     iter  0: A = 1.0000, lambda = 1.0000, b = 0.0000, cond(J) =      inf, |f(x)| = 100.8779
     iter  1: A = 1.2692, lambda = 0.3924, b = 0.0443, cond(J) =  69.5973, |f(x)| = 97.1734
     iter  2: A = 1.6749, lambda = 0.1685, b = 0.1072, cond(J) =  29.5220, |f(x)| = 88.6636
     iter  3: A = 2.5579, lambda = 0.0544, b = 0.2552, cond(J) =  22.9334, |f(x)| = 42.7765
     iter  4: A = 3.0167, lambda = 0.0472, b = 0.3704, cond(J) = 120.2912, |f(x)| = 23.0102
     iter  5: A = 3.3590, lambda = 0.0455, b = 0.4321, cond(J) = 266.5620, |f(x)| = 16.0680
     iter  6: A = 3.6552, lambda = 0.0479, b = 0.4426, cond(J) = 343.8946, |f(x)| = 14.6421
     iter  7: A = 3.9546, lambda = 0.0532, b = 0.4897, cond(J) = 301.4985, |f(x)| = 13.5266
     iter  8: A = 4.1421, lambda = 0.0633, b = 0.6783, cond(J) = 203.9164, |f(x)| = 12.3149
     iter  9: A = 4.3752, lambda = 0.0800, b = 0.9228, cond(J) = 158.2267, |f(x)| = 11.2475
     iter 10: A = 4.6371, lambda = 0.0891, b = 0.9588, cond(J) = 136.6189, |f(x)| = 10.5457
     iter 11: A = 4.7684, lambda = 0.0937, b = 0.9860, cond(J) = 125.4740, |f(x)| = 10.4753
     iter 12: A = 4.7977, lambda = 0.0948, b = 0.9917, cond(J) = 120.1098, |f(x)| = 10.4723
     iter 13: A = 4.8006, lambda = 0.0949, b = 0.9924, cond(J) = 118.9113, |f(x)| = 10.4723
     iter 14: A = 4.8008, lambda = 0.0949, b = 0.9925, cond(J) = 118.7661, |f(x)| = 10.4723
     iter 15: A = 4.8008, lambda = 0.0949, b = 0.9925, cond(J) = 118.7550, |f(x)| = 10.4723
     iter 16: A = 4.8008, lambda = 0.0949, b = 0.9925, cond(J) = 118.7543, |f(x)| = 10.4723
     iter 17: A = 4.8008, lambda = 0.0949, b = 0.9925, cond(J) = 118.7543, |f(x)| = 10.4723
     iter 18: A = 4.8008, lambda = 0.0949, b = 0.9925, cond(J) = 118.7543, |f(x)| = 10.4723
     summary from method 'trust-region/levenberg-marquardt'
     number of iterations: 18
     function evaluations: 25
     Jacobian evaluations: 19
     reason for stopping: small step size
     initial |f(x)| = 100.877904
     final   |f(x)| = 10.472268
     chisq/dof = 1.1306
     A      = 4.80085 +/- 0.17652
     lambda = 0.09488 +/- 0.00527
     b      = 0.99249 +/- 0.04419
     status = success

The approximate values of the parameters are found correctly, and the
chi-squared value indicates a good fit (the chi-squared per degree of
freedom is approximately 1).  In this case the errors on the parameters
can be estimated from the square roots of the diagonal elements of the
covariance matrix.  If the chi-squared value shows a poor fit (i.e.
\chi^2/(n-p) \gg 1 then the error estimates obtained from the covariance
matrix will be too small.  In the example program the error estimates
are multiplied by \sqrt{\chi^2/(n-p)} in this case, a common way of
increasing the errors for a poor fit.  Note that a poor fit will result
from the use of an inappropriate model, and the scaled error estimates
may then be outside the range of validity for Gaussian errors.

Additionally, we see that the condition number of J(x) stays reasonably
small throughout the iteration.  This indicates we could safely switch
to the Cholesky solver for speed improvement, although this particular
system is too small to really benefit.

Fig.  %s shows the fitted curve with the original data.

[image src="fit-exp.png"]

Figure: Exponential fitted curve with data

     #include <stdlib.h>
     #include <stdio.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_blas.h>
     #include <gsl/gsl_multifit_nlinear.h>

     #define N      100    /* number of data points to fit */
     #define TMAX   (40.0) /* time variable in [0,TMAX] */

     struct data {
       size_t n;
       double * t;
       double * y;
     };

     int
     expb_f (const gsl_vector * x, void *data,
             gsl_vector * f)
     {
       size_t n = ((struct data *)data)->n;
       double *t = ((struct data *)data)->t;
       double *y = ((struct data *)data)->y;

       double A = gsl_vector_get (x, 0);
       double lambda = gsl_vector_get (x, 1);
       double b = gsl_vector_get (x, 2);

       size_t i;

       for (i = 0; i < n; i++)
         {
           /* Model Yi = A * exp(-lambda * t_i) + b */
           double Yi = A * exp (-lambda * t[i]) + b;
           gsl_vector_set (f, i, Yi - y[i]);
         }

       return GSL_SUCCESS;
     }

     int
     expb_df (const gsl_vector * x, void *data,
              gsl_matrix * J)
     {
       size_t n = ((struct data *)data)->n;
       double *t = ((struct data *)data)->t;

       double A = gsl_vector_get (x, 0);
       double lambda = gsl_vector_get (x, 1);

       size_t i;

       for (i = 0; i < n; i++)
         {
           /* Jacobian matrix J(i,j) = dfi / dxj, */
           /* where fi = (Yi - yi)/sigma[i],      */
           /*       Yi = A * exp(-lambda * t_i) + b  */
           /* and the xj are the parameters (A,lambda,b) */
           double e = exp(-lambda * t[i]);
           gsl_matrix_set (J, i, 0, e);
           gsl_matrix_set (J, i, 1, -t[i] * A * e);
           gsl_matrix_set (J, i, 2, 1.0);
         }

       return GSL_SUCCESS;
     }

     void
     callback(const size_t iter, void *params,
              const gsl_multifit_nlinear_workspace *w)
     {
       gsl_vector *f = gsl_multifit_nlinear_residual(w);
       gsl_vector *x = gsl_multifit_nlinear_position(w);
       double rcond;

       /* compute reciprocal condition number of J(x) */
       gsl_multifit_nlinear_rcond(&rcond, w);

       fprintf(stderr, "iter %2zu: A = %.4f, lambda = %.4f, b = %.4f, cond(J) = %8.4f, |f(x)| = %.4f\n",
               iter,
               gsl_vector_get(x, 0),
               gsl_vector_get(x, 1),
               gsl_vector_get(x, 2),
               1.0 / rcond,
               gsl_blas_dnrm2(f));
     }

     int
     main (void)
     {
       const gsl_multifit_nlinear_type *T = gsl_multifit_nlinear_trust;
       gsl_multifit_nlinear_workspace *w;
       gsl_multifit_nlinear_fdf fdf;
       gsl_multifit_nlinear_parameters fdf_params =
         gsl_multifit_nlinear_default_parameters();
       const size_t n = N;
       const size_t p = 3;

       gsl_vector *f;
       gsl_matrix *J;
       gsl_matrix *covar = gsl_matrix_alloc (p, p);
       double t[N], y[N], weights[N];
       struct data d = { n, t, y };
       double x_init[3] = { 1.0, 1.0, 0.0 }; /* starting values */
       gsl_vector_view x = gsl_vector_view_array (x_init, p);
       gsl_vector_view wts = gsl_vector_view_array(weights, n);
       gsl_rng * r;
       double chisq, chisq0;
       int status, info;
       size_t i;

       const double xtol = 1e-8;
       const double gtol = 1e-8;
       const double ftol = 0.0;

       gsl_rng_env_setup();
       r = gsl_rng_alloc(gsl_rng_default);

       /* define the function to be minimized */
       fdf.f = expb_f;
       fdf.df = expb_df;   /* set to NULL for finite-difference Jacobian */
       fdf.fvv = NULL;     /* not using geodesic acceleration */
       fdf.n = n;
       fdf.p = p;
       fdf.params = &d;

       /* this is the data to be fitted */
       for (i = 0; i < n; i++)
         {
           double ti = i * TMAX / (n - 1.0);
           double yi = 1.0 + 5 * exp (-0.1 * ti);
           double si = 0.1 * yi;
           double dy = gsl_ran_gaussian(r, si);

           t[i] = ti;
           y[i] = yi + dy;
           weights[i] = 1.0 / (si * si);
           printf ("data: %g %g %g\n", ti, y[i], si);
         };

       /* allocate workspace with default parameters */
       w = gsl_multifit_nlinear_alloc (T, &fdf_params, n, p);

       /* initialize solver with starting point and weights */
       gsl_multifit_nlinear_winit (&x.vector, &wts.vector, &fdf, w);

       /* compute initial cost function */
       f = gsl_multifit_nlinear_residual(w);
       gsl_blas_ddot(f, f, &chisq0);

       /* solve the system with a maximum of 100 iterations */
       status = gsl_multifit_nlinear_driver(100, xtol, gtol, ftol,
                                            callback, NULL, &info, w);

       /* compute covariance of best fit parameters */
       J = gsl_multifit_nlinear_jac(w);
       gsl_multifit_nlinear_covar (J, 0.0, covar);

       /* compute final cost */
       gsl_blas_ddot(f, f, &chisq);

     #define FIT(i) gsl_vector_get(w->x, i)
     #define ERR(i) sqrt(gsl_matrix_get(covar,i,i))

       fprintf(stderr, "summary from method '%s/%s'\n",
               gsl_multifit_nlinear_name(w),
               gsl_multifit_nlinear_trs_name(w));
       fprintf(stderr, "number of iterations: %zu\n",
               gsl_multifit_nlinear_niter(w));
       fprintf(stderr, "function evaluations: %zu\n", fdf.nevalf);
       fprintf(stderr, "Jacobian evaluations: %zu\n", fdf.nevaldf);
       fprintf(stderr, "reason for stopping: %s\n",
               (info == 1) ? "small step size" : "small gradient");
       fprintf(stderr, "initial |f(x)| = %f\n", sqrt(chisq0));
       fprintf(stderr, "final   |f(x)| = %f\n", sqrt(chisq));

       {
         double dof = n - p;
         double c = GSL_MAX_DBL(1, sqrt(chisq / dof));

         fprintf(stderr, "chisq/dof = %g\n", chisq / dof);

         fprintf (stderr, "A      = %.5f +/- %.5f\n", FIT(0), c*ERR(0));
         fprintf (stderr, "lambda = %.5f +/- %.5f\n", FIT(1), c*ERR(1));
         fprintf (stderr, "b      = %.5f +/- %.5f\n", FIT(2), c*ERR(2));
       }

       fprintf (stderr, "status = %s\n", gsl_strerror (status));

       gsl_multifit_nlinear_free (w);
       gsl_matrix_free (covar);
       gsl_rng_free (r);

       return 0;
     }


File: gsl-ref.info,  Node: Geodesic Acceleration Example 1,  Next: Geodesic Acceleration Example 2,  Prev: Exponential Fitting Example,  Up: Examples<32>

41.12.2 Geodesic Acceleration Example 1
---------------------------------------

The following example program minimizes a modified Rosenbrock function,
which is characterized by a narrow canyon with steep walls.  The
starting point is selected high on the canyon wall, so the solver must
first find the canyon bottom and then navigate to the minimum.  The
problem is solved both with and without using geodesic acceleration for
comparison.  The cost function is given by

     Phi(x) = 1/2 (f1^2 + f2^2)
     f1 = 100 ( x2 - x1^2 )
     f2 = 1 - x1

The Jacobian matrix is

     J = [ -200*x1 100  ]
         [   -1     0   ]

In order to use geodesic acceleration, the user must provide the second
directional derivative of each residual in the velocity direction, D_v^2
f_i = \sum_{\alpha\beta} v_{\alpha} v_{\beta} \partial_{\alpha}
\partial_{\beta} f_i.  The velocity vector v is provided by the solver.
For this example, these derivatives are

     fvv = [ -200 v1^2 ]
           [     0     ]

The solution of this minimization problem is

     x* = [ 1 ; 1 ]
     Phi(x*) = 0

The program output is shown below:

     === Solving system without acceleration ===
     NITER         = 53
     NFEV          = 56
     NJEV          = 54
     NAEV          = 0
     initial cost  = 2.250225000000e+04
     final cost    = 6.674986031430e-18
     final x       = (9.999999974165e-01, 9.999999948328e-01)
     final cond(J) = 6.000096055094e+02
     === Solving system with acceleration ===
     NITER         = 15
     NFEV          = 17
     NJEV          = 16
     NAEV          = 16
     initial cost  = 2.250225000000e+04
     final cost    = 7.518932873279e-19
     final x       = (9.999999991329e-01, 9.999999982657e-01)
     final cond(J) = 6.000097233278e+02

[image src="nlfit2.png"]

Figure: Paths taken by solver for Rosenbrock function

We can see that enabling geodesic acceleration requires less than a
third of the number of Jacobian evaluations in order to locate the
minimum.  The path taken by both methods is shown in Fig.  %s.  The
contours show the cost function \Phi(x_1,x_2).  We see that both methods
quickly find the canyon bottom, but the geodesic acceleration method
navigates along the bottom to the solution with significantly fewer
iterations.

The program is given below.

     #include <stdlib.h>
     #include <stdio.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_blas.h>
     #include <gsl/gsl_multifit_nlinear.h>

     int
     func_f (const gsl_vector * x, void *params, gsl_vector * f)
     {
       double x1 = gsl_vector_get(x, 0);
       double x2 = gsl_vector_get(x, 1);

       gsl_vector_set(f, 0, 100.0 * (x2 - x1*x1));
       gsl_vector_set(f, 1, 1.0 - x1);

       return GSL_SUCCESS;
     }

     int
     func_df (const gsl_vector * x, void *params, gsl_matrix * J)
     {
       double x1 = gsl_vector_get(x, 0);

       gsl_matrix_set(J, 0, 0, -200.0*x1);
       gsl_matrix_set(J, 0, 1, 100.0);
       gsl_matrix_set(J, 1, 0, -1.0);
       gsl_matrix_set(J, 1, 1, 0.0);

       return GSL_SUCCESS;
     }

     int
     func_fvv (const gsl_vector * x, const gsl_vector * v,
               void *params, gsl_vector * fvv)
     {
       double v1 = gsl_vector_get(v, 0);

       gsl_vector_set(fvv, 0, -200.0 * v1 * v1);
       gsl_vector_set(fvv, 1, 0.0);

       return GSL_SUCCESS;
     }

     void
     callback(const size_t iter, void *params,
              const gsl_multifit_nlinear_workspace *w)
     {
       gsl_vector * x = gsl_multifit_nlinear_position(w);

       /* print out current location */
       printf("%f %f\n",
              gsl_vector_get(x, 0),
              gsl_vector_get(x, 1));
     }

     void
     solve_system(gsl_vector *x0, gsl_multifit_nlinear_fdf *fdf,
                  gsl_multifit_nlinear_parameters *params)
     {
       const gsl_multifit_nlinear_type *T = gsl_multifit_nlinear_trust;
       const size_t max_iter = 200;
       const double xtol = 1.0e-8;
       const double gtol = 1.0e-8;
       const double ftol = 1.0e-8;
       const size_t n = fdf->n;
       const size_t p = fdf->p;
       gsl_multifit_nlinear_workspace *work =
         gsl_multifit_nlinear_alloc(T, params, n, p);
       gsl_vector * f = gsl_multifit_nlinear_residual(work);
       gsl_vector * x = gsl_multifit_nlinear_position(work);
       int info;
       double chisq0, chisq, rcond;

       /* initialize solver */
       gsl_multifit_nlinear_init(x0, fdf, work);

       /* store initial cost */
       gsl_blas_ddot(f, f, &chisq0);

       /* iterate until convergence */
       gsl_multifit_nlinear_driver(max_iter, xtol, gtol, ftol,
                                   callback, NULL, &info, work);

       /* store final cost */
       gsl_blas_ddot(f, f, &chisq);

       /* store cond(J(x)) */
       gsl_multifit_nlinear_rcond(&rcond, work);

       /* print summary */

       fprintf(stderr, "NITER         = %zu\n", gsl_multifit_nlinear_niter(work));
       fprintf(stderr, "NFEV          = %zu\n", fdf->nevalf);
       fprintf(stderr, "NJEV          = %zu\n", fdf->nevaldf);
       fprintf(stderr, "NAEV          = %zu\n", fdf->nevalfvv);
       fprintf(stderr, "initial cost  = %.12e\n", chisq0);
       fprintf(stderr, "final cost    = %.12e\n", chisq);
       fprintf(stderr, "final x       = (%.12e, %.12e)\n",
               gsl_vector_get(x, 0), gsl_vector_get(x, 1));
       fprintf(stderr, "final cond(J) = %.12e\n", 1.0 / rcond);

       printf("\n\n");

       gsl_multifit_nlinear_free(work);
     }

     int
     main (void)
     {
       const size_t n = 2;
       const size_t p = 2;
       gsl_vector *f = gsl_vector_alloc(n);
       gsl_vector *x = gsl_vector_alloc(p);
       gsl_multifit_nlinear_fdf fdf;
       gsl_multifit_nlinear_parameters fdf_params =
         gsl_multifit_nlinear_default_parameters();

       /* print map of Phi(x1, x2) */
       {
         double x1, x2, chisq;
         double *f1 = gsl_vector_ptr(f, 0);
         double *f2 = gsl_vector_ptr(f, 1);

         for (x1 = -1.2; x1 < 1.3; x1 += 0.1)
           {
             for (x2 = -0.5; x2 < 2.1; x2 += 0.1)
               {
                 gsl_vector_set(x, 0, x1);
                 gsl_vector_set(x, 1, x2);
                 func_f(x, NULL, f);

                 chisq = (*f1) * (*f1) + (*f2) * (*f2);
                 printf("%f %f %f\n", x1, x2, chisq);
               }
             printf("\n");
           }
         printf("\n\n");
       }

       /* define function to be minimized */
       fdf.f = func_f;
       fdf.df = func_df;
       fdf.fvv = func_fvv;
       fdf.n = n;
       fdf.p = p;
       fdf.params = NULL;

       /* starting point */
       gsl_vector_set(x, 0, -0.5);
       gsl_vector_set(x, 1, 1.75);

       fprintf(stderr, "=== Solving system without acceleration ===\n");
       fdf_params.trs = gsl_multifit_nlinear_trs_lm;
       solve_system(x, &fdf, &fdf_params);

       fprintf(stderr, "=== Solving system with acceleration ===\n");
       fdf_params.trs = gsl_multifit_nlinear_trs_lmaccel;
       solve_system(x, &fdf, &fdf_params);

       gsl_vector_free(f);
       gsl_vector_free(x);

       return 0;
     }


File: gsl-ref.info,  Node: Geodesic Acceleration Example 2,  Next: Comparing TRS Methods Example,  Prev: Geodesic Acceleration Example 1,  Up: Examples<32>

41.12.3 Geodesic Acceleration Example 2
---------------------------------------

The following example fits a set of data to a Gaussian model using the
Levenberg-Marquardt method with geodesic acceleration.  The cost
function is

     Phi(x) = 1/2 \sum_i f_i^2
     f_i = y_i - Y(a,b,c;t_i)

where y_i is the measured data point at time t_i, and the model is
specified by

     Y(a,b,c;t) = a exp(-1/2 ((t-b)/c)^2)

The parameters a,b,c represent the amplitude, mean, and width of the
Gaussian respectively.  The program below generates the y_i data on
[0,1] using the values a = 5, b = 0.4, c = 0.15 and adding random noise.
The i-th row of the Jacobian is

     J(i,:) = ( -e_i  -(a/c)*z_i*e_i  -(a/c)*z_i^2*e_i )

where

     z_i = (t_i - b) / c
     e_i = \exp(-1/2 z_i^2)

In order to use geodesic acceleration, we need the second directional
derivative of the residuals in the velocity direction, D_v^2 f_i =
\sum_{\alpha\beta} v_{\alpha} v_{\beta} \partial_{\alpha}
\partial_{\beta} f_i, where v is provided by the solver.  To compute
this, it is helpful to make a table of all second derivatives of the
residuals f_i with respect to each combination of model parameters.
This table is

The lower half of the table is omitted since it is symmetric.  Then, the
second directional derivative of f_i is

     D_v^2 f_i = v_a^2 (d/da)^2 f_i + 2 v_a v_b (d/da) (d/db) f_i + 2 v_a v_c (d/da) (d/dc) f_i + v_b^2 (d/db)^2 f_i + 2 v_b v_c (d/db) (d/dc) f_i + v_c^2 (d/dc)^2 f_i

The factors of 2 come from the symmetry of the mixed second partial
derivatives.  The iteration is started using the initial guess a = 1, b
= 0, c = 1.  The program output is shown below:

     iter  0: a = 1.0000, b = 0.0000, c = 1.0000, |a|/|v| = 0.0000 cond(J) =      inf, |f(x)| = 35.4785
     iter  1: a = 1.5708, b = 0.5321, c = 0.5219, |a|/|v| = 0.3093 cond(J) =  29.0443, |f(x)| = 31.1042
     iter  2: a = 1.7387, b = 0.4040, c = 0.4568, |a|/|v| = 0.1199 cond(J) =   3.5256, |f(x)| = 28.7217
     iter  3: a = 2.2340, b = 0.3829, c = 0.3053, |a|/|v| = 0.3308 cond(J) =   4.5121, |f(x)| = 23.8074
     iter  4: a = 3.2275, b = 0.3952, c = 0.2243, |a|/|v| = 0.2784 cond(J) =   8.6499, |f(x)| = 15.6003
     iter  5: a = 4.3347, b = 0.3974, c = 0.1752, |a|/|v| = 0.2029 cond(J) =  15.1732, |f(x)| = 7.5908
     iter  6: a = 4.9352, b = 0.3992, c = 0.1536, |a|/|v| = 0.1001 cond(J) =  26.6621, |f(x)| = 4.8402
     iter  7: a = 5.0716, b = 0.3994, c = 0.1498, |a|/|v| = 0.0166 cond(J) =  34.6922, |f(x)| = 4.7103
     iter  8: a = 5.0828, b = 0.3994, c = 0.1495, |a|/|v| = 0.0012 cond(J) =  36.5422, |f(x)| = 4.7095
     iter  9: a = 5.0831, b = 0.3994, c = 0.1495, |a|/|v| = 0.0000 cond(J) =  36.6929, |f(x)| = 4.7095
     iter 10: a = 5.0831, b = 0.3994, c = 0.1495, |a|/|v| = 0.0000 cond(J) =  36.6975, |f(x)| = 4.7095
     iter 11: a = 5.0831, b = 0.3994, c = 0.1495, |a|/|v| = 0.0000 cond(J) =  36.6976, |f(x)| = 4.7095
     NITER         = 11
     NFEV          = 18
     NJEV          = 12
     NAEV          = 17
     initial cost  = 1.258724737288e+03
     final cost    = 2.217977560180e+01
     final x       = (5.083101559156e+00, 3.994484109594e-01, 1.494898e-01)
     final cond(J) = 3.669757713403e+01

We see the method converges after 11 iterations.  For comparison the
standard Levenberg-Marquardt method requires 26 iterations and so the
Gaussian fitting problem benefits substantially from the geodesic
acceleration correction.  The column marked ‘|a|/|v|’ above shows the
ratio of the acceleration term to the velocity term as the iteration
progresses.  Larger values of this ratio indicate that the geodesic
acceleration correction term is contributing substantial information to
the solver relative to the standard LM velocity step.

The data and fitted model are shown in Fig.  %s.

[image src="nlfit2b.png"]

Figure: Gaussian model fitted to data

The program is given below.

     #include <stdlib.h>
     #include <stdio.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_blas.h>
     #include <gsl/gsl_multifit_nlinear.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>

     struct data
     {
       double *t;
       double *y;
       size_t n;
     };

     /* model function: a * exp( -1/2 * [ (t - b) / c ]^2 ) */
     double
     gaussian(const double a, const double b, const double c, const double t)
     {
       const double z = (t - b) / c;
       return (a * exp(-0.5 * z * z));
     }

     int
     func_f (const gsl_vector * x, void *params, gsl_vector * f)
     {
       struct data *d = (struct data *) params;
       double a = gsl_vector_get(x, 0);
       double b = gsl_vector_get(x, 1);
       double c = gsl_vector_get(x, 2);
       size_t i;

       for (i = 0; i < d->n; ++i)
         {
           double ti = d->t[i];
           double yi = d->y[i];
           double y = gaussian(a, b, c, ti);

           gsl_vector_set(f, i, yi - y);
         }

       return GSL_SUCCESS;
     }

     int
     func_df (const gsl_vector * x, void *params, gsl_matrix * J)
     {
       struct data *d = (struct data *) params;
       double a = gsl_vector_get(x, 0);
       double b = gsl_vector_get(x, 1);
       double c = gsl_vector_get(x, 2);
       size_t i;

       for (i = 0; i < d->n; ++i)
         {
           double ti = d->t[i];
           double zi = (ti - b) / c;
           double ei = exp(-0.5 * zi * zi);

           gsl_matrix_set(J, i, 0, -ei);
           gsl_matrix_set(J, i, 1, -(a / c) * ei * zi);
           gsl_matrix_set(J, i, 2, -(a / c) * ei * zi * zi);
         }

       return GSL_SUCCESS;
     }

     int
     func_fvv (const gsl_vector * x, const gsl_vector * v,
               void *params, gsl_vector * fvv)
     {
       struct data *d = (struct data *) params;
       double a = gsl_vector_get(x, 0);
       double b = gsl_vector_get(x, 1);
       double c = gsl_vector_get(x, 2);
       double va = gsl_vector_get(v, 0);
       double vb = gsl_vector_get(v, 1);
       double vc = gsl_vector_get(v, 2);
       size_t i;

       for (i = 0; i < d->n; ++i)
         {
           double ti = d->t[i];
           double zi = (ti - b) / c;
           double ei = exp(-0.5 * zi * zi);
           double Dab = -zi * ei / c;
           double Dac = -zi * zi * ei / c;
           double Dbb = a * ei / (c * c) * (1.0 - zi*zi);
           double Dbc = a * zi * ei / (c * c) * (2.0 - zi*zi);
           double Dcc = a * zi * zi * ei / (c * c) * (3.0 - zi*zi);
           double sum;

           sum = 2.0 * va * vb * Dab +
                 2.0 * va * vc * Dac +
                       vb * vb * Dbb +
                 2.0 * vb * vc * Dbc +
                       vc * vc * Dcc;

           gsl_vector_set(fvv, i, sum);
         }

       return GSL_SUCCESS;
     }

     void
     callback(const size_t iter, void *params,
              const gsl_multifit_nlinear_workspace *w)
     {
       gsl_vector *f = gsl_multifit_nlinear_residual(w);
       gsl_vector *x = gsl_multifit_nlinear_position(w);
       double avratio = gsl_multifit_nlinear_avratio(w);
       double rcond;

       (void) params; /* not used */

       /* compute reciprocal condition number of J(x) */
       gsl_multifit_nlinear_rcond(&rcond, w);

       fprintf(stderr, "iter %2zu: a = %.4f, b = %.4f, c = %.4f, |a|/|v| = %.4f cond(J) = %8.4f, |f(x)| = %.4f\n",
               iter,
               gsl_vector_get(x, 0),
               gsl_vector_get(x, 1),
               gsl_vector_get(x, 2),
               avratio,
               1.0 / rcond,
               gsl_blas_dnrm2(f));
     }

     void
     solve_system(gsl_vector *x, gsl_multifit_nlinear_fdf *fdf,
                  gsl_multifit_nlinear_parameters *params)
     {
       const gsl_multifit_nlinear_type *T = gsl_multifit_nlinear_trust;
       const size_t max_iter = 200;
       const double xtol = 1.0e-8;
       const double gtol = 1.0e-8;
       const double ftol = 1.0e-8;
       const size_t n = fdf->n;
       const size_t p = fdf->p;
       gsl_multifit_nlinear_workspace *work =
         gsl_multifit_nlinear_alloc(T, params, n, p);
       gsl_vector * f = gsl_multifit_nlinear_residual(work);
       gsl_vector * y = gsl_multifit_nlinear_position(work);
       int info;
       double chisq0, chisq, rcond;

       /* initialize solver */
       gsl_multifit_nlinear_init(x, fdf, work);

       /* store initial cost */
       gsl_blas_ddot(f, f, &chisq0);

       /* iterate until convergence */
       gsl_multifit_nlinear_driver(max_iter, xtol, gtol, ftol,
                                   callback, NULL, &info, work);

       /* store final cost */
       gsl_blas_ddot(f, f, &chisq);

       /* store cond(J(x)) */
       gsl_multifit_nlinear_rcond(&rcond, work);

       gsl_vector_memcpy(x, y);

       /* print summary */

       fprintf(stderr, "NITER         = %zu\n", gsl_multifit_nlinear_niter(work));
       fprintf(stderr, "NFEV          = %zu\n", fdf->nevalf);
       fprintf(stderr, "NJEV          = %zu\n", fdf->nevaldf);
       fprintf(stderr, "NAEV          = %zu\n", fdf->nevalfvv);
       fprintf(stderr, "initial cost  = %.12e\n", chisq0);
       fprintf(stderr, "final cost    = %.12e\n", chisq);
       fprintf(stderr, "final x       = (%.12e, %.12e, %12e)\n",
               gsl_vector_get(x, 0), gsl_vector_get(x, 1), gsl_vector_get(x, 2));
       fprintf(stderr, "final cond(J) = %.12e\n", 1.0 / rcond);

       gsl_multifit_nlinear_free(work);
     }

     int
     main (void)
     {
       const size_t n = 300;  /* number of data points to fit */
       const size_t p = 3;    /* number of model parameters */
       const double a = 5.0;  /* amplitude */
       const double b = 0.4;  /* center */
       const double c = 0.15; /* width */
       const gsl_rng_type * T = gsl_rng_default;
       gsl_vector *f = gsl_vector_alloc(n);
       gsl_vector *x = gsl_vector_alloc(p);
       gsl_multifit_nlinear_fdf fdf;
       gsl_multifit_nlinear_parameters fdf_params =
         gsl_multifit_nlinear_default_parameters();
       struct data fit_data;
       gsl_rng * r;
       size_t i;

       gsl_rng_env_setup ();
       r = gsl_rng_alloc (T);

       fit_data.t = malloc(n * sizeof(double));
       fit_data.y = malloc(n * sizeof(double));
       fit_data.n = n;

       /* generate synthetic data with noise */
       for (i = 0; i < n; ++i)
         {
           double t = (double)i / (double) n;
           double y0 = gaussian(a, b, c, t);
           double dy = gsl_ran_gaussian (r, 0.1 * y0);

           fit_data.t[i] = t;
           fit_data.y[i] = y0 + dy;
         }

       /* define function to be minimized */
       fdf.f = func_f;
       fdf.df = func_df;
       fdf.fvv = func_fvv;
       fdf.n = n;
       fdf.p = p;
       fdf.params = &fit_data;

       /* starting point */
       gsl_vector_set(x, 0, 1.0);
       gsl_vector_set(x, 1, 0.0);
       gsl_vector_set(x, 2, 1.0);

       fdf_params.trs = gsl_multifit_nlinear_trs_lmaccel;
       solve_system(x, &fdf, &fdf_params);

       /* print data and model */
       {
         double A = gsl_vector_get(x, 0);
         double B = gsl_vector_get(x, 1);
         double C = gsl_vector_get(x, 2);

         for (i = 0; i < n; ++i)
           {
             double ti = fit_data.t[i];
             double yi = fit_data.y[i];
             double fi = gaussian(A, B, C, ti);

             printf("%f %f %f\n", ti, yi, fi);
           }
       }

       gsl_vector_free(f);
       gsl_vector_free(x);
       gsl_rng_free(r);

       return 0;
     }


File: gsl-ref.info,  Node: Comparing TRS Methods Example,  Next: Large Nonlinear Least Squares Example,  Prev: Geodesic Acceleration Example 2,  Up: Examples<32>

41.12.4 Comparing TRS Methods Example
-------------------------------------

The following program compares all available nonlinear least squares
trust-region subproblem (TRS) methods on the Branin function, a common
optimization test problem.  The cost function is

     \Phi(x) &= 1/2 (f_1^2 + f_2^2)
     f_1 &= x_2 + a_1 x_1^2 + a_2 x_1 + a_3
     f_2 &= sqrt(a_4) sqrt(1 + (1 - a_5) cos(x_1))

with a_1 = -{5.1 \over 4 \pi^2}, a_2 = {5 \over \pi}, a_3 = -6, a_4 =
10, a_5 = {1 \over 8\pi}.  There are three minima of this function in
the range (x_1,x_2) \in [-5,15] \times [-5,15].  The program below uses
the starting point (x_1,x_2) = (6,14.5) and calculates the solution with
all available nonlinear least squares TRS methods.  The program output
is shown below:

     Method                    NITER  NFEV  NJEV  Initial Cost  Final cost   Final cond(J) Final x
     levenberg-marquardt       20     27    21    1.9874e+02    3.9789e-01   6.1399e+07    (-3.14e+00, 1.23e+01)
     levenberg-marquardt+accel 27     36    28    1.9874e+02    3.9789e-01   1.4465e+07    (3.14e+00, 2.27e+00)
     dogleg                    23     64    23    1.9874e+02    3.9789e-01   5.0692e+08    (3.14e+00, 2.28e+00)
     double-dogleg             24     69    24    1.9874e+02    3.9789e-01   3.4879e+07    (3.14e+00, 2.27e+00)
     2D-subspace               23     54    24    1.9874e+02    3.9789e-01   2.5142e+07    (3.14e+00, 2.27e+00)

The first row of output above corresponds to standard
Levenberg-Marquardt, while the second row includes geodesic
acceleration.  We see that the standard LM method converges to the
minimum at (-\pi,12.275) and also uses the least number of iterations
and Jacobian evaluations.  All other methods converge to the minimum
(\pi,2.275) and perform similarly in terms of number of Jacobian
evaluations.  We see that J is fairly ill-conditioned at both minima,
indicating that the QR (or SVD) solver is the best choice for this
problem.  Since there are only two parameters in this optimization
problem, we can easily visualize the paths taken by each method, which
are shown in Fig.  %s.  The figure shows contours of the cost function
\Phi(x_1,x_2) which exhibits three global minima in the range [-5,15]
\times [-5,15].  The paths taken by each solver are shown as colored
lines.

[image src="nlfit3.png"]

Figure: Paths taken for different TRS methods for the Branin function

The program is given below.

     #include <stdlib.h>
     #include <stdio.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_blas.h>
     #include <gsl/gsl_multifit_nlinear.h>

     /* parameters to model */
     struct model_params
     {
       double a1;
       double a2;
       double a3;
       double a4;
       double a5;
     };

     /* Branin function */
     int
     func_f (const gsl_vector * x, void *params, gsl_vector * f)
     {
       struct model_params *par = (struct model_params *) params;
       double x1 = gsl_vector_get(x, 0);
       double x2 = gsl_vector_get(x, 1);
       double f1 = x2 + par->a1 * x1 * x1 + par->a2 * x1 + par->a3;
       double f2 = sqrt(par->a4) * sqrt(1.0 + (1.0 - par->a5) * cos(x1));

       gsl_vector_set(f, 0, f1);
       gsl_vector_set(f, 1, f2);

       return GSL_SUCCESS;
     }

     int
     func_df (const gsl_vector * x, void *params, gsl_matrix * J)
     {
       struct model_params *par = (struct model_params *) params;
       double x1 = gsl_vector_get(x, 0);
       double f2 = sqrt(par->a4) * sqrt(1.0 + (1.0 - par->a5) * cos(x1));

       gsl_matrix_set(J, 0, 0, 2.0 * par->a1 * x1 + par->a2);
       gsl_matrix_set(J, 0, 1, 1.0);

       gsl_matrix_set(J, 1, 0, -0.5 * par->a4 / f2 * (1.0 - par->a5) * sin(x1));
       gsl_matrix_set(J, 1, 1, 0.0);

       return GSL_SUCCESS;
     }

     int
     func_fvv (const gsl_vector * x, const gsl_vector * v,
               void *params, gsl_vector * fvv)
     {
       struct model_params *par = (struct model_params *) params;
       double x1 = gsl_vector_get(x, 0);
       double v1 = gsl_vector_get(v, 0);
       double c = cos(x1);
       double s = sin(x1);
       double f2 = sqrt(par->a4) * sqrt(1.0 + (1.0 - par->a5) * c);
       double t = 0.5 * par->a4 * (1.0 - par->a5) / f2;

       gsl_vector_set(fvv, 0, 2.0 * par->a1 * v1 * v1);
       gsl_vector_set(fvv, 1, -t * (c + s*s/f2) * v1 * v1);

       return GSL_SUCCESS;
     }

     void
     callback(const size_t iter, void *params,
              const gsl_multifit_nlinear_workspace *w)
     {
       gsl_vector * x = gsl_multifit_nlinear_position(w);
       double x1 = gsl_vector_get(x, 0);
       double x2 = gsl_vector_get(x, 1);

       /* print out current location */
       printf("%f %f\n", x1, x2);
     }

     void
     solve_system(gsl_vector *x0, gsl_multifit_nlinear_fdf *fdf,
                  gsl_multifit_nlinear_parameters *params)
     {
       const gsl_multifit_nlinear_type *T = gsl_multifit_nlinear_trust;
       const size_t max_iter = 200;
       const double xtol = 1.0e-8;
       const double gtol = 1.0e-8;
       const double ftol = 1.0e-8;
       const size_t n = fdf->n;
       const size_t p = fdf->p;
       gsl_multifit_nlinear_workspace *work =
         gsl_multifit_nlinear_alloc(T, params, n, p);
       gsl_vector * f = gsl_multifit_nlinear_residual(work);
       gsl_vector * x = gsl_multifit_nlinear_position(work);
       int info;
       double chisq0, chisq, rcond;

       printf("# %s/%s\n",
              gsl_multifit_nlinear_name(work),
              gsl_multifit_nlinear_trs_name(work));

       /* initialize solver */
       gsl_multifit_nlinear_init(x0, fdf, work);

       /* store initial cost */
       gsl_blas_ddot(f, f, &chisq0);

       /* iterate until convergence */
       gsl_multifit_nlinear_driver(max_iter, xtol, gtol, ftol,
                                   callback, NULL, &info, work);

       /* store final cost */
       gsl_blas_ddot(f, f, &chisq);

       /* store cond(J(x)) */
       gsl_multifit_nlinear_rcond(&rcond, work);

       /* print summary */
       fprintf(stderr, "%-25s %-6zu %-5zu %-5zu %-13.4e %-12.4e %-13.4e (%.2e, %.2e)\n",
               gsl_multifit_nlinear_trs_name(work),
               gsl_multifit_nlinear_niter(work),
               fdf->nevalf,
               fdf->nevaldf,
               chisq0,
               chisq,
               1.0 / rcond,
               gsl_vector_get(x, 0),
               gsl_vector_get(x, 1));

       printf("\n\n");

       gsl_multifit_nlinear_free(work);
     }

     int
     main (void)
     {
       const size_t n = 2;
       const size_t p = 2;
       gsl_vector *f = gsl_vector_alloc(n);
       gsl_vector *x = gsl_vector_alloc(p);
       gsl_multifit_nlinear_fdf fdf;
       gsl_multifit_nlinear_parameters fdf_params =
         gsl_multifit_nlinear_default_parameters();
       struct model_params params;

       params.a1 = -5.1 / (4.0 * M_PI * M_PI);
       params.a2 = 5.0 / M_PI;
       params.a3 = -6.0;
       params.a4 = 10.0;
       params.a5 = 1.0 / (8.0 * M_PI);

       /* print map of Phi(x1, x2) */
       {
         double x1, x2, chisq;

         for (x1 = -5.0; x1 < 15.0; x1 += 0.1)
           {
             for (x2 = -5.0; x2 < 15.0; x2 += 0.1)
               {
                 gsl_vector_set(x, 0, x1);
                 gsl_vector_set(x, 1, x2);
                 func_f(x, &params, f);

                 gsl_blas_ddot(f, f, &chisq);

                 printf("%f %f %f\n", x1, x2, chisq);
               }
             printf("\n");
           }
         printf("\n\n");
       }

       /* define function to be minimized */
       fdf.f = func_f;
       fdf.df = func_df;
       fdf.fvv = func_fvv;
       fdf.n = n;
       fdf.p = p;
       fdf.params = &params;

       /* starting point */
       gsl_vector_set(x, 0, 6.0);
       gsl_vector_set(x, 1, 14.5);

       fprintf(stderr, "%-25s %-6s %-5s %-5s %-13s %-12s %-13s %-15s\n",
               "Method", "NITER", "NFEV", "NJEV", "Initial Cost",
               "Final cost", "Final cond(J)", "Final x");

       fdf_params.trs = gsl_multifit_nlinear_trs_lm;
       solve_system(x, &fdf, &fdf_params);

       fdf_params.trs = gsl_multifit_nlinear_trs_lmaccel;
       solve_system(x, &fdf, &fdf_params);

       fdf_params.trs = gsl_multifit_nlinear_trs_dogleg;
       solve_system(x, &fdf, &fdf_params);

       fdf_params.trs = gsl_multifit_nlinear_trs_ddogleg;
       solve_system(x, &fdf, &fdf_params);

       fdf_params.trs = gsl_multifit_nlinear_trs_subspace2D;
       solve_system(x, &fdf, &fdf_params);

       gsl_vector_free(f);
       gsl_vector_free(x);

       return 0;
     }


File: gsl-ref.info,  Node: Large Nonlinear Least Squares Example,  Prev: Comparing TRS Methods Example,  Up: Examples<32>

41.12.5 Large Nonlinear Least Squares Example
---------------------------------------------

The following program illustrates the large nonlinear least squares
solvers on a system with significant sparse structure in the Jacobian.
The cost function is

     \Phi(x) &= 1/2 \sum_{i=1}^{p+1} f_i^2
     f_i &= \sqrt{\alpha} (x_i - 1), 1 \le i \le p
     f_{p+1} &= ||x||^2 - 1/4

with \alpha = 10^{-5}.  The residual f_{p+1} imposes a constraint on the
p parameters x, to ensure that ||x||^2 \approx {1 \over 4}.  The
(p+1)-by-p Jacobian for this system is

     J(x) = [ \sqrt{alpha} I_p; 2 x^T ]

and the normal equations matrix is

     J^T J = \alpha I_p + 4 x x^T

Finally, the second directional derivative of f for the geodesic
acceleration method is

     fvv = [     0     ]
           [ 2 ||v||^2 ]

Since the upper p-by-p block of J is diagonal, this sparse structure
should be exploited in the nonlinear solver.  For comparison, the
following program solves the system for p = 2000 using the dense direct
Cholesky solver based on the normal equations matrix J^T J, as well as
the iterative Steihaug-Toint solver, based on sparse matrix-vector
products J u and J^T u.  The program output is shown below:

     Method                    NITER NFEV NJUEV NJTJEV NAEV Init Cost  Final cost cond(J) Final |x|^2 Time (s)
     levenberg-marquardt       25    31   26    26     0    7.1218e+18 1.9555e-02 447.50  2.5044e-01  46.28
     levenberg-marquardt+accel 22    23   45    23     22   7.1218e+18 1.9555e-02 447.64  2.5044e-01  33.92
     dogleg                    37    87   36    36     0    7.1218e+18 1.9555e-02 447.59  2.5044e-01  56.05
     double-dogleg             35    88   34    34     0    7.1218e+18 1.9555e-02 447.62  2.5044e-01  52.65
     2D-subspace               37    88   36    36     0    7.1218e+18 1.9555e-02 447.71  2.5044e-01  59.75
     steihaug-toint            35    88   345   0      0    7.1218e+18 1.9555e-02 inf     2.5044e-01  0.09

The first five rows use methods based on factoring the dense J^T J
matrix while the last row uses the iterative Steihaug-Toint method.
While the number of Jacobian matrix-vector products (NJUEV) is less for
the dense methods, the added time to construct and factor the J^T J
matrix (NJTJEV) results in a much larger runtime than the iterative
method (see last column).

The program is given below.

     #include <stdlib.h>
     #include <stdio.h>
     #include <sys/time.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_matrix.h>
     #include <gsl/gsl_blas.h>
     #include <gsl/gsl_multilarge_nlinear.h>
     #include <gsl/gsl_spblas.h>
     #include <gsl/gsl_spmatrix.h>

     /* parameters for functions */
     struct model_params
     {
       double alpha;
       gsl_spmatrix *J;
     };

     /* penalty function */
     int
     penalty_f (const gsl_vector * x, void *params, gsl_vector * f)
     {
       struct model_params *par = (struct model_params *) params;
       const double sqrt_alpha = sqrt(par->alpha);
       const size_t p = x->size;
       size_t i;
       double sum = 0.0;

       for (i = 0; i < p; ++i)
         {
           double xi = gsl_vector_get(x, i);

           gsl_vector_set(f, i, sqrt_alpha*(xi - 1.0));

           sum += xi * xi;
         }

       gsl_vector_set(f, p, sum - 0.25);

       return GSL_SUCCESS;
     }

     int
     penalty_df (CBLAS_TRANSPOSE_t TransJ, const gsl_vector * x,
                 const gsl_vector * u, void * params, gsl_vector * v,
                 gsl_matrix * JTJ)
     {
       struct model_params *par = (struct model_params *) params;
       const size_t p = x->size;
       size_t j;

       /* store 2*x in last row of J */
       for (j = 0; j < p; ++j)
         {
           double xj = gsl_vector_get(x, j);
           gsl_spmatrix_set(par->J, p, j, 2.0 * xj);
         }

       /* compute v = op(J) u */
       if (v)
         gsl_spblas_dgemv(TransJ, 1.0, par->J, u, 0.0, v);

       if (JTJ)
         {
           gsl_vector_view diag = gsl_matrix_diagonal(JTJ);

           /* compute J^T J = [ alpha*I_p + 4 x x^T ] */
           gsl_matrix_set_zero(JTJ);

           /* store 4 x x^T in lower half of JTJ */
           gsl_blas_dsyr(CblasLower, 4.0, x, JTJ);

           /* add alpha to diag(JTJ) */
           gsl_vector_add_constant(&diag.vector, par->alpha);
         }

       return GSL_SUCCESS;
     }

     int
     penalty_fvv (const gsl_vector * x, const gsl_vector * v,
                  void *params, gsl_vector * fvv)
     {
       const size_t p = x->size;
       double normv = gsl_blas_dnrm2(v);

       gsl_vector_set_zero(fvv);
       gsl_vector_set(fvv, p, 2.0 * normv * normv);

       (void)params; /* avoid unused parameter warning */

       return GSL_SUCCESS;
     }

     void
     solve_system(const gsl_vector *x0, gsl_multilarge_nlinear_fdf *fdf,
                  gsl_multilarge_nlinear_parameters *params)
     {
       const gsl_multilarge_nlinear_type *T = gsl_multilarge_nlinear_trust;
       const size_t max_iter = 200;
       const double xtol = 1.0e-8;
       const double gtol = 1.0e-8;
       const double ftol = 1.0e-8;
       const size_t n = fdf->n;
       const size_t p = fdf->p;
       gsl_multilarge_nlinear_workspace *work =
         gsl_multilarge_nlinear_alloc(T, params, n, p);
       gsl_vector * f = gsl_multilarge_nlinear_residual(work);
       gsl_vector * x = gsl_multilarge_nlinear_position(work);
       int info;
       double chisq0, chisq, rcond, xsq;
       struct timeval tv0, tv1;

       gettimeofday(&tv0, NULL);

       /* initialize solver */
       gsl_multilarge_nlinear_init(x0, fdf, work);

       /* store initial cost */
       gsl_blas_ddot(f, f, &chisq0);

       /* iterate until convergence */
       gsl_multilarge_nlinear_driver(max_iter, xtol, gtol, ftol,
                                     NULL, NULL, &info, work);

       gettimeofday(&tv1, NULL);

       /* store final cost */
       gsl_blas_ddot(f, f, &chisq);

       /* compute final ||x||^2 */
       gsl_blas_ddot(x, x, &xsq);

       /* store cond(J(x)) */
       gsl_multilarge_nlinear_rcond(&rcond, work);

       /* print summary */
       fprintf(stderr, "%-25s %-5zu %-4zu %-5zu %-6zu %-4zu %-10.4e %-10.4e %-7.2f %-11.4e %.2f\n",
               gsl_multilarge_nlinear_trs_name(work),
               gsl_multilarge_nlinear_niter(work),
               fdf->nevalf,
               fdf->nevaldfu,
               fdf->nevaldf2,
               fdf->nevalfvv,
               chisq0,
               chisq,
               1.0 / rcond,
               xsq,
               (tv1.tv_sec - tv0.tv_sec) + 1.0e-6 * (tv1.tv_usec - tv0.tv_usec));

       gsl_multilarge_nlinear_free(work);
     }

     int
     main (void)
     {
       const size_t p = 2000;
       const size_t n = p + 1;
       gsl_vector *f = gsl_vector_alloc(n);
       gsl_vector *x = gsl_vector_alloc(p);

       /* allocate sparse Jacobian matrix with 2*p non-zero elements in triplet format */
       gsl_spmatrix *J = gsl_spmatrix_alloc_nzmax(n, p, 2 * p, GSL_SPMATRIX_TRIPLET);

       gsl_multilarge_nlinear_fdf fdf;
       gsl_multilarge_nlinear_parameters fdf_params =
         gsl_multilarge_nlinear_default_parameters();
       struct model_params params;
       size_t i;

       params.alpha = 1.0e-5;
       params.J = J;

       /* define function to be minimized */
       fdf.f = penalty_f;
       fdf.df = penalty_df;
       fdf.fvv = penalty_fvv;
       fdf.n = n;
       fdf.p = p;
       fdf.params = &params;

       for (i = 0; i < p; ++i)
         {
           /* starting point */
           gsl_vector_set(x, i, i + 1.0);

           /* store sqrt(alpha)*I_p in upper p-by-p block of J */
           gsl_spmatrix_set(J, i, i, sqrt(params.alpha));
         }

       fprintf(stderr, "%-25s %-4s %-4s %-5s %-6s %-4s %-10s %-10s %-7s %-11s %-10s\n",
               "Method", "NITER", "NFEV", "NJUEV", "NJTJEV", "NAEV", "Init Cost",
               "Final cost", "cond(J)", "Final |x|^2", "Time (s)");

       fdf_params.scale = gsl_multilarge_nlinear_scale_levenberg;

       fdf_params.trs = gsl_multilarge_nlinear_trs_lm;
       solve_system(x, &fdf, &fdf_params);

       fdf_params.trs = gsl_multilarge_nlinear_trs_lmaccel;
       solve_system(x, &fdf, &fdf_params);

       fdf_params.trs = gsl_multilarge_nlinear_trs_dogleg;
       solve_system(x, &fdf, &fdf_params);

       fdf_params.trs = gsl_multilarge_nlinear_trs_ddogleg;
       solve_system(x, &fdf, &fdf_params);

       fdf_params.trs = gsl_multilarge_nlinear_trs_subspace2D;
       solve_system(x, &fdf, &fdf_params);

       fdf_params.trs = gsl_multilarge_nlinear_trs_cgst;
       solve_system(x, &fdf, &fdf_params);

       gsl_vector_free(f);
       gsl_vector_free(x);
       gsl_spmatrix_free(J);

       return 0;
     }


File: gsl-ref.info,  Node: References and Further Reading<34>,  Prev: Examples<32>,  Up: Nonlinear Least-Squares Fitting

41.13 References and Further Reading
====================================

The following publications are relevant to the algorithms described in
this section,

   * J.J. Moré, `The Levenberg-Marquardt Algorithm: Implementation and
     Theory', Lecture Notes in Mathematics, v630 (1978), ed G. Watson.

   * H. B. Nielsen, “Damping Parameter in Marquardt’s Method”, IMM
     Department of Mathematical Modeling, DTU, Tech.  Report
     IMM-REP-1999-05 (1999).

   * K. Madsen and H. B. Nielsen, “Introduction to Optimization and Data
     Fitting”, IMM Department of Mathematical Modeling, DTU, 2010.

   * J. E. Dennis and R. B. Schnabel, Numerical Methods for
     Unconstrained Optimization and Nonlinear Equations, SIAM, 1996.

   * M. K. Transtrum, B. B. Machta, and J. P. Sethna, Geometry of
     nonlinear least squares with applications to sloppy models and
     optimization, Phys.  Rev.  E 83, 036701, 2011.

   * M. K. Transtrum and J. P. Sethna, Improvements to the
     Levenberg-Marquardt algorithm for nonlinear least-squares
     minimization, arXiv:1201.5885, 2012.

   * J.J. Moré, B.S. Garbow, K.E. Hillstrom, “Testing Unconstrained
     Optimization Software”, ACM Transactions on Mathematical Software,
     Vol 7, No 1 (1981), p 17–41.

   * H. B. Nielsen, “UCTP Test Problems for Unconstrained Optimization”,
     IMM Department of Mathematical Modeling, DTU, Tech.  Report
     IMM-REP-2000-17 (2000).


File: gsl-ref.info,  Node: Basis Splines,  Next: Sparse Matrices,  Prev: Nonlinear Least-Squares Fitting,  Up: Top

42 Basis Splines
****************

This chapter describes functions for the computation of smoothing basis
splines (B-splines).  A smoothing spline differs from an interpolating
spline in that the resulting curve is not required to pass through each
datapoint.  For information about interpolating splines, see *note
Interpolation: 98c.

The header file ‘gsl_bspline.h’ contains the prototypes for the bspline
functions and related declarations.

* Menu:

* Overview: Overview<7>.
* Initializing the B-splines solver::
* Constructing the knots vector::
* Evaluation of B-splines::
* Evaluation of B-spline derivatives::
* Working with the Greville abscissae::
* Examples: Examples<33>.
* References and Further Reading: References and Further Reading<35>.


File: gsl-ref.info,  Node: Overview<7>,  Next: Initializing the B-splines solver,  Up: Basis Splines

42.1 Overview
=============

B-splines are commonly used as basis functions to fit smoothing curves
to large data sets.  To do this, the abscissa axis is broken up into
some number of intervals, where the endpoints of each interval are
called `breakpoints'.  These breakpoints are then converted to `knots'
by imposing various continuity and smoothness conditions at each
interface.  Given a nondecreasing knot vector

     t = \{t_0, t_1, \dots, t_{n+k-1}\}

the n basis splines of order k are defined by

     B_(i,1)(x) = (1, t_i <= x < t_(i+1)
                  (0, else
     B_(i,k)(x) = [(x - t_i)/(t_(i+k-1) - t_i)] B_(i,k-1)(x)
                   + [(t_(i+k) - x)/(t_(i+k) - t_(i+1))] B_(i+1,k-1)(x)

for i = 0, \ldots, n-1.  The common case of cubic B-splines is given by
k = 4.  The above recurrence relation can be evaluated in a numerically
stable way by the de Boor algorithm.

If we define appropriate knots on an interval [a,b] then the B-spline
basis functions form a complete set on that interval.  Therefore we can
expand a smoothing function as

     f(x) = \sum_{i=0}^{n-1} c_i B_{i,k}(x)

given enough (x_j, f(x_j)) data pairs.  The coefficients c_i can be
readily obtained from a least-squares fit.


File: gsl-ref.info,  Node: Initializing the B-splines solver,  Next: Constructing the knots vector,  Prev: Overview<7>,  Up: Basis Splines

42.2 Initializing the B-splines solver
======================================

 -- Type: gsl_bspline_workspace

     The computation of B-spline functions requires a preallocated
     workspace.

 -- Function: gsl_bspline_workspace * gsl_bspline_alloc (const size_t k,
          const size_t nbreak)

     This function allocates a workspace for computing B-splines of
     order ‘k’.  The number of breakpoints is given by ‘nbreak’.  This
     leads to n = nbreak + k - 2 basis functions.  Cubic B-splines are
     specified by k = 4.  The size of the workspace is O(2k^2 + 5k +
     nbreak).

 -- Function: void gsl_bspline_free (gsl_bspline_workspace * w)

     This function frees the memory associated with the workspace ‘w’.


File: gsl-ref.info,  Node: Constructing the knots vector,  Next: Evaluation of B-splines,  Prev: Initializing the B-splines solver,  Up: Basis Splines

42.3 Constructing the knots vector
==================================

 -- Function: int gsl_bspline_knots (const gsl_vector * breakpts,
          gsl_bspline_workspace * w)

     This function computes the knots associated with the given
     breakpoints and stores them internally in ‘w->knots’.

 -- Function: int gsl_bspline_knots_uniform (const double a, const
          double b, gsl_bspline_workspace * w)

     This function assumes uniformly spaced breakpoints on [a,b] and
     constructs the corresponding knot vector using the previously
     specified ‘nbreak’ parameter.  The knots are stored in ‘w->knots’.


File: gsl-ref.info,  Node: Evaluation of B-splines,  Next: Evaluation of B-spline derivatives,  Prev: Constructing the knots vector,  Up: Basis Splines

42.4 Evaluation of B-splines
============================

 -- Function: int gsl_bspline_eval (const double x, gsl_vector * B,
          gsl_bspline_workspace * w)

     This function evaluates all B-spline basis functions at the
     position ‘x’ and stores them in the vector ‘B’, so that the i-th
     element is B_i(x).  The vector ‘B’ must be of length n = nbreak + k
     - 2.  This value may also be obtained by calling *note
     gsl_bspline_ncoeffs(): bdf.  Computing all the basis functions at
     once is more efficient than computing them individually, due to the
     nature of the defining recurrence relation.

 -- Function: int gsl_bspline_eval_nonzero (const double x, gsl_vector
          * Bk, size_t * istart, size_t * iend, gsl_bspline_workspace
          * w)

     This function evaluates all potentially nonzero B-spline basis
     functions at the position ‘x’ and stores them in the vector ‘Bk’,
     so that the i-th element is B_{(istart+i)}(x).  The last element of
     ‘Bk’ is B_{iend}(x).  The vector ‘Bk’ must be of length k.  By
     returning only the nonzero basis functions, this function allows
     quantities involving linear combinations of the B_i(x) to be
     computed without unnecessary terms (such linear combinations occur,
     for example, when evaluating an interpolated function).

 -- Function: size_t gsl_bspline_ncoeffs (gsl_bspline_workspace * w)

     This function returns the number of B-spline coefficients given by
     n = nbreak + k - 2.


File: gsl-ref.info,  Node: Evaluation of B-spline derivatives,  Next: Working with the Greville abscissae,  Prev: Evaluation of B-splines,  Up: Basis Splines

42.5 Evaluation of B-spline derivatives
=======================================

 -- Function: int gsl_bspline_deriv_eval (const double x, const
          size_t nderiv, gsl_matrix * dB, gsl_bspline_workspace * w)

     This function evaluates all B-spline basis function derivatives of
     orders 0 through ‘nderiv’ (inclusive) at the position ‘x’ and
     stores them in the matrix ‘dB’.  The (i,j)-th element of ‘dB’ is
     d^jB_i(x)/dx^j.  The matrix ‘dB’ must be of size n = nbreak + k - 2
     by nderiv + 1.  The value n may also be obtained by calling *note
     gsl_bspline_ncoeffs(): bdf.  Note that function evaluations are
     included as the zeroth order derivatives in ‘dB’.  Computing all
     the basis function derivatives at once is more efficient than
     computing them individually, due to the nature of the defining
     recurrence relation.

 -- Function: int gsl_bspline_deriv_eval_nonzero (const double x, const
          size_t nderiv, gsl_matrix * dB, size_t * istart, size_t
          * iend, gsl_bspline_workspace * w)

     This function evaluates all potentially nonzero B-spline basis
     function derivatives of orders 0 through ‘nderiv’ (inclusive) at
     the position ‘x’ and stores them in the matrix ‘dB’.  The (i,j)-th
     element of ‘dB’ is d^jB_{(istart+i)}(x)/dx^j.  The last row of ‘dB’
     contains d^jB_{iend}(x)/dx^j.  The matrix ‘dB’ must be of size k by
     at least nderiv + 1.  Note that function evaluations are included
     as the zeroth order derivatives in ‘dB’.  By returning only the
     nonzero basis functions, this function allows quantities involving
     linear combinations of the B_i(x) and their derivatives to be
     computed without unnecessary terms.


File: gsl-ref.info,  Node: Working with the Greville abscissae,  Next: Examples<33>,  Prev: Evaluation of B-spline derivatives,  Up: Basis Splines

42.6 Working with the Greville abscissae
========================================

The Greville abscissae are defined to be the mean location of k-1
consecutive knots in the knot vector for each basis spline function of
order k.  With the first and last knots in the *note
gsl_bspline_workspace: bd7. knot vector excluded, there are *note
gsl_bspline_ncoeffs(): bdf. Greville abscissae for any given B-spline
basis.  These values are often used in B-spline collocation applications
and may also be called Marsden-Schoenberg points.

 -- Function: double gsl_bspline_greville_abscissa (size_t i,
          gsl_bspline_workspace * w)

     Returns the location of the i-th Greville abscissa for the given
     B-spline basis.  For the ill-defined case when k = 1, the
     implementation chooses to return breakpoint interval midpoints.


File: gsl-ref.info,  Node: Examples<33>,  Next: References and Further Reading<35>,  Prev: Working with the Greville abscissae,  Up: Basis Splines

42.7 Examples
=============

The following program computes a linear least squares fit to data using
cubic B-spline basis functions with uniform breakpoints.  The data is
generated from the curve y(x) = \cos{(x)} \exp{(-x/10)} on the interval
[0, 15] with Gaussian noise added.

     #include <stdio.h>
     #include <stdlib.h>
     #include <math.h>
     #include <gsl/gsl_bspline.h>
     #include <gsl/gsl_multifit.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_statistics.h>

     /* number of data points to fit */
     #define N        200

     /* number of fit coefficients */
     #define NCOEFFS  12

     /* nbreak = ncoeffs + 2 - k = ncoeffs - 2 since k = 4 */
     #define NBREAK   (NCOEFFS - 2)

     int
     main (void)
     {
       const size_t n = N;
       const size_t ncoeffs = NCOEFFS;
       const size_t nbreak = NBREAK;
       size_t i, j;
       gsl_bspline_workspace *bw;
       gsl_vector *B;
       double dy;
       gsl_rng *r;
       gsl_vector *c, *w;
       gsl_vector *x, *y;
       gsl_matrix *X, *cov;
       gsl_multifit_linear_workspace *mw;
       double chisq, Rsq, dof, tss;

       gsl_rng_env_setup();
       r = gsl_rng_alloc(gsl_rng_default);

       /* allocate a cubic bspline workspace (k = 4) */
       bw = gsl_bspline_alloc(4, nbreak);
       B = gsl_vector_alloc(ncoeffs);

       x = gsl_vector_alloc(n);
       y = gsl_vector_alloc(n);
       X = gsl_matrix_alloc(n, ncoeffs);
       c = gsl_vector_alloc(ncoeffs);
       w = gsl_vector_alloc(n);
       cov = gsl_matrix_alloc(ncoeffs, ncoeffs);
       mw = gsl_multifit_linear_alloc(n, ncoeffs);

       /* this is the data to be fitted */
       for (i = 0; i < n; ++i)
         {
           double sigma;
           double xi = (15.0 / (N - 1)) * i;
           double yi = cos(xi) * exp(-0.1 * xi);

           sigma = 0.1 * yi;
           dy = gsl_ran_gaussian(r, sigma);
           yi += dy;

           gsl_vector_set(x, i, xi);
           gsl_vector_set(y, i, yi);
           gsl_vector_set(w, i, 1.0 / (sigma * sigma));

           printf("%f %f\n", xi, yi);
         }

       /* use uniform breakpoints on [0, 15] */
       gsl_bspline_knots_uniform(0.0, 15.0, bw);

       /* construct the fit matrix X */
       for (i = 0; i < n; ++i)
         {
           double xi = gsl_vector_get(x, i);

           /* compute B_j(xi) for all j */
           gsl_bspline_eval(xi, B, bw);

           /* fill in row i of X */
           for (j = 0; j < ncoeffs; ++j)
             {
               double Bj = gsl_vector_get(B, j);
               gsl_matrix_set(X, i, j, Bj);
             }
         }

       /* do the fit */
       gsl_multifit_wlinear(X, w, y, c, cov, &chisq, mw);

       dof = n - ncoeffs;
       tss = gsl_stats_wtss(w->data, 1, y->data, 1, y->size);
       Rsq = 1.0 - chisq / tss;

       fprintf(stderr, "chisq/dof = %e, Rsq = %f\n",
                        chisq / dof, Rsq);

       printf("\n\n");

       /* output the smoothed curve */
       {
         double xi, yi, yerr;

         for (xi = 0.0; xi < 15.0; xi += 0.1)
           {
             gsl_bspline_eval(xi, B, bw);
             gsl_multifit_linear_est(B, c, cov, &yi, &yerr);
             printf("%f %f\n", xi, yi);
           }
       }

       gsl_rng_free(r);
       gsl_bspline_free(bw);
       gsl_vector_free(B);
       gsl_vector_free(x);
       gsl_vector_free(y);
       gsl_matrix_free(X);
       gsl_vector_free(c);
       gsl_vector_free(w);
       gsl_matrix_free(cov);
       gsl_multifit_linear_free(mw);

       return 0;
     } /* main() */

The output is shown below:

     $ ./a.out > bspline.txt
     chisq/dof = 1.118217e+00, Rsq = 0.989771

The data and fitted model are shown in Fig.  %s.

[image src="bspline.png"]

Figure: Data (black) and fitted model (red)


File: gsl-ref.info,  Node: References and Further Reading<35>,  Prev: Examples<33>,  Up: Basis Splines

42.8 References and Further Reading
===================================

Further information on the algorithms described in this section can be
found in the following book,

   * C. de Boor, `A Practical Guide to Splines' (1978), Springer-Verlag,
     ISBN 0-387-90356-9.

Further information of Greville abscissae and B-spline collocation can
be found in the following paper,

   * Richard W. Johnson, Higher order B-spline collocation at the
     Greville abscissae.  `Applied Numerical Mathematics'.  vol.: 52,
     2005, 63–75.

A large collection of B-spline routines is available in the PPPACK
library available at ‘http://www.netlib.org/pppack’, which is also part
of SLATEC.


File: gsl-ref.info,  Node: Sparse Matrices,  Next: Sparse BLAS Support,  Prev: Basis Splines,  Up: Top

43 Sparse Matrices
******************

This chapter describes functions for the construction and manipulation
of sparse matrices, matrices which are populated primarily with zeros
and contain only a few non-zero elements.  Sparse matrices often appear
in the solution of partial differential equations.  It is beneficial to
use specialized data structures and algorithms for storing and working
with sparse matrices, since dense matrix algorithms and structures can
be prohibitively slow and use huge amounts of memory when applied to
sparse matrices.

The header file ‘gsl_spmatrix.h’ contains the prototypes for the sparse
matrix functions and related declarations.

* Menu:

* Data types: Data types<2>.
* Sparse Matrix Storage Formats::
* Overview: Overview<8>.
* Allocation::
* Accessing Matrix Elements::
* Initializing Matrix Elements::
* Reading and Writing Matrices::
* Copying Matrices::
* Exchanging Rows and Columns::
* Matrix Operations::
* Matrix Properties::
* Finding Maximum and Minimum Elements::
* Compressed Format::
* Conversion Between Sparse and Dense Matrices::
* Examples: Examples<34>.
* References and Further Reading: References and Further Reading<36>.


File: gsl-ref.info,  Node: Data types<2>,  Next: Sparse Matrix Storage Formats,  Up: Sparse Matrices

43.1 Data types
===============

All the functions are available for each of the standard data-types.
The versions for ‘double’ have the prefix ‘gsl_spmatrix’, Similarly the
versions for single-precision ‘float’ arrays have the prefix
‘gsl_spmatrix_float’.  The full list of available types is given below,

Prefix                               Type
                                     
--------------------------------------------------------------
                                     
gsl_spmatrix                         double
                                     
                                     
gsl_spmatrix_float                   float
                                     
                                     
gsl_spmatrix_long_double             long double
                                     
                                     
gsl_spmatrix_int                     int
                                     
                                     
gsl_spmatrix_uint                    unsigned int
                                     
                                     
gsl_spmatrix_long                    long
                                     
                                     
gsl_spmatrix_ulong                   unsigned long
                                     
                                     
gsl_spmatrix_short                   short
                                     
                                     
gsl_spmatrix_ushort                  unsigned short
                                     
                                     
gsl_spmatrix_char                    char
                                     
                                     
gsl_spmatrix_uchar                   unsigned char
                                     
                                     
gsl_spmatrix_complex                 complex double
                                     
                                     
gsl_spmatrix_complex_float           complex float
                                     
                                     
gsl_spmatrix_complex_long_double     complex long double
                                     


File: gsl-ref.info,  Node: Sparse Matrix Storage Formats,  Next: Overview<8>,  Prev: Data types<2>,  Up: Sparse Matrices

43.2 Sparse Matrix Storage Formats
==================================

GSL currently supports three storage formats for sparse matrices: the
coordinate (COO) representation, compressed sparse column (CSC) and
compressed sparse row (CSR) formats.  These are discussed in more detail
below.  In order to illustrate the different storage formats, the
following sections will reference this M-by-N sparse matrix, with M=4
and N=5:

     \begin{pmatrix} 9 & 0 & 0 & 0 & -3 \\ 4 & 7 & 0 & 0 & 0 \\ 0 & 8 & -1 & 8 & 0 \\ 4 & 0 & 5 & 6 & 0 \end{pmatrix}

The number of non-zero elements in the matrix, also abbreviated as ‘nnz’
is equal to 10 in this case.

* Menu:

* Coordinate Storage (COO): Coordinate Storage COO.
* Compressed Sparse Column (CSC): Compressed Sparse Column CSC.
* Compressed Sparse Row (CSR): Compressed Sparse Row CSR.


File: gsl-ref.info,  Node: Coordinate Storage COO,  Next: Compressed Sparse Column CSC,  Up: Sparse Matrix Storage Formats

43.2.1 Coordinate Storage (COO)
-------------------------------

The coordinate storage format, also known as `triplet format', stores
triplets (i,j,x) for each non-zero element of the matrix.  This notation
means that the (i,j) element of the matrix A is A_{ij} = x.  The matrix
is stored using three arrays of the same length, representing the row
indices, column indices, and matrix data.  For the reference matrix
above, one possible storage format is:

data     9      7      4      8      -3     -1     8      5      6      4
                                                                        
                                                                        
row      0      1      1      2      0      2      2      3      3      3
                                                                        
                                                                        
col      0      1      0      1      4      2      3      2      3      0
                                                                        

Note that this representation is not unique - the coordinate triplets
may appear in any ordering and would still represent the same sparse
matrix.  The length of the three arrays is equal to the number of
non-zero elements in the matrix, ‘nnz’, which in this case is 10.  The
coordinate format is extremely convenient for sparse matrix `assembly',
the process of adding new elements, or changing existing elements, in a
sparse matrix.  However, it is generally not suitable for the efficient
implementation of matrix-matrix products, or matrix factorization
algorithms.  For these applications it is better to use one of the
compressed formats discussed below.

In order to faciliate efficient sparse matrix assembly, GSL stores the
coordinate data in a balanced binary search tree, specifically an AVL
tree, in addition to the three arrays discussed above.  This allows GSL
to efficiently determine whether an entry (i,j) already exists in the
matrix, and to replace an existing matrix entry with a new value,
without needing to search unsorted arrays.


File: gsl-ref.info,  Node: Compressed Sparse Column CSC,  Next: Compressed Sparse Row CSR,  Prev: Coordinate Storage COO,  Up: Sparse Matrix Storage Formats

43.2.2 Compressed Sparse Column (CSC)
-------------------------------------

Compressed sparse column storage stores each column of non-zero values
in the sparse matrix in a continuous memory block, keeping pointers to
the beginning of each column in that memory block, and storing the row
indices of each non-zero element.  For the reference matrix above, these
arrays look like

data        9      4      4      7      8      -1     5      8      6      -3
                                                                           
                                                                           
row         0      1      3      1      2      2      3      2      3      0
                                                                           
                                                                           
col_ptr     0      3      5      7      9      10
                                               

The ‘data’ and ‘row’ arrays are of length ‘nnz’ and are the same as the
COO storage format.  The ‘col_ptr’ array has length N+1, and
‘col_ptr[j]’ gives the index in ‘data’ of the start of column ‘j’.
Therefore, the j-th column of the matrix is stored in
‘data[col_ptr[j]]’, ‘data[col_ptr[j] + 1]’, …, ‘data[col_ptr[j+1] - 1]’.
The last element of ‘col_ptr’ is ‘nnz’.


File: gsl-ref.info,  Node: Compressed Sparse Row CSR,  Prev: Compressed Sparse Column CSC,  Up: Sparse Matrix Storage Formats

43.2.3 Compressed Sparse Row (CSR)
----------------------------------

Compressed row storage stores each row of non-zero values in a
continuous memory block, keeping pointers to the beginning of each row
in the block and storing the column indices of each non-zero element.
For the reference matrix above, these arrays look like

data        9      -3     4      7      8      -1     8      4      5      6
                                                                           
                                                                           
col         0      4      0      1      1      2      3      0      2      3
                                                                           
                                                                           
row_ptr     0      2      4      7      10
                                        

The ‘data’ and ‘col’ arrays are of length ‘nnz’ and are the same as the
COO storage format.  The ‘row_ptr’ array has length M+1, and
‘row_ptr[i]’ gives the index in ‘data’ of the start of row ‘i’.
Therefore, the i-th row of the matrix is stored in ‘data[row_ptr[i]]’,
‘data[row_ptr[i] + 1]’, …, ‘data[row_ptr[i+1] - 1]’.  The last element
of ‘row_ptr’ is ‘nnz’.


File: gsl-ref.info,  Node: Overview<8>,  Next: Allocation,  Prev: Sparse Matrix Storage Formats,  Up: Sparse Matrices

43.3 Overview
=============

These routines provide support for constructing and manipulating sparse
matrices in GSL, using an API similar to *note gsl_matrix: 3a1.  The
basic structure is called *note gsl_spmatrix: bf4.

 -- Type: gsl_spmatrix

     This structure is defined as:

          typedef struct
          {
            size_t size1;
            size_t size2;
            size_t *i;
            double *data;
            size_t *p;
            size_t nzmax;
            size_t nz;
            gsl_spmatrix_tree *tree_data;
            void *work;
            size_t sptype;
          } gsl_spmatrix;

     This defines a ‘size1’-by-‘size2’ sparse matrix.  The number of
     non-zero elements currently in the matrix is given by ‘nz’.  For
     the triplet representation, ‘i’, ‘p’, and ‘data’ are arrays of size
     ‘nz’ which contain the row indices, column indices, and element
     value, respectively.  So if data[k] = A(i,j), then i = i[k] and j =
     p[k].

     For compressed column storage, ‘i’ and ‘data’ are arrays of size
     ‘nz’ containing the row indices and element values, identical to
     the triplet case.  ‘p’ is an array of size ‘size2’ + 1 where ‘p[j]’
     points to the index in ‘data’ of the start of column ‘j’.  Thus, if
     data[k] = A(i,j), then i = i[k] and p[j] <= k < p[j+1].

     For compressed row storage, ‘i’ and ‘data’ are arrays of size ‘nz’
     containing the column indices and element values, identical to the
     triplet case.  ‘p’ is an array of size ‘size1’ + 1 where ‘p[i]’
     points to the index in ‘data’ of the start of row ‘i’.  Thus, if
     data[k] = A(i,j), then j = i[k] and p[i] <= k < p[i+1].

     The parameter ‘tree_data’ is a binary tree structure used in the
     triplet representation, specifically a balanced AVL tree.  This
     speeds up element searches and duplicate detection during the
     matrix assembly process.  The parameter ‘work’ is additional
     workspace needed for various operations like converting from
     triplet to compressed storage.  ‘sptype’ indicates the type of
     storage format being used (COO, CSC or CSR).

     The compressed storage format defined above makes it very simple to
     interface with sophisticated external linear solver libraries which
     accept compressed storage input.  The user can simply pass the
     arrays ‘i’, ‘p’, and ‘data’ as the inputs to external libraries.


File: gsl-ref.info,  Node: Allocation,  Next: Accessing Matrix Elements,  Prev: Overview<8>,  Up: Sparse Matrices

43.4 Allocation
===============

The functions for allocating memory for a sparse matrix follow the style
of ‘malloc()’ and ‘free()’.  They also perform their own error checking.
If there is insufficient memory available to allocate a matrix then the
functions call the GSL error handler with an error code of *note
GSL_ENOMEM: 2a. in addition to returning a null pointer.

 -- Function: gsl_spmatrix * gsl_spmatrix_alloc (const size_t n1, const
          size_t n2)

     This function allocates a sparse matrix of size ‘n1’-by-‘n2’ and
     initializes it to all zeros.  If the size of the matrix is not
     known at allocation time, both ‘n1’ and ‘n2’ may be set to 1, and
     they will automatically grow as elements are added to the matrix.
     This function sets the matrix to the triplet representation, which
     is the easiest for adding and accessing matrix elements.  This
     function tries to make a reasonable guess for the number of
     non-zero elements (‘nzmax’) which will be added to the matrix by
     assuming a sparse density of 10\%.  The function *note
     gsl_spmatrix_alloc_nzmax(): bf7. can be used if this number is
     known more accurately.  The workspace is of size O(nzmax).

 -- Function: gsl_spmatrix * gsl_spmatrix_alloc_nzmax (const size_t n1,
          const size_t n2, const size_t nzmax, const size_t sptype)

     This function allocates a sparse matrix of size ‘n1’-by-‘n2’ and
     initializes it to all zeros.  If the size of the matrix is not
     known at allocation time, both ‘n1’ and ‘n2’ may be set to 1, and
     they will automatically grow as elements are added to the matrix.
     The parameter ‘nzmax’ specifies the maximum number of non-zero
     elements which will be added to the matrix.  It does not need to be
     precisely known in advance, since storage space will automatically
     grow using *note gsl_spmatrix_realloc(): bf8. if ‘nzmax’ is not
     large enough.  Accurate knowledge of this parameter reduces the
     number of reallocation calls required.  The parameter ‘sptype’
     specifies the storage format of the sparse matrix.  Possible values
     are

      -- Macro: GSL_SPMATRIX_COO

          This flag specifies coordinate (triplet) storage.

      -- Macro: GSL_SPMATRIX_CSC

          This flag specifies compressed sparse column storage.

      -- Macro: GSL_SPMATRIX_CSR

          This flag specifies compressed sparse row storage.

     The allocated *note gsl_spmatrix: bf4. structure is of size
     O(nzmax).

 -- Function: int gsl_spmatrix_realloc (const size_t nzmax, gsl_spmatrix
          * m)

     This function reallocates the storage space for ‘m’ to accomodate
     ‘nzmax’ non-zero elements.  It is typically called internally by
     *note gsl_spmatrix_set(): bfc. if the user wants to add more
     elements to the sparse matrix than the previously specified
     ‘nzmax’.

     Input matrix formats supported: *note COO: bee, *note CSC: bf0,
     *note CSR: bf2.

 -- Function: void gsl_spmatrix_free (gsl_spmatrix * m)

     This function frees the memory associated with the sparse matrix
     ‘m’.

     Input matrix formats supported: *note COO: bee, *note CSC: bf0,
     *note CSR: bf2.


File: gsl-ref.info,  Node: Accessing Matrix Elements,  Next: Initializing Matrix Elements,  Prev: Allocation,  Up: Sparse Matrices

43.5 Accessing Matrix Elements
==============================

 -- Function: double gsl_spmatrix_get (const gsl_spmatrix * m, const
          size_t i, const size_t j)

     This function returns element (‘i’, ‘j’) of the matrix ‘m’.

     Input matrix formats supported: *note COO: bee, *note CSC: bf0,
     *note CSR: bf2.

 -- Function: int gsl_spmatrix_set (gsl_spmatrix * m, const size_t i,
          const size_t j, const double x)

     This function sets element (‘i’, ‘j’) of the matrix ‘m’ to the
     value ‘x’.

     Input matrix formats supported: *note COO: bee.

 -- Function: double * gsl_spmatrix_ptr (gsl_spmatrix * m, const
          size_t i, const size_t j)

     This function returns a pointer to the (‘i’, ‘j’) element of the
     matrix ‘m’.  If the (‘i’, ‘j’) element is not explicitly stored in
     the matrix, a null pointer is returned.

     Input matrix formats supported: *note COO: bee, *note CSC: bf0,
     *note CSR: bf2.


File: gsl-ref.info,  Node: Initializing Matrix Elements,  Next: Reading and Writing Matrices,  Prev: Accessing Matrix Elements,  Up: Sparse Matrices

43.6 Initializing Matrix Elements
=================================

Since the sparse matrix format only stores the non-zero elements, it is
automatically initialized to zero upon allocation.  The function *note
gsl_spmatrix_set_zero(): c02. may be used to re-initialize a matrix to
zero after elements have been added to it.

 -- Function: int gsl_spmatrix_set_zero (gsl_spmatrix * m)

     This function sets (or resets) all the elements of the matrix ‘m’
     to zero.  For CSC and CSR matrices, the cost of this operation is
     O(1).  For COO matrices, the binary tree structure must be
     dismantled, so the cost is O(nz).

     Input matrix formats supported: *note COO: bee, *note CSC: bf0,
     *note CSR: bf2.


File: gsl-ref.info,  Node: Reading and Writing Matrices,  Next: Copying Matrices,  Prev: Initializing Matrix Elements,  Up: Sparse Matrices

43.7 Reading and Writing Matrices
=================================

 -- Function: int gsl_spmatrix_fwrite (FILE * stream, const gsl_spmatrix
          * m)

     This function writes the elements of the matrix ‘m’ to the stream
     ‘stream’ in binary format.  The return value is 0 for success and
     ‘GSL_EFAILED’ if there was a problem writing to the file.  Since
     the data is written in the native binary format it may not be
     portable between different architectures.

     Input matrix formats supported: *note COO: bee, *note CSC: bf0,
     *note CSR: bf2.

 -- Function: int gsl_spmatrix_fread (FILE * stream, gsl_spmatrix * m)

     This function reads into the matrix ‘m’ from the open stream
     ‘stream’ in binary format.  The matrix ‘m’ must be preallocated
     with the correct storage format, dimensions and have a sufficiently
     large ‘nzmax’ in order to read in all matrix elements, otherwise
     ‘GSL_EBADLEN’ is returned.  The return value is 0 for success and
     ‘GSL_EFAILED’ if there was a problem reading from the file.  The
     data is assumed to have been written in the native binary format on
     the same architecture.

     Input matrix formats supported: *note COO: bee, *note CSC: bf0,
     *note CSR: bf2.

 -- Function: int gsl_spmatrix_fprintf (FILE * stream, const
          gsl_spmatrix * m, const char * format)

     This function writes the elements of the matrix ‘m’ line-by-line to
     the stream ‘stream’ using the format specifier ‘format’, which
     should be one of the ‘%g’, ‘%e’ or ‘%f’ formats for floating point
     numbers.  The function returns 0 for success and ‘GSL_EFAILED’ if
     there was a problem writing to the file.  The input matrix ‘m’ may
     be in any storage format, and the output file will be written in
     MatrixMarket format.

     Input matrix formats supported: *note COO: bee, *note CSC: bf0,
     *note CSR: bf2.

 -- Function: gsl_spmatrix * gsl_spmatrix_fscanf (FILE * stream)

     This function reads sparse matrix data in the MatrixMarket format
     from the stream ‘stream’ and stores it in a newly allocated matrix
     which is returned in *note COO: bee. format.  The function returns
     0 for success and ‘GSL_EFAILED’ if there was a problem reading from
     the file.  The user should free the returned matrix when it is no
     longer needed.


File: gsl-ref.info,  Node: Copying Matrices,  Next: Exchanging Rows and Columns,  Prev: Reading and Writing Matrices,  Up: Sparse Matrices

43.8 Copying Matrices
=====================

 -- Function: int gsl_spmatrix_memcpy (gsl_spmatrix * dest, const
          gsl_spmatrix * src)

     This function copies the elements of the sparse matrix ‘src’ into
     ‘dest’.  The two matrices must have the same dimensions and be in
     the same storage format.

     Input matrix formats supported: *note COO: bee, *note CSC: bf0,
     *note CSR: bf2.


File: gsl-ref.info,  Node: Exchanging Rows and Columns,  Next: Matrix Operations,  Prev: Copying Matrices,  Up: Sparse Matrices

43.9 Exchanging Rows and Columns
================================

 -- Function: int gsl_spmatrix_transpose_memcpy (gsl_spmatrix * dest,
          const gsl_spmatrix * src)

     This function copies the transpose of the sparse matrix ‘src’ into
     ‘dest’.  The dimensions of ‘dest’ must match the transpose of the
     matrix ‘src’.  Also, both matrices must use the same sparse storage
     format.

     Input matrix formats supported: *note COO: bee, *note CSC: bf0,
     *note CSR: bf2.

 -- Function: int gsl_spmatrix_transpose (gsl_spmatrix * m)

     This function replaces the matrix ‘m’ by its transpose, but changes
     the storage format for compressed matrix inputs.  Since compressed
     column storage is the transpose of compressed row storage, this
     function simply converts a CSC matrix to CSR and vice versa.  This
     is the most efficient way to transpose a compressed storage matrix,
     but the user should note that the storage format of their
     compressed matrix will change on output.  For COO matrix inputs,
     the output matrix is also in COO storage.

     Input matrix formats supported: *note COO: bee, *note CSC: bf0,
     *note CSR: bf2.


File: gsl-ref.info,  Node: Matrix Operations,  Next: Matrix Properties,  Prev: Exchanging Rows and Columns,  Up: Sparse Matrices

43.10 Matrix Operations
=======================

 -- Function: int gsl_spmatrix_scale (gsl_spmatrix * m, const double x)

     This function scales all elements of the matrix ‘m’ by the constant
     factor ‘x’.  The result m(i,j) \leftarrow x m(i,j) is stored in
     ‘m’.

     Input matrix formats supported: *note COO: bee, *note CSC: bf0,
     *note CSR: bf2.

 -- Function: int gsl_spmatrix_scale_columns (gsl_spmatrix * A, const
          gsl_vector * x)

     This function scales the columns of the M-by-N sparse matrix ‘A’ by
     the elements of the vector ‘x’, of length N. The j-th column of ‘A’
     is multiplied by ‘x[j]’.  This is equivalent to forming

          A \rightarrow A X

     where X = \textrm{diag}(x).

     Input matrix formats supported: *note COO: bee, *note CSC: bf0,
     *note CSR: bf2.

 -- Function: int gsl_spmatrix_scale_rows (gsl_spmatrix * A, const
          gsl_vector * x)

     This function scales the rows of the M-by-N sparse matrix ‘A’ by
     the elements of the vector ‘x’, of length M. The i-th row of ‘A’ is
     multiplied by ‘x[i]’.  This is equivalent to forming

          A \rightarrow X A

     where X = \textrm{diag}(x).

     Input matrix formats supported: *note COO: bee, *note CSC: bf0,
     *note CSR: bf2.

 -- Function: int gsl_spmatrix_add (gsl_spmatrix * c, const gsl_spmatrix
          * a, const gsl_spmatrix * b)

     This function computes the sum c = a + b.  The three matrices must
     have the same dimensions.

     Input matrix formats supported: *note CSC: bf0, *note CSR: bf2.

 -- Function: int gsl_spmatrix_add_to_dense (gsl_matrix * a, const
          gsl_spmatrix * b)

     This function adds the elements of the sparse matrix ‘b’ to the
     elements of the dense matrix ‘a’.  The result a(i,j) \leftarrow
     a(i,j) + b(i,j) is stored in ‘a’ and ‘b’ remains unchanged.  The
     two matrices must have the same dimensions.

     Input matrix formats supported: *note COO: bee, *note CSC: bf0,
     *note CSR: bf2.


File: gsl-ref.info,  Node: Matrix Properties,  Next: Finding Maximum and Minimum Elements,  Prev: Matrix Operations,  Up: Sparse Matrices

43.11 Matrix Properties
=======================

 -- Function: const char * gsl_spmatrix_type (const gsl_spmatrix * m)

     This function returns a string describing the sparse storage format
     of the matrix ‘m’.  For example:

          printf ("matrix is '%s' format.\n", gsl_spmatrix_type (m));

     would print something like:

          matrix is 'CSR' format.

     Input matrix formats supported: *note COO: bee, *note CSC: bf0,
     *note CSR: bf2.

 -- Function: size_t gsl_spmatrix_nnz (const gsl_spmatrix * m)

     This function returns the number of non-zero elements in ‘m’.

     Input matrix formats supported: *note COO: bee, *note CSC: bf0,
     *note CSR: bf2.

 -- Function: int gsl_spmatrix_equal (const gsl_spmatrix * a, const
          gsl_spmatrix * b)

     This function returns 1 if the matrices ‘a’ and ‘b’ are equal (by
     comparison of element values) and 0 otherwise.  The matrices ‘a’
     and ‘b’ must be in the same sparse storage format for comparison.

     Input matrix formats supported: *note COO: bee, *note CSC: bf0,
     *note CSR: bf2.


File: gsl-ref.info,  Node: Finding Maximum and Minimum Elements,  Next: Compressed Format,  Prev: Matrix Properties,  Up: Sparse Matrices

43.12 Finding Maximum and Minimum Elements
==========================================

 -- Function: int gsl_spmatrix_minmax (const gsl_spmatrix * m, double
          * min_out, double * max_out)

     This function returns the minimum and maximum elements of the
     matrix ‘m’, storing them in ‘min_out’ and ‘max_out’, and searching
     only the non-zero values.

     Input matrix formats supported: *note COO: bee, *note CSC: bf0,
     *note CSR: bf2.

 -- Function: int gsl_spmatrix_min_index (const gsl_spmatrix * m, size_t
          * imin, size_t * jmin)

     This function returns the indices of the minimum value in the
     matrix ‘m’, searching only the non-zero values, and storing them in
     ‘imin’ and ‘jmin’.  When there are several equal minimum elements
     then the first element found is returned.

     Input matrix formats supported: *note COO: bee, *note CSC: bf0,
     *note CSR: bf2.


File: gsl-ref.info,  Node: Compressed Format,  Next: Conversion Between Sparse and Dense Matrices,  Prev: Finding Maximum and Minimum Elements,  Up: Sparse Matrices

43.13 Compressed Format
=======================

These routines calculate a compressed matrix from a coordinate
representation.

 -- Function: int gsl_spmatrix_csc (gsl_spmatrix * dest, const
          gsl_spmatrix * src)

     This function creates a sparse matrix in *note compressed sparse
     column: bf0. format from the input sparse matrix ‘src’ which must
     be in COO format.  The compressed matrix is stored in ‘dest’.

     Input matrix formats supported: *note COO: bee.

 -- Function: int gsl_spmatrix_csr (gsl_spmatrix * dest, const
          gsl_spmatrix * src)

     This function creates a sparse matrix in *note compressed sparse
     row: bf2. format from the input sparse matrix ‘src’ which must be
     in COO format.  The compressed matrix is stored in ‘dest’.

     Input matrix formats supported: *note COO: bee.

 -- Function: gsl_spmatrix * gsl_spmatrix_compress (const gsl_spmatrix
          * src, const int sptype)

     This function allocates a new sparse matrix, and stores ‘src’ into
     it using the format specified by ‘sptype’.  The input ‘sptype’ can
     be one of *note GSL_SPMATRIX_COO: bf9, *note GSL_SPMATRIX_CSC: bfa,
     or *note GSL_SPMATRIX_CSR: bfb.  A pointer to the newly allocated
     matrix is returned, and must be freed by the caller when no longer
     needed.


File: gsl-ref.info,  Node: Conversion Between Sparse and Dense Matrices,  Next: Examples<34>,  Prev: Compressed Format,  Up: Sparse Matrices

43.14 Conversion Between Sparse and Dense Matrices
==================================================

The *note gsl_spmatrix: bf4. structure can be converted into the dense
*note gsl_matrix: 3a1. format and vice versa with the following
routines.

 -- Function: int gsl_spmatrix_d2sp (gsl_spmatrix * S, const gsl_matrix
          * A)

     This function converts the dense matrix ‘A’ into sparse COO format
     and stores the result in ‘S’.

     Input matrix formats supported: *note COO: bee.

 -- Function: int gsl_spmatrix_sp2d (gsl_matrix * A, const gsl_spmatrix
          * S)

     This function converts the sparse matrix ‘S’ into a dense matrix
     and stores the result in ‘A’.

     Input matrix formats supported: *note COO: bee, *note CSC: bf0,
     *note CSR: bf2.


File: gsl-ref.info,  Node: Examples<34>,  Next: References and Further Reading<36>,  Prev: Conversion Between Sparse and Dense Matrices,  Up: Sparse Matrices

43.15 Examples
==============

The following example program builds a 5-by-4 sparse matrix and prints
it in coordinate, compressed column, and compressed row format.  The
matrix which is constructed is

     [ 0    0  3.1  4.6 ]
     [ 1    0  7.2   0  ]
     [ 0    0   0    0  ]
     [ 2.1 2.9  0   8.5 ]
     [ 4.1  0   0    0  ]

The output of the program is:

     printing all matrix elements:
     A(0,0) = 0
     A(0,1) = 0
     A(0,2) = 3.1
     A(0,3) = 4.6
     A(1,0) = 1
     .
     .
     .
     A(4,0) = 4.1
     A(4,1) = 0
     A(4,2) = 0
     A(4,3) = 0
     matrix in triplet format (i,j,Aij):
     (0, 2, 3.1)
     (0, 3, 4.6)
     (1, 0, 1.0)
     (1, 2, 7.2)
     (3, 0, 2.1)
     (3, 1, 2.9)
     (3, 3, 8.5)
     (4, 0, 4.1)
     matrix in compressed column format:
     i = [ 1, 3, 4, 3, 0, 1, 0, 3, ]
     p = [ 0, 3, 4, 6, 8, ]
     d = [ 1, 2.1, 4.1, 2.9, 3.1, 7.2, 4.6, 8.5, ]
     matrix in compressed row format:
     i = [ 2, 3, 0, 2, 0, 1, 3, 0, ]
     p = [ 0, 2, 4, 4, 7, 8, ]
     d = [ 3.1, 4.6, 1, 7.2, 2.1, 2.9, 8.5, 4.1, ]

We see in the compressed column output, the data array stores each
column contiguously, the array i stores the row index of the
corresponding data element, and the array p stores the index of the
start of each column in the data array.  Similarly, for the compressed
row output, the data array stores each row contiguously, the array i
stores the column index of the corresponding data element, and the p
array stores the index of the start of each row in the data array.

     #include <stdio.h>
     #include <stdlib.h>

     #include <gsl/gsl_spmatrix.h>

     int
     main()
     {
       gsl_spmatrix *A = gsl_spmatrix_alloc(5, 4); /* triplet format */
       gsl_spmatrix *B, *C;
       size_t i, j;

       /* build the sparse matrix */
       gsl_spmatrix_set(A, 0, 2, 3.1);
       gsl_spmatrix_set(A, 0, 3, 4.6);
       gsl_spmatrix_set(A, 1, 0, 1.0);
       gsl_spmatrix_set(A, 1, 2, 7.2);
       gsl_spmatrix_set(A, 3, 0, 2.1);
       gsl_spmatrix_set(A, 3, 1, 2.9);
       gsl_spmatrix_set(A, 3, 3, 8.5);
       gsl_spmatrix_set(A, 4, 0, 4.1);

       printf("printing all matrix elements:\n");
       for (i = 0; i < 5; ++i)
         for (j = 0; j < 4; ++j)
           printf("A(%zu,%zu) = %g\n", i, j,
                  gsl_spmatrix_get(A, i, j));

       /* print out elements in triplet format */
       printf("matrix in triplet format (i,j,Aij):\n");
       gsl_spmatrix_fprintf(stdout, A, "%.1f");

       /* convert to compressed column format */
       B = gsl_spmatrix_ccs(A);

       printf("matrix in compressed column format:\n");
       printf("i = [ ");
       for (i = 0; i < B->nz; ++i)
         printf("%d, ", B->i[i]);
       printf("]\n");

       printf("p = [ ");
       for (i = 0; i < B->size2 + 1; ++i)
         printf("%d, ", B->p[i]);
       printf("]\n");

       printf("d = [ ");
       for (i = 0; i < B->nz; ++i)
         printf("%g, ", B->data[i]);
       printf("]\n");

       /* convert to compressed row format */
       C = gsl_spmatrix_crs(A);

       printf("matrix in compressed row format:\n");
       printf("i = [ ");
       for (i = 0; i < C->nz; ++i)
         printf("%d, ", C->i[i]);
       printf("]\n");

       printf("p = [ ");
       for (i = 0; i < C->size1 + 1; ++i)
         printf("%d, ", C->p[i]);
       printf("]\n");

       printf("d = [ ");
       for (i = 0; i < C->nz; ++i)
         printf("%g, ", C->data[i]);
       printf("]\n");

       gsl_spmatrix_free(A);
       gsl_spmatrix_free(B);
       gsl_spmatrix_free(C);

       return 0;
     }


File: gsl-ref.info,  Node: References and Further Reading<36>,  Prev: Examples<34>,  Up: Sparse Matrices

43.16 References and Further Reading
====================================

The algorithms used by these functions are described in the following
sources,

   * Davis, T. A., Direct Methods for Sparse Linear Systems, SIAM, 2006.

   * CSparse software library,
     ‘https://www.cise.ufl.edu/research/sparse/CSparse’


File: gsl-ref.info,  Node: Sparse BLAS Support,  Next: Sparse Linear Algebra,  Prev: Sparse Matrices,  Up: Top

44 Sparse BLAS Support
**********************

The Sparse Basic Linear Algebra Subprograms (BLAS) define a set of
fundamental operations on vectors and sparse matrices which can be used
to create optimized higher-level linear algebra functionality.  GSL
supports a limited number of BLAS operations for sparse matrices.

The header file ‘gsl_spblas.h’ contains the prototypes for the sparse
BLAS functions and related declarations.

* Menu:

* Sparse BLAS operations::
* References and Further Reading: References and Further Reading<37>.


File: gsl-ref.info,  Node: Sparse BLAS operations,  Next: References and Further Reading<37>,  Up: Sparse BLAS Support

44.1 Sparse BLAS operations
===========================

 -- Function: int gsl_spblas_dgemv (const CBLAS_TRANSPOSE_t TransA,
          const double alpha, const gsl_spmatrix * A, const gsl_vector
          * x, const double beta, gsl_vector * y)

     This function computes the matrix-vector product and sum y
     \leftarrow \alpha op(A) x + \beta y, where op(A) = A, A^T for
     ‘TransA’ = ‘CblasNoTrans’, ‘CblasTrans’.  In-place computations are
     not supported, so ‘x’ and ‘y’ must be distinct vectors.  The matrix
     ‘A’ may be in triplet or compressed format.

 -- Function: int gsl_spblas_dgemm (const double alpha, const
          gsl_spmatrix * A, const gsl_spmatrix * B, gsl_spmatrix * C)

     This function computes the sparse matrix-matrix product C = \alpha
     A B. The matrices must be in compressed format.


File: gsl-ref.info,  Node: References and Further Reading<37>,  Prev: Sparse BLAS operations,  Up: Sparse BLAS Support

44.2 References and Further Reading
===================================

The algorithms used by these functions are described in the following
sources:

   * Davis, T. A., Direct Methods for Sparse Linear Systems, SIAM, 2006.

   * CSparse software library,
     ‘https://www.cise.ufl.edu/research/sparse/CSparse’


File: gsl-ref.info,  Node: Sparse Linear Algebra,  Next: Physical Constants,  Prev: Sparse BLAS Support,  Up: Top

45 Sparse Linear Algebra
************************

This chapter describes functions for solving sparse linear systems of
equations.  The library provides linear algebra routines which operate
directly on the *note gsl_spmatrix: bf4. and *note gsl_vector: 35f.
objects.

The functions described in this chapter are declared in the header file
‘gsl_splinalg.h’.

* Menu:

* Overview: Overview<9>.
* Sparse Iterative Solvers::
* Examples: Examples<35>.
* References and Further Reading: References and Further Reading<38>.


File: gsl-ref.info,  Node: Overview<9>,  Next: Sparse Iterative Solvers,  Up: Sparse Linear Algebra

45.1 Overview
=============

This chapter is primarily concerned with the solution of the linear
system

     A x = b

where A is a general square n-by-n non-singular sparse matrix, x is an
unknown n-by-1 vector, and b is a given n-by-1 right hand side vector.
There exist many methods for solving such sparse linear systems, which
broadly fall into either direct or iterative categories.  Direct methods
include LU and QR decompositions, while iterative methods start with an
initial guess for the vector x and update the guess through iteration
until convergence.  GSL does not currently provide any direct sparse
solvers.


File: gsl-ref.info,  Node: Sparse Iterative Solvers,  Next: Examples<35>,  Prev: Overview<9>,  Up: Sparse Linear Algebra

45.2 Sparse Iterative Solvers
=============================

* Menu:

* Overview: Overview<10>.
* Types of Sparse Iterative Solvers::
* Iterating the Sparse Linear System::


File: gsl-ref.info,  Node: Overview<10>,  Next: Types of Sparse Iterative Solvers,  Up: Sparse Iterative Solvers

45.2.1 Overview
---------------

Many practical iterative methods of solving large n-by-n sparse linear
systems involve projecting an approximate solution for ‘x’ onto a
subspace of {\bf R}^n.  If we define a m-dimensional subspace {\cal K}
as the subspace of approximations to the solution ‘x’, then m
constraints must be imposed to determine the next approximation.  These
m constraints define another m-dimensional subspace denoted by {\cal L}.
The subspace dimension m is typically chosen to be much smaller than n
in order to reduce the computational effort needed to generate the next
approximate solution vector.  The many iterative algorithms which exist
differ mainly in their choice of {\cal K} and {\cal L}.


File: gsl-ref.info,  Node: Types of Sparse Iterative Solvers,  Next: Iterating the Sparse Linear System,  Prev: Overview<10>,  Up: Sparse Iterative Solvers

45.2.2 Types of Sparse Iterative Solvers
----------------------------------------

The sparse linear algebra library provides the following types of
iterative solvers:

 -- Type: gsl_splinalg_itersolve_type

      -- Variable: gsl_splinalg_itersolve_gmres

          This specifies the Generalized Minimum Residual Method
          (GMRES). This is a projection method using {\cal K} = {\cal
          K}_m and {\cal L} = A {\cal K}_m where {\cal K}_m is the m-th
          Krylov subspace

               K_m = span( r_0, A r_0, ..., A^(m-1) r_0)

          and r_0 = b - A x_0 is the residual vector of the initial
          guess x_0.  If m is set equal to n, then the Krylov subspace
          is {\bf R}^n and GMRES will provide the exact solution ‘x’.
          However, the goal is for the method to arrive at a very good
          approximation to ‘x’ using a much smaller subspace {\cal K}_m.
          By default, the GMRES method selects m = MIN(n,10) but the
          user may specify a different value for m.  The GMRES storage
          requirements grow as O(n(m+1)) and the number of flops grow as
          O(4 m^2 n - 4 m^3 / 3).

          In the below function *note gsl_splinalg_itersolve_iterate():
          c31, one GMRES iteration is defined as projecting the
          approximate solution vector onto each Krylov subspace {\cal
          K}_1, ..., {\cal K}_m, and so m can be kept smaller by
          “restarting” the method and calling *note
          gsl_splinalg_itersolve_iterate(): c31. multiple times,
          providing the updated approximation ‘x’ to each new call.  If
          the method is not adequately converging, the user may try
          increasing the parameter m.

          GMRES is considered a robust general purpose iterative solver,
          however there are cases where the method stagnates if the
          matrix is not positive-definite and fails to reduce the
          residual until the very last projection onto the subspace
          {\cal K}_n = {\bf R}^n.  In these cases, preconditioning the
          linear system can help, but GSL does not currently provide any
          preconditioners.


File: gsl-ref.info,  Node: Iterating the Sparse Linear System,  Prev: Types of Sparse Iterative Solvers,  Up: Sparse Iterative Solvers

45.2.3 Iterating the Sparse Linear System
-----------------------------------------

The following functions are provided to allocate storage for the sparse
linear solvers and iterate the system to a solution.

 -- Function: gsl_splinalg_itersolve * gsl_splinalg_itersolve_alloc
          (const gsl_splinalg_itersolve_type * T, const size_t n, const
          size_t m)

     This function allocates a workspace for the iterative solution of
     ‘n’-by-‘n’ sparse matrix systems.  The iterative solver type is
     specified by ‘T’.  The argument ‘m’ specifies the size of the
     solution candidate subspace {\cal K}_m.  The dimension ‘m’ may be
     set to 0 in which case a reasonable default value is used.

 -- Function: void gsl_splinalg_itersolve_free (gsl_splinalg_itersolve
          * w)

     This function frees the memory associated with the workspace ‘w’.

 -- Function: const char * gsl_splinalg_itersolve_name (const
          gsl_splinalg_itersolve * w)

     This function returns a string pointer to the name of the solver.

 -- Function: int gsl_splinalg_itersolve_iterate (const gsl_spmatrix
          * A, const gsl_vector * b, const double tol, gsl_vector * x,
          gsl_splinalg_itersolve * w)

     This function performs one iteration of the iterative method for
     the sparse linear system specfied by the matrix ‘A’, right hand
     side vector ‘b’ and solution vector ‘x’.  On input, ‘x’ must be set
     to an initial guess for the solution.  On output, ‘x’ is updated to
     give the current solution estimate.  The parameter ‘tol’ specifies
     the relative tolerance between the residual norm and norm of ‘b’ in
     order to check for convergence.  When the following condition is
     satisfied:

          || A x - b || <= tol * || b ||

     the method has converged, the function returns ‘GSL_SUCCESS’ and
     the final solution is provided in ‘x’.  Otherwise, the function
     returns ‘GSL_CONTINUE’ to signal that more iterations are required.
     Here, || \cdot || represents the Euclidean norm.  The input matrix
     ‘A’ may be in triplet or compressed format.

 -- Function: double gsl_splinalg_itersolve_normr (const
          gsl_splinalg_itersolve * w)

     This function returns the current residual norm ||r|| = ||A x -
     b||, which is updated after each call to *note
     gsl_splinalg_itersolve_iterate(): c31.


File: gsl-ref.info,  Node: Examples<35>,  Next: References and Further Reading<38>,  Prev: Sparse Iterative Solvers,  Up: Sparse Linear Algebra

45.3 Examples
=============

This example program demonstrates the sparse linear algebra routines on
the solution of a simple 1D Poisson equation on [0,1]:

     u''(x) = f(x) = -\pi^2 \sin(\pi x)

with boundary conditions u(0) = u(1) = 0.  The analytic solution of this
simple problem is u(x) = \sin{\pi x}.  We will solve this problem by
finite differencing the left hand side to give

     1/h^2 ( u_(i+1) - 2 u_i + u_(i-1) ) = f_i

Defining a grid of N points, h = 1/(N-1).  In the finite difference
equation above, u_0 = u_{N-1} = 0 are known from the boundary
conditions, so we will only put the equations for i = 1, ..., N-2 into
the matrix system.  The resulting (N-2) \times (N-2) matrix equation is

An example program which constructs and solves this system is given
below.  The system is solved using the iterative GMRES solver.  Here is
the output of the program:

     iter 0 residual = 4.297275996844e-11
     Converged

showing that the method converged in a single iteration.  The calculated
solution is shown in Fig.  %s.

[image src="sparse_poisson.png"]

Figure: Solution of PDE

     #include <stdio.h>
     #include <stdlib.h>
     #include <math.h>

     #include <gsl/gsl_math.h>
     #include <gsl/gsl_vector.h>
     #include <gsl/gsl_spmatrix.h>
     #include <gsl/gsl_splinalg.h>

     int
     main()
     {
       const size_t N = 100;                       /* number of grid points */
       const size_t n = N - 2;                     /* subtract 2 to exclude boundaries */
       const double h = 1.0 / (N - 1.0);           /* grid spacing */
       gsl_spmatrix *A = gsl_spmatrix_alloc(n ,n); /* triplet format */
       gsl_spmatrix *C;                            /* compressed format */
       gsl_vector *f = gsl_vector_alloc(n);        /* right hand side vector */
       gsl_vector *u = gsl_vector_alloc(n);        /* solution vector */
       size_t i;

       /* construct the sparse matrix for the finite difference equation */

       /* construct first row */
       gsl_spmatrix_set(A, 0, 0, -2.0);
       gsl_spmatrix_set(A, 0, 1, 1.0);

       /* construct rows [1:n-2] */
       for (i = 1; i < n - 1; ++i)
         {
           gsl_spmatrix_set(A, i, i + 1, 1.0);
           gsl_spmatrix_set(A, i, i, -2.0);
           gsl_spmatrix_set(A, i, i - 1, 1.0);
         }

       /* construct last row */
       gsl_spmatrix_set(A, n - 1, n - 1, -2.0);
       gsl_spmatrix_set(A, n - 1, n - 2, 1.0);

       /* scale by h^2 */
       gsl_spmatrix_scale(A, 1.0 / (h * h));

       /* construct right hand side vector */
       for (i = 0; i < n; ++i)
         {
           double xi = (i + 1) * h;
           double fi = -M_PI * M_PI * sin(M_PI * xi);
           gsl_vector_set(f, i, fi);
         }

       /* convert to compressed column format */
       C = gsl_spmatrix_ccs(A);

       /* now solve the system with the GMRES iterative solver */
       {
         const double tol = 1.0e-6;  /* solution relative tolerance */
         const size_t max_iter = 10; /* maximum iterations */
         const gsl_splinalg_itersolve_type *T = gsl_splinalg_itersolve_gmres;
         gsl_splinalg_itersolve *work =
           gsl_splinalg_itersolve_alloc(T, n, 0);
         size_t iter = 0;
         double residual;
         int status;

         /* initial guess u = 0 */
         gsl_vector_set_zero(u);

         /* solve the system A u = f */
         do
           {
             status = gsl_splinalg_itersolve_iterate(C, f, tol, u, work);

             /* print out residual norm ||A*u - f|| */
             residual = gsl_splinalg_itersolve_normr(work);
             fprintf(stderr, "iter %zu residual = %.12e\n", iter, residual);

             if (status == GSL_SUCCESS)
               fprintf(stderr, "Converged\n");
           }
         while (status == GSL_CONTINUE && ++iter < max_iter);

         /* output solution */
         for (i = 0; i < n; ++i)
           {
             double xi = (i + 1) * h;
             double u_exact = sin(M_PI * xi);
             double u_gsl = gsl_vector_get(u, i);

             printf("%f %.12e %.12e\n", xi, u_gsl, u_exact);
           }

         gsl_splinalg_itersolve_free(work);
       }

       gsl_spmatrix_free(A);
       gsl_spmatrix_free(C);
       gsl_vector_free(f);
       gsl_vector_free(u);

       return 0;
     } /* main() */


File: gsl-ref.info,  Node: References and Further Reading<38>,  Prev: Examples<35>,  Up: Sparse Linear Algebra

45.4 References and Further Reading
===================================

The implementation of the GMRES iterative solver closely follows the
publications

   * H. F. Walker, Implementation of the GMRES method using Householder
     transformations, SIAM J. Sci.  Stat.  Comput.  9(1), 1988.

   * Y. Saad, Iterative methods for sparse linear systems, 2nd edition,
     SIAM, 2003.


File: gsl-ref.info,  Node: Physical Constants,  Next: IEEE floating-point arithmetic,  Prev: Sparse Linear Algebra,  Up: Top

46 Physical Constants
*********************

This chapter describes macros for the values of physical constants, such
as the speed of light, c, and gravitational constant, G. The values are
available in different unit systems, including the standard MKSA system
(meters, kilograms, seconds, amperes) and the CGSM system (centimeters,
grams, seconds, gauss), which is commonly used in Astronomy.

The definitions of constants in the MKSA system are available in the
file ‘gsl_const_mksa.h’.  The constants in the CGSM system are defined
in ‘gsl_const_cgsm.h’.  Dimensionless constants, such as the fine
structure constant, which are pure numbers are defined in
‘gsl_const_num.h’.

The full list of constants is described briefly below.  Consult the
header files themselves for the values of the constants used in the
library.

* Menu:

* Fundamental Constants::
* Astronomy and Astrophysics::
* Atomic and Nuclear Physics::
* Measurement of Time::
* Imperial Units::
* Speed and Nautical Units::
* Printers Units::
* Volume, Area and Length: Volume Area and Length.
* Mass and Weight::
* Thermal Energy and Power::
* Pressure::
* Viscosity::
* Light and Illumination::
* Radioactivity::
* Force and Energy::
* Prefixes::
* Examples: Examples<36>.
* References and Further Reading: References and Further Reading<39>.


File: gsl-ref.info,  Node: Fundamental Constants,  Next: Astronomy and Astrophysics,  Up: Physical Constants

46.1 Fundamental Constants
==========================

 -- Macro: GSL_CONST_MKSA_SPEED_OF_LIGHT

     The speed of light in vacuum, c.

 -- Macro: GSL_CONST_MKSA_VACUUM_PERMEABILITY

     The permeability of free space, \mu_0.  This constant is defined in
     the MKSA system only.

 -- Macro: GSL_CONST_MKSA_VACUUM_PERMITTIVITY

     The permittivity of free space, \epsilon_0.  This constant is
     defined in the MKSA system only.

 -- Macro: GSL_CONST_MKSA_PLANCKS_CONSTANT_H

     Planck’s constant, h.

 -- Macro: GSL_CONST_MKSA_PLANCKS_CONSTANT_HBAR

     Planck’s constant divided by 2\pi, \hbar.

 -- Macro: GSL_CONST_NUM_AVOGADRO

     Avogadro’s number, N_a.

 -- Macro: GSL_CONST_MKSA_FARADAY

     The molar charge of 1 Faraday.

 -- Macro: GSL_CONST_MKSA_BOLTZMANN

     The Boltzmann constant, k.

 -- Macro: GSL_CONST_MKSA_MOLAR_GAS

     The molar gas constant, R_0.

 -- Macro: GSL_CONST_MKSA_STANDARD_GAS_VOLUME

     The standard gas volume, V_0.

 -- Macro: GSL_CONST_MKSA_STEFAN_BOLTZMANN_CONSTANT

     The Stefan-Boltzmann radiation constant, \sigma.

 -- Macro: GSL_CONST_MKSA_GAUSS

     The magnetic field of 1 Gauss.


File: gsl-ref.info,  Node: Astronomy and Astrophysics,  Next: Atomic and Nuclear Physics,  Prev: Fundamental Constants,  Up: Physical Constants

46.2 Astronomy and Astrophysics
===============================

 -- Macro: GSL_CONST_MKSA_ASTRONOMICAL_UNIT

     The length of 1 astronomical unit (mean earth-sun distance), au.

 -- Macro: GSL_CONST_MKSA_GRAVITATIONAL_CONSTANT

     The gravitational constant, G.

 -- Macro: GSL_CONST_MKSA_LIGHT_YEAR

     The distance of 1 light-year, ly.

 -- Macro: GSL_CONST_MKSA_PARSEC

     The distance of 1 parsec, pc.

 -- Macro: GSL_CONST_MKSA_GRAV_ACCEL

     The standard gravitational acceleration on Earth, g.

 -- Macro: GSL_CONST_MKSA_SOLAR_MASS

     The mass of the Sun.


File: gsl-ref.info,  Node: Atomic and Nuclear Physics,  Next: Measurement of Time,  Prev: Astronomy and Astrophysics,  Up: Physical Constants

46.3 Atomic and Nuclear Physics
===============================

 -- Macro: GSL_CONST_MKSA_ELECTRON_CHARGE

     The charge of the electron, e.

 -- Macro: GSL_CONST_MKSA_ELECTRON_VOLT

     The energy of 1 electron volt, eV.

 -- Macro: GSL_CONST_MKSA_UNIFIED_ATOMIC_MASS

     The unified atomic mass, amu.

 -- Macro: GSL_CONST_MKSA_MASS_ELECTRON

     The mass of the electron, m_e.

 -- Macro: GSL_CONST_MKSA_MASS_MUON

     The mass of the muon, m_\mu.

 -- Macro: GSL_CONST_MKSA_MASS_PROTON

     The mass of the proton, m_p.

 -- Macro: GSL_CONST_MKSA_MASS_NEUTRON

     The mass of the neutron, m_n.

 -- Macro: GSL_CONST_NUM_FINE_STRUCTURE

     The electromagnetic fine structure constant \alpha.

 -- Macro: GSL_CONST_MKSA_RYDBERG

     The Rydberg constant, Ry, in units of energy.  This is related to
     the Rydberg inverse wavelength R_\infty by Ry = h c R_\infty.

 -- Macro: GSL_CONST_MKSA_BOHR_RADIUS

     The Bohr radius, a_0.

 -- Macro: GSL_CONST_MKSA_ANGSTROM

     The length of 1 angstrom.

 -- Macro: GSL_CONST_MKSA_BARN

     The area of 1 barn.

 -- Macro: GSL_CONST_MKSA_BOHR_MAGNETON

     The Bohr Magneton, \mu_B.

 -- Macro: GSL_CONST_MKSA_NUCLEAR_MAGNETON

     The Nuclear Magneton, \mu_N.

 -- Macro: GSL_CONST_MKSA_ELECTRON_MAGNETIC_MOMENT

     The absolute value of the magnetic moment of the electron, \mu_e.
     The physical magnetic moment of the electron is negative.

 -- Macro: GSL_CONST_MKSA_PROTON_MAGNETIC_MOMENT

     The magnetic moment of the proton, \mu_p.

 -- Macro: GSL_CONST_MKSA_THOMSON_CROSS_SECTION

     The Thomson cross section, \sigma_T.

 -- Macro: GSL_CONST_MKSA_DEBYE

     The electric dipole moment of 1 Debye, D.


File: gsl-ref.info,  Node: Measurement of Time,  Next: Imperial Units,  Prev: Atomic and Nuclear Physics,  Up: Physical Constants

46.4 Measurement of Time
========================

 -- Macro: GSL_CONST_MKSA_MINUTE

     The number of seconds in 1 minute.

 -- Macro: GSL_CONST_MKSA_HOUR

     The number of seconds in 1 hour.

 -- Macro: GSL_CONST_MKSA_DAY

     The number of seconds in 1 day.

 -- Macro: GSL_CONST_MKSA_WEEK

     The number of seconds in 1 week.


File: gsl-ref.info,  Node: Imperial Units,  Next: Speed and Nautical Units,  Prev: Measurement of Time,  Up: Physical Constants

46.5 Imperial Units
===================

 -- Macro: GSL_CONST_MKSA_INCH

     The length of 1 inch.

 -- Macro: GSL_CONST_MKSA_FOOT

     The length of 1 foot.

 -- Macro: GSL_CONST_MKSA_YARD

     The length of 1 yard.

 -- Macro: GSL_CONST_MKSA_MILE

     The length of 1 mile.

 -- Macro: GSL_CONST_MKSA_MIL

     The length of 1 mil (1/1000th of an inch).


File: gsl-ref.info,  Node: Speed and Nautical Units,  Next: Printers Units,  Prev: Imperial Units,  Up: Physical Constants

46.6 Speed and Nautical Units
=============================

 -- Macro: GSL_CONST_MKSA_KILOMETERS_PER_HOUR

     The speed of 1 kilometer per hour.

 -- Macro: GSL_CONST_MKSA_MILES_PER_HOUR

     The speed of 1 mile per hour.

 -- Macro: GSL_CONST_MKSA_NAUTICAL_MILE

     The length of 1 nautical mile.

 -- Macro: GSL_CONST_MKSA_FATHOM

     The length of 1 fathom.

 -- Macro: GSL_CONST_MKSA_KNOT

     The speed of 1 knot.


File: gsl-ref.info,  Node: Printers Units,  Next: Volume Area and Length,  Prev: Speed and Nautical Units,  Up: Physical Constants

46.7 Printers Units
===================

 -- Macro: GSL_CONST_MKSA_POINT

     The length of 1 printer’s point (1/72 inch).

 -- Macro: GSL_CONST_MKSA_TEXPOINT

     The length of 1 TeX point (1/72.27 inch).


File: gsl-ref.info,  Node: Volume Area and Length,  Next: Mass and Weight,  Prev: Printers Units,  Up: Physical Constants

46.8 Volume, Area and Length
============================

 -- Macro: GSL_CONST_MKSA_MICRON

     The length of 1 micron.

 -- Macro: GSL_CONST_MKSA_HECTARE

     The area of 1 hectare.

 -- Macro: GSL_CONST_MKSA_ACRE

     The area of 1 acre.

 -- Macro: GSL_CONST_MKSA_LITER

     The volume of 1 liter.

 -- Macro: GSL_CONST_MKSA_US_GALLON

     The volume of 1 US gallon.

 -- Macro: GSL_CONST_MKSA_CANADIAN_GALLON

     The volume of 1 Canadian gallon.

 -- Macro: GSL_CONST_MKSA_UK_GALLON

     The volume of 1 UK gallon.

 -- Macro: GSL_CONST_MKSA_QUART

     The volume of 1 quart.

 -- Macro: GSL_CONST_MKSA_PINT

     The volume of 1 pint.


File: gsl-ref.info,  Node: Mass and Weight,  Next: Thermal Energy and Power,  Prev: Volume Area and Length,  Up: Physical Constants

46.9 Mass and Weight
====================

 -- Macro: GSL_CONST_MKSA_POUND_MASS

     The mass of 1 pound.

 -- Macro: GSL_CONST_MKSA_OUNCE_MASS

     The mass of 1 ounce.

 -- Macro: GSL_CONST_MKSA_TON

     The mass of 1 ton.

 -- Macro: GSL_CONST_MKSA_METRIC_TON

     The mass of 1 metric ton (1000 kg).

 -- Macro: GSL_CONST_MKSA_UK_TON

     The mass of 1 UK ton.

 -- Macro: GSL_CONST_MKSA_TROY_OUNCE

     The mass of 1 troy ounce.

 -- Macro: GSL_CONST_MKSA_CARAT

     The mass of 1 carat.

 -- Macro: GSL_CONST_MKSA_GRAM_FORCE

     The force of 1 gram weight.

 -- Macro: GSL_CONST_MKSA_POUND_FORCE

     The force of 1 pound weight.

 -- Macro: GSL_CONST_MKSA_KILOPOUND_FORCE

     The force of 1 kilopound weight.

 -- Macro: GSL_CONST_MKSA_POUNDAL

     The force of 1 poundal.


File: gsl-ref.info,  Node: Thermal Energy and Power,  Next: Pressure,  Prev: Mass and Weight,  Up: Physical Constants

46.10 Thermal Energy and Power
==============================

 -- Macro: GSL_CONST_MKSA_CALORIE

     The energy of 1 calorie.

 -- Macro: GSL_CONST_MKSA_BTU

     The energy of 1 British Thermal Unit, btu.

 -- Macro: GSL_CONST_MKSA_THERM

     The energy of 1 Therm.

 -- Macro: GSL_CONST_MKSA_HORSEPOWER

     The power of 1 horsepower.


File: gsl-ref.info,  Node: Pressure,  Next: Viscosity,  Prev: Thermal Energy and Power,  Up: Physical Constants

46.11 Pressure
==============

 -- Macro: GSL_CONST_MKSA_BAR

     The pressure of 1 bar.

 -- Macro: GSL_CONST_MKSA_STD_ATMOSPHERE

     The pressure of 1 standard atmosphere.

 -- Macro: GSL_CONST_MKSA_TORR

     The pressure of 1 torr.

 -- Macro: GSL_CONST_MKSA_METER_OF_MERCURY

     The pressure of 1 meter of mercury.

 -- Macro: GSL_CONST_MKSA_INCH_OF_MERCURY

     The pressure of 1 inch of mercury.

 -- Macro: GSL_CONST_MKSA_INCH_OF_WATER

     The pressure of 1 inch of water.

 -- Macro: GSL_CONST_MKSA_PSI

     The pressure of 1 pound per square inch.


File: gsl-ref.info,  Node: Viscosity,  Next: Light and Illumination,  Prev: Pressure,  Up: Physical Constants

46.12 Viscosity
===============

 -- Macro: GSL_CONST_MKSA_POISE

     The dynamic viscosity of 1 poise.

 -- Macro: GSL_CONST_MKSA_STOKES

     The kinematic viscosity of 1 stokes.


File: gsl-ref.info,  Node: Light and Illumination,  Next: Radioactivity,  Prev: Viscosity,  Up: Physical Constants

46.13 Light and Illumination
============================

 -- Macro: GSL_CONST_MKSA_STILB

     The luminance of 1 stilb.

 -- Macro: GSL_CONST_MKSA_LUMEN

     The luminous flux of 1 lumen.

 -- Macro: GSL_CONST_MKSA_LUX

     The illuminance of 1 lux.

 -- Macro: GSL_CONST_MKSA_PHOT

     The illuminance of 1 phot.

 -- Macro: GSL_CONST_MKSA_FOOTCANDLE

     The illuminance of 1 footcandle.

 -- Macro: GSL_CONST_MKSA_LAMBERT

     The luminance of 1 lambert.

 -- Macro: GSL_CONST_MKSA_FOOTLAMBERT

     The luminance of 1 footlambert.


File: gsl-ref.info,  Node: Radioactivity,  Next: Force and Energy,  Prev: Light and Illumination,  Up: Physical Constants

46.14 Radioactivity
===================

 -- Macro: GSL_CONST_MKSA_CURIE

     The activity of 1 curie.

 -- Macro: GSL_CONST_MKSA_ROENTGEN

     The exposure of 1 roentgen.

 -- Macro: GSL_CONST_MKSA_RAD

     The absorbed dose of 1 rad.


File: gsl-ref.info,  Node: Force and Energy,  Next: Prefixes,  Prev: Radioactivity,  Up: Physical Constants

46.15 Force and Energy
======================

 -- Macro: GSL_CONST_MKSA_NEWTON

     The SI unit of force, 1 Newton.

 -- Macro: GSL_CONST_MKSA_DYNE

     The force of 1 Dyne = 10^{-5} Newton.

 -- Macro: GSL_CONST_MKSA_JOULE

     The SI unit of energy, 1 Joule.

 -- Macro: GSL_CONST_MKSA_ERG

     The energy 1 erg = 10^{-7} Joule.


File: gsl-ref.info,  Node: Prefixes,  Next: Examples<36>,  Prev: Force and Energy,  Up: Physical Constants

46.16 Prefixes
==============

These constants are dimensionless scaling factors.

 -- Macro: GSL_CONST_NUM_YOTTA

     10^{24}

 -- Macro: GSL_CONST_NUM_ZETTA

     10^{21}

 -- Macro: GSL_CONST_NUM_EXA

     10^{18}

 -- Macro: GSL_CONST_NUM_PETA

     10^{15}

 -- Macro: GSL_CONST_NUM_TERA

     10^{12}

 -- Macro: GSL_CONST_NUM_GIGA

     10^9

 -- Macro: GSL_CONST_NUM_MEGA

     10^6

 -- Macro: GSL_CONST_NUM_KILO

     10^3

 -- Macro: GSL_CONST_NUM_MILLI

     10^{-3}

 -- Macro: GSL_CONST_NUM_MICRO

     10^{-6}

 -- Macro: GSL_CONST_NUM_NANO

     10^{-9}

 -- Macro: GSL_CONST_NUM_PICO

     10^{-12}

 -- Macro: GSL_CONST_NUM_FEMTO

     10^{-15}

 -- Macro: GSL_CONST_NUM_ATTO

     10^{-18}

 -- Macro: GSL_CONST_NUM_ZEPTO

     10^{-21}

 -- Macro: GSL_CONST_NUM_YOCTO

     10^{-24}


File: gsl-ref.info,  Node: Examples<36>,  Next: References and Further Reading<39>,  Prev: Prefixes,  Up: Physical Constants

46.17 Examples
==============

The following program demonstrates the use of the physical constants in
a calculation.  In this case, the goal is to calculate the range of
light-travel times from Earth to Mars.

The required data is the average distance of each planet from the Sun in
astronomical units (the eccentricities and inclinations of the orbits
will be neglected for the purposes of this calculation).  The average
radius of the orbit of Mars is 1.52 astronomical units, and for the
orbit of Earth it is 1 astronomical unit (by definition).  These values
are combined with the MKSA values of the constants for the speed of
light and the length of an astronomical unit to produce a result for the
shortest and longest light-travel times in seconds.  The figures are
converted into minutes before being displayed.

     #include <stdio.h>
     #include <gsl/gsl_const_mksa.h>

     int
     main (void)
     {
       double c  = GSL_CONST_MKSA_SPEED_OF_LIGHT;
       double au = GSL_CONST_MKSA_ASTRONOMICAL_UNIT;
       double minutes = GSL_CONST_MKSA_MINUTE;

       /* distance stored in meters */
       double r_earth = 1.00 * au;
       double r_mars  = 1.52 * au;

       double t_min, t_max;

       t_min = (r_mars - r_earth) / c;
       t_max = (r_mars + r_earth) / c;

       printf ("light travel time from Earth to Mars:\n");
       printf ("minimum = %.1f minutes\n", t_min / minutes);
       printf ("maximum = %.1f minutes\n", t_max / minutes);

       return 0;
     }

Here is the output from the program,

     light travel time from Earth to Mars:
     minimum = 4.3 minutes
     maximum = 21.0 minutes


File: gsl-ref.info,  Node: References and Further Reading<39>,  Prev: Examples<36>,  Up: Physical Constants

46.18 References and Further Reading
====================================

The authoritative sources for physical constants are the 2006 CODATA
recommended values, published in the article below.  Further information
on the values of physical constants is also available from the NIST
website.

   * P.J. Mohr, B.N. Taylor, D.B. Newell, “CODATA Recommended Values of
     the Fundamental Physical Constants: 2006”, Reviews of Modern
     Physics, 80(2), pp.  633–730 (2008).

   * ‘http://www.physics.nist.gov/cuu/Constants/index.html’

   * ‘http://physics.nist.gov/Pubs/SP811/appenB9.html’


File: gsl-ref.info,  Node: IEEE floating-point arithmetic,  Next: Debugging Numerical Programs,  Prev: Physical Constants,  Up: Top

47 IEEE floating-point arithmetic
*********************************

This chapter describes functions for examining the representation of
floating point numbers and controlling the floating point environment of
your program.  The functions described in this chapter are declared in
the header file ‘gsl_ieee_utils.h’.

* Menu:

* Representation of floating point numbers::
* Setting up your IEEE environment::
* References and Further Reading: References and Further Reading<40>.


File: gsl-ref.info,  Node: Representation of floating point numbers,  Next: Setting up your IEEE environment,  Up: IEEE floating-point arithmetic

47.1 Representation of floating point numbers
=============================================

The IEEE Standard for Binary Floating-Point Arithmetic defines binary
formats for single and double precision numbers.  Each number is
composed of three parts: a `sign bit' (s), an `exponent' (E) and a
`fraction' (f).  The numerical value of the combination (s,E,f) is given
by the following formula,

     (-1)^s (1.fffff...) 2^E

The sign bit is either zero or one.  The exponent ranges from a minimum
value E_{min} to a maximum value E_{max} depending on the precision.
The exponent is converted to an unsigned number e, known as the `biased
exponent', for storage by adding a `bias' parameter,

     e = E + bias

The sequence fffff... represents the digits of the binary fraction f.
The binary digits are stored in `normalized form', by adjusting the
exponent to give a leading digit of 1.  Since the leading digit is
always 1 for normalized numbers it is assumed implicitly and does not
have to be stored.  Numbers smaller than 2^{E_{min}} are be stored in
`denormalized form' with a leading zero,

     (-1)^s (0.fffff...) 2^(E_min)

This allows gradual underflow down to 2^{E_{min} - p} for p bits of
precision.  A zero is encoded with the special exponent of 2^{E_{min}-1}
and infinities with the exponent of 2^{E_{max}+1}.

The format for single precision numbers uses 32 bits divided in the
following way:

     seeeeeeeefffffffffffffffffffffff

     s = sign bit, 1 bit
     e = exponent, 8 bits  (E_min=-126, E_max=127, bias=127)
     f = fraction, 23 bits

The format for double precision numbers uses 64 bits divided in the
following way:

     seeeeeeeeeeeffffffffffffffffffffffffffffffffffffffffffffffffffff

     s = sign bit, 1 bit
     e = exponent, 11 bits  (E_min=-1022, E_max=1023, bias=1023)
     f = fraction, 52 bits

It is often useful to be able to investigate the behavior of a
calculation at the bit-level and the library provides functions for
printing the IEEE representations in a human-readable form.

 -- Function: void gsl_ieee_fprintf_float (FILE * stream, const float
          * x)
 -- Function: void gsl_ieee_fprintf_double (FILE * stream, const double
          * x)

     These functions output a formatted version of the IEEE
     floating-point number pointed to by ‘x’ to the stream ‘stream’.  A
     pointer is used to pass the number indirectly, to avoid any
     undesired promotion from ‘float’ to ‘double’.  The output takes one
     of the following forms,

     ‘NaN’

          the Not-a-Number symbol

     ‘Inf, -Inf’

          positive or negative infinity

     ‘1.fffff...*2^E, -1.fffff...*2^E’

          a normalized floating point number

     ‘0.fffff...*2^E, -0.fffff...*2^E’

          a denormalized floating point number

     ‘0, -0’

          positive or negative zero

     The output can be used directly in GNU Emacs Calc mode by preceding
     it with ‘2#’ to indicate binary.

 -- Function: void gsl_ieee_printf_float (const float * x)
 -- Function: void gsl_ieee_printf_double (const double * x)

     These functions output a formatted version of the IEEE
     floating-point number pointed to by ‘x’ to the stream ‘stdout’.

The following program demonstrates the use of the functions by printing
the single and double precision representations of the fraction 1/3.
For comparison the representation of the value promoted from single to
double precision is also printed.

     #include <stdio.h>
     #include <gsl/gsl_ieee_utils.h>

     int
     main (void)
     {
       float f = 1.0/3.0;
       double d = 1.0/3.0;

       double fd = f; /* promote from float to double */

       printf (" f="); gsl_ieee_printf_float(&f);
       printf ("\n");

       printf ("fd="); gsl_ieee_printf_double(&fd);
       printf ("\n");

       printf (" d="); gsl_ieee_printf_double(&d);
       printf ("\n");

       return 0;
     }

The binary representation of 1/3 is 0.01010101....  The output below
shows that the IEEE format normalizes this fraction to give a leading
digit of 1:

      f= 1.01010101010101010101011*2^-2
     fd= 1.0101010101010101010101100000000000000000000000000000*2^-2
      d= 1.0101010101010101010101010101010101010101010101010101*2^-2

The output also shows that a single-precision number is promoted to
double-precision by adding zeros in the binary representation.


File: gsl-ref.info,  Node: Setting up your IEEE environment,  Next: References and Further Reading<40>,  Prev: Representation of floating point numbers,  Up: IEEE floating-point arithmetic

47.2 Setting up your IEEE environment
=====================================

The IEEE standard defines several `modes' for controlling the behavior
of floating point operations.  These modes specify the important
properties of computer arithmetic: the direction used for rounding (e.g.
whether numbers should be rounded up, down or to the nearest number),
the rounding precision and how the program should handle arithmetic
exceptions, such as division by zero.

Many of these features can now be controlled via standard functions such
as ‘fpsetround()’, which should be used whenever they are available.
Unfortunately in the past there has been no universal API for
controlling their behavior—each system has had its own low-level way of
accessing them.  To help you write portable programs GSL allows you to
specify modes in a platform-independent way using the environment
variable *note GSL_IEEE_MODE: cca.  The library then takes care of all
the necessary machine-specific initializations for you when you call the
function *note gsl_ieee_env_setup(): ccb.

 -- Macro: GSL_IEEE_MODE

     Environment variable which specifies IEEE mode.

 -- Function: void gsl_ieee_env_setup ()

     This function reads the environment variable *note GSL_IEEE_MODE:
     cca. and attempts to set up the corresponding specified IEEE modes.
     The environment variable should be a list of keywords, separated by
     commas, like this:

          GSL_IEEE_MODE = "keyword, keyword, ..."

     where ‘keyword’ is one of the following mode-names:

          single-precision
          double-precision
          extended-precision
          round-to-nearest
          round-down
          round-up
          round-to-zero
          mask-all
          mask-invalid
          mask-denormalized
          mask-division-by-zero
          mask-overflow
          mask-underflow
          trap-inexact
          trap-common

     If *note GSL_IEEE_MODE: cca. is empty or undefined then the
     function returns immediately and no attempt is made to change the
     system’s IEEE mode.  When the modes from *note GSL_IEEE_MODE: cca.
     are turned on the function prints a short message showing the new
     settings to remind you that the results of the program will be
     affected.

     If the requested modes are not supported by the platform being used
     then the function calls the error handler and returns an error code
     of ‘GSL_EUNSUP’.

     When options are specified using this method, the resulting mode is
     based on a default setting of the highest available precision
     (double precision or extended precision, depending on the platform)
     in round-to-nearest mode, with all exceptions enabled apart from
     the INEXACT exception.  The INEXACT exception is generated whenever
     rounding occurs, so it must generally be disabled in typical
     scientific calculations.  All other floating-point exceptions are
     enabled by default, including underflows and the use of
     denormalized numbers, for safety.  They can be disabled with the
     individual ‘mask-’ settings or together using ‘mask-all’.

     The following adjusted combination of modes is convenient for many
     purposes:

          GSL_IEEE_MODE="double-precision,"\
                          "mask-underflow,"\
                            "mask-denormalized"

     This choice ignores any errors relating to small numbers (either
     denormalized, or underflowing to zero) but traps overflows,
     division by zero and invalid operations.

     Note that on the x86 series of processors this function sets both
     the original x87 mode and the newer MXCSR mode, which controls SSE
     floating-point operations.  The SSE floating-point units do not
     have a precision-control bit, and always work in double-precision.
     The single-precision and extended-precision keywords have no effect
     in this case.

To demonstrate the effects of different rounding modes consider the
following program which computes e, the base of natural logarithms, by
summing a rapidly-decreasing series,

     e = 1 + 1/2! + 1/3! + 1/4! + ...
       = 2.71828182846...

     #include <stdio.h>
     #include <gsl/gsl_math.h>
     #include <gsl/gsl_ieee_utils.h>

     int
     main (void)
     {
       double x = 1, oldsum = 0, sum = 0;
       int i = 0;

       gsl_ieee_env_setup (); /* read GSL_IEEE_MODE */

       do
         {
           i++;

           oldsum = sum;
           sum += x;
           x = x / i;

           printf ("i=%2d sum=%.18f error=%g\n",
                   i, sum, sum - M_E);

           if (i > 30)
              break;
         }
       while (sum != oldsum);

       return 0;
     }

Here are the results of running the program in ‘round-to-nearest’ mode.
This is the IEEE default so it isn’t really necessary to specify it
here:

     $ GSL_IEEE_MODE="round-to-nearest" ./a.out
     i= 1 sum=1.000000000000000000 error=-1.71828
     i= 2 sum=2.000000000000000000 error=-0.718282
     ....
     i=18 sum=2.718281828459045535 error=4.44089e-16
     i=19 sum=2.718281828459045535 error=4.44089e-16

After nineteen terms the sum converges to within 4 \times 10^{-16} of
the correct value.  If we now change the rounding mode to ‘round-down’
the final result is less accurate:

     $ GSL_IEEE_MODE="round-down" ./a.out
     i= 1 sum=1.000000000000000000 error=-1.71828
     ....
     i=19 sum=2.718281828459041094 error=-3.9968e-15

The result is about 4 \times 10^{-15} below the correct value, an order
of magnitude worse than the result obtained in the ‘round-to-nearest’
mode.

If we change to rounding mode to ‘round-up’ then the final result is
higher than the correct value (when we add each term to the sum the
final result is always rounded up, which increases the sum by at least
one tick until the added term underflows to zero).  To avoid this
problem we would need to use a safer converge criterion, such as ‘while
(fabs(sum - oldsum) > epsilon)’, with a suitably chosen value of
epsilon.

Finally we can see the effect of computing the sum using
single-precision rounding, in the default ‘round-to-nearest’ mode.  In
this case the program thinks it is still using double precision numbers
but the CPU rounds the result of each floating point operation to
single-precision accuracy.  This simulates the effect of writing the
program using single-precision ‘float’ variables instead of ‘double’
variables.  The iteration stops after about half the number of
iterations and the final result is much less accurate:

     $ GSL_IEEE_MODE="single-precision" ./a.out
     ....
     i=12 sum=2.718281984329223633 error=1.5587e-07

with an error of O(10^{-7}), which corresponds to single precision
accuracy (about 1 part in 10^7).  Continuing the iterations further does
not decrease the error because all the subsequent results are rounded to
the same value.


File: gsl-ref.info,  Node: References and Further Reading<40>,  Prev: Setting up your IEEE environment,  Up: IEEE floating-point arithmetic

47.3 References and Further Reading
===================================

The reference for the IEEE standard is,

   * ANSI/IEEE Std 754-1985, IEEE Standard for Binary Floating-Point
     Arithmetic.

A more pedagogical introduction to the standard can be found in the
following paper,

   * David Goldberg: What Every Computer Scientist Should Know About
     Floating-Point Arithmetic.  `ACM Computing Surveys', Vol.: 23, No.:
     1 (March 1991), pages 5–48.

   * Corrigendum: `ACM Computing Surveys', Vol.: 23, No.: 3 (September
     1991), page 413.  and see also the sections by B. A. Wichmann and
     Charles B. Dunham in Surveyor’s Forum: “What Every Computer
     Scientist Should Know About Floating-Point Arithmetic”.  `ACM
     Computing Surveys', Vol.: 24, No.: 3 (September 1992), page 319.

A detailed textbook on IEEE arithmetic and its practical use is
available from SIAM Press,

   * Michael L. Overton, `Numerical Computing with IEEE Floating Point
     Arithmetic', SIAM Press, ISBN 0898715717.


File: gsl-ref.info,  Node: Debugging Numerical Programs,  Next: Contributors to GSL,  Prev: IEEE floating-point arithmetic,  Up: Top

48 Debugging Numerical Programs
*******************************

This chapter describes some tips and tricks for debugging numerical
programs which use GSL.

* Menu:

* Using gdb::
* Examining floating point registers::
* Handling floating point exceptions::
* GCC warning options for numerical programs::
* References and Further Reading: References and Further Reading<41>.


File: gsl-ref.info,  Node: Using gdb,  Next: Examining floating point registers,  Up: Debugging Numerical Programs

48.1 Using gdb
==============

Any errors reported by the library are passed to the function
‘gsl_error()’.  By running your programs under gdb and setting a
breakpoint in this function you can automatically catch any library
errors.  You can add a breakpoint for every session by putting:

     break gsl_error

into your ‘.gdbinit’ file in the directory where your program is
started.

If the breakpoint catches an error then you can use a backtrace (‘bt’)
to see the call-tree, and the arguments which possibly caused the error.
By moving up into the calling function you can investigate the values of
variables at that point.  Here is an example from the program
‘fft/test_trap’, which contains the following line:

     status = gsl_fft_complex_wavetable_alloc (0, &complex_wavetable);

The function *note gsl_fft_complex_wavetable_alloc(): 5f7. takes the
length of an FFT as its first argument.  When this line is executed an
error will be generated because the length of an FFT is not allowed to
be zero.

To debug this problem we start ‘gdb’, using the file ‘.gdbinit’ to
define a breakpoint in ‘gsl_error()’:

     $ gdb test_trap

     GDB is free software and you are welcome to distribute copies
     of it under certain conditions; type "show copying" to see
     the conditions.  There is absolutely no warranty for GDB;
     type "show warranty" for details.  GDB 4.16 (i586-debian-linux),
     Copyright 1996 Free Software Foundation, Inc.

     Breakpoint 1 at 0x8050b1e: file error.c, line 14.

When we run the program this breakpoint catches the error and shows the
reason for it:

     (gdb) run
     Starting program: test_trap

     Breakpoint 1, gsl_error (reason=0x8052b0d
         "length n must be positive integer",
         file=0x8052b04 "c_init.c", line=108, gsl_errno=1)
         at error.c:14
     14        if (gsl_error_handler)

The first argument of ‘gsl_error()’ is always a string describing the
error.  Now we can look at the backtrace to see what caused the problem:

     (gdb) bt
     #0  gsl_error (reason=0x8052b0d
         "length n must be positive integer",
         file=0x8052b04 "c_init.c", line=108, gsl_errno=1)
         at error.c:14
     #1  0x8049376 in gsl_fft_complex_wavetable_alloc (n=0,
         wavetable=0xbffff778) at c_init.c:108
     #2  0x8048a00 in main (argc=1, argv=0xbffff9bc)
         at test_trap.c:94
     #3  0x80488be in ___crt_dummy__ ()

We can see that the error was generated in the function *note
gsl_fft_complex_wavetable_alloc(): 5f7. when it was called with an
argument of ‘n = 0’.  The original call came from line 94 in the file
‘test_trap.c’.

By moving up to the level of the original call we can find the line that
caused the error:

     (gdb) up
     #1  0x8049376 in gsl_fft_complex_wavetable_alloc (n=0,
         wavetable=0xbffff778) at c_init.c:108
     108   GSL_ERROR ("length n must be positive integer", GSL_EDOM);
     (gdb) up
     #2  0x8048a00 in main (argc=1, argv=0xbffff9bc)
       at test_trap.c:94
     94    status = gsl_fft_complex_wavetable_alloc (0,
             &complex_wavetable);

Thus we have found the line that caused the problem.  From this point we
could also print out the values of other variables such as
‘complex_wavetable’.


File: gsl-ref.info,  Node: Examining floating point registers,  Next: Handling floating point exceptions,  Prev: Using gdb,  Up: Debugging Numerical Programs

48.2 Examining floating point registers
=======================================

The contents of floating point registers can be examined using the
command ‘info float’ (on supported platforms):

     (gdb) info float
          st0: 0xc4018b895aa17a945000  Valid Normal -7.838871e+308
          st1: 0x3ff9ea3f50e4d7275000  Valid Normal 0.0285946
          st2: 0x3fe790c64ce27dad4800  Valid Normal 6.7415931e-08
          st3: 0x3ffaa3ef0df6607d7800  Spec  Normal 0.0400229
          st4: 0x3c028000000000000000  Valid Normal 4.4501477e-308
          st5: 0x3ffef5412c22219d9000  Zero  Normal 0.9580257
          st6: 0x3fff8000000000000000  Valid Normal 1
          st7: 0xc4028b65a1f6d243c800  Valid Normal -1.566206e+309
        fctrl: 0x0272 53 bit; NEAR; mask DENOR UNDER LOS;
        fstat: 0xb9ba flags 0001; top 7; excep DENOR OVERF UNDER LOS
         ftag: 0x3fff
          fip: 0x08048b5c
          fcs: 0x051a0023
       fopoff: 0x08086820
       fopsel: 0x002b

Individual registers can be examined using the variables ‘$reg’, where
‘reg’ is the register name:

     (gdb) p $st1
     $1 = 0.02859464454261210347719


File: gsl-ref.info,  Node: Handling floating point exceptions,  Next: GCC warning options for numerical programs,  Prev: Examining floating point registers,  Up: Debugging Numerical Programs

48.3 Handling floating point exceptions
=======================================

It is possible to stop the program whenever a ‘SIGFPE’ floating point
exception occurs.  This can be useful for finding the cause of an
unexpected infinity or ‘NaN’.  The current handler settings can be shown
with the command ‘info signal SIGFPE’:

     (gdb) info signal SIGFPE
     Signal  Stop  Print  Pass to program Description
     SIGFPE  Yes   Yes    Yes             Arithmetic exception

Unless the program uses a signal handler the default setting should be
changed so that SIGFPE is not passed to the program, as this would cause
it to exit.  The command ‘handle SIGFPE stop nopass’ prevents this:

     (gdb) handle SIGFPE stop nopass
     Signal  Stop  Print  Pass to program Description
     SIGFPE  Yes   Yes    No              Arithmetic exception

Depending on the platform it may be necessary to instruct the kernel to
generate signals for floating point exceptions.  For programs using GSL
this can be achieved using the *note GSL_IEEE_MODE: cca. environment
variable in conjunction with the function *note gsl_ieee_env_setup():
ccb. as described in *note IEEE floating-point arithmetic: cc2.:

     (gdb) set env GSL_IEEE_MODE=double-precision


File: gsl-ref.info,  Node: GCC warning options for numerical programs,  Next: References and Further Reading<41>,  Prev: Handling floating point exceptions,  Up: Debugging Numerical Programs

48.4 GCC warning options for numerical programs
===============================================

Writing reliable numerical programs in C requires great care.  The
following GCC warning options are recommended when compiling numerical
programs:

     gcc -ansi -pedantic -Werror -Wall -W
       -Wmissing-prototypes -Wstrict-prototypes
       -Wconversion -Wshadow -Wpointer-arith
       -Wcast-qual -Wcast-align
       -Wwrite-strings -Wnested-externs
       -fshort-enums -fno-common -Dinline= -g -O2

For details of each option consult the manual `Using and Porting GCC'.
The following table gives a brief explanation of what types of errors
these options catch.

‘-ansi -pedantic’

     Use ANSI C, and reject any non-ANSI extensions.  These flags help
     in writing portable programs that will compile on other systems.

‘-Werror’

     Consider warnings to be errors, so that compilation stops.  This
     prevents warnings from scrolling off the top of the screen and
     being lost.  You won’t be able to compile the program until it is
     completely warning-free.

‘-Wall’

     This turns on a set of warnings for common programming problems.
     You need ‘-Wall’, but it is not enough on its own.

‘-O2’

     Turn on optimization.  The warnings for uninitialized variables in
     ‘-Wall’ rely on the optimizer to analyze the code.  If there is no
     optimization then these warnings aren’t generated.

‘-W’

     This turns on some extra warnings not included in ‘-Wall’, such as
     missing return values and comparisons between signed and unsigned
     integers.

‘-Wmissing-prototypes -Wstrict-prototypes’

     Warn if there are any missing or inconsistent prototypes.  Without
     prototypes it is harder to detect problems with incorrect
     arguments.

‘-Wconversion’

     The main use of this option is to warn about conversions from
     signed to unsigned integers.  For example, ‘unsigned int x = -1’.
     If you need to perform such a conversion you can use an explicit
     cast.

‘-Wshadow’

     This warns whenever a local variable shadows another local
     variable.  If two variables have the same name then it is a
     potential source of confusion.

‘-Wpointer-arith -Wcast-qual -Wcast-align’

     These options warn if you try to do pointer arithmetic for types
     which don’t have a size, such as ‘void’, if you remove a ‘const’
     cast from a pointer, or if you cast a pointer to a type which has a
     different size, causing an invalid alignment.

‘-Wwrite-strings’

     This option gives string constants a ‘const’ qualifier so that it
     will be a compile-time error to attempt to overwrite them.

‘-fshort-enums’

     This option makes the type of ‘enum’ as short as possible.
     Normally this makes an ‘enum’ different from an ‘int’.
     Consequently any attempts to assign a pointer-to-int to a
     pointer-to-enum will generate a cast-alignment warning.

‘-fno-common’

     This option prevents global variables being simultaneously defined
     in different object files (you get an error at link time).  Such a
     variable should be defined in one file and referred to in other
     files with an ‘extern’ declaration.

‘-Wnested-externs’

     This warns if an ‘extern’ declaration is encountered within a
     function.

‘-Dinline=’

     The ‘inline’ keyword is not part of ANSI C. Thus if you want to use
     ‘-ansi’ with a program which uses inline functions you can use this
     preprocessor definition to remove the ‘inline’ keywords.

‘-g’

     It always makes sense to put debugging symbols in the executable so
     that you can debug it using ‘gdb’.  The only effect of debugging
     symbols is to increase the size of the file, and you can use the
     ‘strip’ command to remove them later if necessary.


File: gsl-ref.info,  Node: References and Further Reading<41>,  Prev: GCC warning options for numerical programs,  Up: Debugging Numerical Programs

48.5 References and Further Reading
===================================

The following books are essential reading for anyone writing and
debugging numerical programs with ‘gcc’ and ‘gdb’.

   * R.M. Stallman, `Using and Porting GNU CC', Free Software
     Foundation, ISBN 1882114388

   * R.M. Stallman, R.H. Pesch, `Debugging with GDB: The GNU
     Source-Level Debugger', Free Software Foundation, ISBN 1882114779

For a tutorial introduction to the GNU C Compiler and related programs,
see

   * B.J. Gough, ‘http://www.network-theory.co.uk/gcc/intro/’,’ `An
     Introduction to GCC', Network Theory Ltd, ISBN 0954161793


File: gsl-ref.info,  Node: Contributors to GSL,  Next: Autoconf Macros,  Prev: Debugging Numerical Programs,  Up: Top

49 Contributors to GSL
**********************

(See the ‘AUTHORS’ file in the distribution for up-to-date information.)

Mark Galassi

     Conceived GSL (with James Theiler) and wrote the design document.
     Wrote the simulated annealing package and the relevant chapter in
     the manual.

James Theiler

     Conceived GSL (with Mark Galassi).  Wrote the random number
     generators and the relevant chapter in this manual.

Jim Davies

     Wrote the statistical routines and the relevant chapter in this
     manual.

Brian Gough

     FFTs, numerical integration, random number generators and
     distributions, root finding, minimization and fitting, polynomial
     solvers, complex numbers, physical constants, permutations, vector
     and matrix functions, histograms, statistics, ieee-utils, revised
     CBLAS Level 2 & 3, matrix decompositions, eigensystems, cumulative
     distribution functions, testing, documentation and releases.

Reid Priedhorsky

     Wrote and documented the initial version of the root finding
     routines while at Los Alamos National Laboratory, Mathematical
     Modeling and Analysis Group.

Gerard Jungman

     Special Functions, Series acceleration, ODEs, BLAS, Linear Algebra,
     Eigensystems, Hankel Transforms.

Patrick Alken

     Implementation of nonsymmetric and generalized eigensystems,
     B-splines, linear and nonlinear least squares, matrix
     decompositions, associated Legendre functions, running statistics,
     sparse matrices, and sparse linear algebra.

Mike Booth

     Wrote the Monte Carlo library.

Jorma Olavi Tähtinen

     Wrote the initial complex arithmetic functions.

Thomas Walter

     Wrote the initial heapsort routines and Cholesky decomposition.

Fabrice Rossi

     Multidimensional minimization.

Carlo Perassi

     Implementation of the random number generators in Knuth’s
     `Seminumerical Algorithms', 3rd Ed.

Szymon Jaroszewicz

     Wrote the routines for generating combinations.

Nicolas Darnis

     Wrote the cyclic functions and the initial functions for canonical
     permutations.

Jason H. Stover

     Wrote the major cumulative distribution functions.

Ivo Alxneit

     Wrote the routines for wavelet transforms.

Tuomo Keskitalo

     Improved the implementation of the ODE solvers and wrote the
     ode-initval2 routines.

Lowell Johnson

     Implementation of the Mathieu functions.

Rhys Ulerich

     Wrote the multiset routines.

Pavel Holoborodko

     Wrote the fixed order Gauss-Legendre quadrature routines.

Pedro Gonnet

     Wrote the CQUAD integration routines.

Thanks to Nigel Lowry for help in proofreading the manual.

The non-symmetric eigensystems routines contain code based on the LAPACK
linear algebra library.  LAPACK is distributed under the following
license:

     Copyright (c) 1992-2006 The University of Tennessee.  All rights reserved.

     Redistribution and use in source and binary forms, with or without
     modification, are permitted provided that the following conditions are
     met:

     * Redistributions of source code must retain the above copyright
       notice, this list of conditions and the following disclaimer.

     * Redistributions in binary form must reproduce the above copyright
       notice, this list of conditions and the following disclaimer listed
       in this license in the documentation and/or other materials
       provided with the distribution.

     * Neither the name of the copyright holders nor the names of its
       contributors may be used to endorse or promote products derived from
       this software without specific prior written permission.

     THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
     ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
     LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
     A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
     OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
     SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
     LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
     DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
     THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
     (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
     OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.


File: gsl-ref.info,  Node: Autoconf Macros,  Next: GSL CBLAS Library,  Prev: Contributors to GSL,  Up: Top

50 Autoconf Macros
******************

For applications using ‘autoconf’ the standard macro ‘AC_CHECK_LIB’ can
be used to link with GSL automatically from a ‘configure’ script.  The
library itself depends on the presence of a CBLAS and math library as
well, so these must also be located before linking with the main
‘libgsl’ file.  The following commands should be placed in the
‘configure.ac’ file to perform these tests:

     AC_CHECK_LIB([m],[cos])
     AC_CHECK_LIB([gslcblas],[cblas_dgemm])
     AC_CHECK_LIB([gsl],[gsl_blas_dgemm])

It is important to check for ‘libm’ and ‘libgslcblas’ before ‘libgsl’,
otherwise the tests will fail.  Assuming the libraries are found the
output during the configure stage looks like this:

     checking for cos in -lm... yes
     checking for cblas_dgemm in -lgslcblas... yes
     checking for gsl_blas_dgemm in -lgsl... yes

If the library is found then the tests will define the macros
‘HAVE_LIBGSL’, ‘HAVE_LIBGSLCBLAS’, ‘HAVE_LIBM’ and add the options
‘-lgsl -lgslcblas -lm’ to the variable ‘LIBS’.

The tests above will find any version of the library.  They are suitable
for general use, where the versions of the functions are not important.
An alternative macro is available in the file ‘gsl.m4’ to test for a
specific version of the library.  To use this macro simply add the
following line to your ‘configure.in’ file instead of the tests above:

     AX_PATH_GSL(GSL_VERSION,
                [action-if-found],
                [action-if-not-found])

The argument ‘GSL_VERSION’ should be the two or three digit
‘major.minor’ or ‘major.minor.micro’ version number of the release you
require.  A suitable choice for ‘action-if-not-found’ is:

     AC_MSG_ERROR(could not find required version of GSL)

Then you can add the variables ‘GSL_LIBS’ and ‘GSL_CFLAGS’ to your
Makefile.am files to obtain the correct compiler flags.  ‘GSL_LIBS’ is
equal to the output of the ‘gsl-config --libs’ command and ‘GSL_CFLAGS’
is equal to ‘gsl-config --cflags’ command.  For example:

     libfoo_la_LDFLAGS = -lfoo $(GSL_LIBS) -lgslcblas

Note that the macro ‘AX_PATH_GSL’ needs to use the C compiler so it
should appear in the ‘configure.in’ file before the macro
‘AC_LANG_CPLUSPLUS’ for programs that use C++.

To test for ‘inline’ the following test should be placed in your
‘configure.in’ file:

     AC_C_INLINE

     if test "$ac_cv_c_inline" != no ; then
       AC_DEFINE(HAVE_INLINE,1)
       AC_SUBST(HAVE_INLINE)
     fi

and the macro will then be defined in the compilation flags or by
including the file ‘config.h’ before any library headers.

The following autoconf test will check for ‘extern inline’:

     dnl Check for "extern inline", using a modified version
     dnl of the test for AC_C_INLINE from acspecific.mt
     dnl
     AC_CACHE_CHECK([for extern inline], ac_cv_c_extern_inline,
     [ac_cv_c_extern_inline=no
     AC_TRY_COMPILE([extern $ac_cv_c_inline double foo(double x);
     extern $ac_cv_c_inline double foo(double x) { return x+1.0; };
     double foo (double x) { return x + 1.0; };],
     [  foo(1.0)  ],
     [ac_cv_c_extern_inline="yes"])
     ])

     if test "$ac_cv_c_extern_inline" != no ; then
       AC_DEFINE(HAVE_INLINE,1)
       AC_SUBST(HAVE_INLINE)
     fi

The substitution of portability functions can be made automatically if
you use ‘autoconf’.  For example, to test whether the BSD function
‘hypot()’ is available you can include the following line in the
configure file ‘configure.in’ for your application:

     AC_CHECK_FUNCS(hypot)

and place the following macro definitions in the file ‘config.h.in’:

     /* Substitute gsl_hypot for missing system hypot */

     #ifndef HAVE_HYPOT
     #define hypot gsl_hypot
     #endif

The application source files can then use the include command ‘#include
<config.h>’ to substitute *note gsl_hypot(): 1a. for each occurrence of
‘hypot()’ when ‘hypot()’ is not available.


File: gsl-ref.info,  Node: GSL CBLAS Library,  Next: GNU General Public License,  Prev: Autoconf Macros,  Up: Top

51 GSL CBLAS Library
********************

The prototypes for the low-level CBLAS functions are declared in the
file ‘gsl_cblas.h’.  For the definition of the functions consult the
documentation available from Netlib (*note see BLAS References and
Further Reading: 4e3.).

* Menu:

* Level 1: Level 1<2>.
* Level 2: Level 2<2>.
* Level 3: Level 3<2>.
* Examples: Examples<37>.


File: gsl-ref.info,  Node: Level 1<2>,  Next: Level 2<2>,  Up: GSL CBLAS Library

51.1 Level 1
============

 -- Function: float cblas_sdsdot (const int N, const float alpha, const
          float * x, const int incx, const float * y, const int incy)

 -- Function: double cblas_dsdot (const int N, const float * x, const
          int incx, const float * y, const int incy)

 -- Function: float cblas_sdot (const int N, const float * x, const
          int incx, const float * y, const int incy)

 -- Function: double cblas_ddot (const int N, const double * x, const
          int incx, const double * y, const int incy)

 -- Function: void cblas_cdotu_sub (const int N, const void * x, const
          int incx, const void * y, const int incy, void * dotu)

 -- Function: void cblas_cdotc_sub (const int N, const void * x, const
          int incx, const void * y, const int incy, void * dotc)

 -- Function: void cblas_zdotu_sub (const int N, const void * x, const
          int incx, const void * y, const int incy, void * dotu)

 -- Function: void cblas_zdotc_sub (const int N, const void * x, const
          int incx, const void * y, const int incy, void * dotc)

 -- Function: float cblas_snrm2 (const int N, const float * x, const
          int incx)

 -- Function: float cblas_sasum (const int N, const float * x, const
          int incx)

 -- Function: double cblas_dnrm2 (const int N, const double * x, const
          int incx)

 -- Function: double cblas_dasum (const int N, const double * x, const
          int incx)

 -- Function: float cblas_scnrm2 (const int N, const void * x, const
          int incx)

 -- Function: float cblas_scasum (const int N, const void * x, const
          int incx)

 -- Function: double cblas_dznrm2 (const int N, const void * x, const
          int incx)

 -- Function: double cblas_dzasum (const int N, const void * x, const
          int incx)

 -- Function: CBLAS_INDEX cblas_isamax (const int N, const float * x,
          const int incx)

 -- Function: CBLAS_INDEX cblas_idamax (const int N, const double * x,
          const int incx)

 -- Function: CBLAS_INDEX cblas_icamax (const int N, const void * x,
          const int incx)

 -- Function: CBLAS_INDEX cblas_izamax (const int N, const void * x,
          const int incx)

 -- Function: void cblas_sswap (const int N, float * x, const int incx,
          float * y, const int incy)

 -- Function: void cblas_scopy (const int N, const float * x, const
          int incx, float * y, const int incy)

 -- Function: void cblas_saxpy (const int N, const float alpha, const
          float * x, const int incx, float * y, const int incy)

 -- Function: void cblas_dswap (const int N, double * x, const int incx,
          double * y, const int incy)

 -- Function: void cblas_dcopy (const int N, const double * x, const
          int incx, double * y, const int incy)

 -- Function: void cblas_daxpy (const int N, const double alpha, const
          double * x, const int incx, double * y, const int incy)

 -- Function: void cblas_cswap (const int N, void * x, const int incx,
          void * y, const int incy)

 -- Function: void cblas_ccopy (const int N, const void * x, const
          int incx, void * y, const int incy)

 -- Function: void cblas_caxpy (const int N, const void * alpha, const
          void * x, const int incx, void * y, const int incy)

 -- Function: void cblas_zswap (const int N, void * x, const int incx,
          void * y, const int incy)

 -- Function: void cblas_zcopy (const int N, const void * x, const
          int incx, void * y, const int incy)

 -- Function: void cblas_zaxpy (const int N, const void * alpha, const
          void * x, const int incx, void * y, const int incy)

 -- Function: void cblas_srotg (float * a, float * b, float * c, float
          * s)

 -- Function: void cblas_srotmg (float * d1, float * d2, float * b1,
          const float b2, float * P)

 -- Function: void cblas_srot (const int N, float * x, const int incx,
          float * y, const int incy, const float c, const float s)

 -- Function: void cblas_srotm (const int N, float * x, const int incx,
          float * y, const int incy, const float * P)

 -- Function: void cblas_drotg (double * a, double * b, double * c,
          double * s)

 -- Function: void cblas_drotmg (double * d1, double * d2, double * b1,
          const double b2, double * P)

 -- Function: void cblas_drot (const int N, double * x, const int incx,
          double * y, const int incy, const double c, const double s)

 -- Function: void cblas_drotm (const int N, double * x, const int incx,
          double * y, const int incy, const double * P)

 -- Function: void cblas_sscal (const int N, const float alpha, float
          * x, const int incx)

 -- Function: void cblas_dscal (const int N, const double alpha, double
          * x, const int incx)

 -- Function: void cblas_cscal (const int N, const void * alpha, void
          * x, const int incx)

 -- Function: void cblas_zscal (const int N, const void * alpha, void
          * x, const int incx)

 -- Function: void cblas_csscal (const int N, const float alpha, void
          * x, const int incx)

 -- Function: void cblas_zdscal (const int N, const double alpha, void
          * x, const int incx)


File: gsl-ref.info,  Node: Level 2<2>,  Next: Level 3<2>,  Prev: Level 1<2>,  Up: GSL CBLAS Library

51.2 Level 2
============

 -- Function: void cblas_sgemv (const enum CBLAS_ORDER order, const enum
          CBLAS_TRANSPOSE TransA, const int M, const int N, const
          float alpha, const float * A, const int lda, const float * x,
          const int incx, const float beta, float * y, const int incy)

 -- Function: void cblas_sgbmv (const enum CBLAS_ORDER order, const enum
          CBLAS_TRANSPOSE TransA, const int M, const int N, const
          int KL, const int KU, const float alpha, const float * A,
          const int lda, const float * x, const int incx, const
          float beta, float * y, const int incy)

 -- Function: void cblas_strmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const float * A, const int lda,
          float * x, const int incx)

 -- Function: void cblas_stbmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const int K, const float * A,
          const int lda, float * x, const int incx)

 -- Function: void cblas_stpmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const float * Ap, float * x,
          const int incx)

 -- Function: void cblas_strsv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const float * A, const int lda,
          float * x, const int incx)

 -- Function: void cblas_stbsv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const int K, const float * A,
          const int lda, float * x, const int incx)

 -- Function: void cblas_stpsv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const float * Ap, float * x,
          const int incx)

 -- Function: void cblas_dgemv (const enum CBLAS_ORDER order, const enum
          CBLAS_TRANSPOSE TransA, const int M, const int N, const
          double alpha, const double * A, const int lda, const double
          * x, const int incx, const double beta, double * y, const
          int incy)

 -- Function: void cblas_dgbmv (const enum CBLAS_ORDER order, const enum
          CBLAS_TRANSPOSE TransA, const int M, const int N, const
          int KL, const int KU, const double alpha, const double * A,
          const int lda, const double * x, const int incx, const
          double beta, double * y, const int incy)

 -- Function: void cblas_dtrmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const double * A, const int lda,
          double * x, const int incx)

 -- Function: void cblas_dtbmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const int K, const double * A,
          const int lda, double * x, const int incx)

 -- Function: void cblas_dtpmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const double * Ap, double * x,
          const int incx)

 -- Function: void cblas_dtrsv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const double * A, const int lda,
          double * x, const int incx)

 -- Function: void cblas_dtbsv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const int K, const double * A,
          const int lda, double * x, const int incx)

 -- Function: void cblas_dtpsv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const double * Ap, double * x,
          const int incx)

 -- Function: void cblas_cgemv (const enum CBLAS_ORDER order, const enum
          CBLAS_TRANSPOSE TransA, const int M, const int N, const void
          * alpha, const void * A, const int lda, const void * x, const
          int incx, const void * beta, void * y, const int incy)

 -- Function: void cblas_cgbmv (const enum CBLAS_ORDER order, const enum
          CBLAS_TRANSPOSE TransA, const int M, const int N, const
          int KL, const int KU, const void * alpha, const void * A,
          const int lda, const void * x, const int incx, const void
          * beta, void * y, const int incy)

 -- Function: void cblas_ctrmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const void * A, const int lda,
          void * x, const int incx)

 -- Function: void cblas_ctbmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const int K, const void * A,
          const int lda, void * x, const int incx)

 -- Function: void cblas_ctpmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const void * Ap, void * x, const
          int incx)

 -- Function: void cblas_ctrsv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const void * A, const int lda,
          void * x, const int incx)

 -- Function: void cblas_ctbsv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const int K, const void * A,
          const int lda, void * x, const int incx)

 -- Function: void cblas_ctpsv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const void * Ap, void * x, const
          int incx)

 -- Function: void cblas_zgemv (const enum CBLAS_ORDER order, const enum
          CBLAS_TRANSPOSE TransA, const int M, const int N, const void
          * alpha, const void * A, const int lda, const void * x, const
          int incx, const void * beta, void * y, const int incy)

 -- Function: void cblas_zgbmv (const enum CBLAS_ORDER order, const enum
          CBLAS_TRANSPOSE TransA, const int M, const int N, const
          int KL, const int KU, const void * alpha, const void * A,
          const int lda, const void * x, const int incx, const void
          * beta, void * y, const int incy)

 -- Function: void cblas_ztrmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const void * A, const int lda,
          void * x, const int incx)

 -- Function: void cblas_ztbmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const int K, const void * A,
          const int lda, void * x, const int incx)

 -- Function: void cblas_ztpmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const void * Ap, void * x, const
          int incx)

 -- Function: void cblas_ztrsv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const void * A, const int lda,
          void * x, const int incx)

 -- Function: void cblas_ztbsv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const int K, const void * A,
          const int lda, void * x, const int incx)

 -- Function: void cblas_ztpsv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE TransA, const enum
          CBLAS_DIAG Diag, const int N, const void * Ap, void * x, const
          int incx)

 -- Function: void cblas_ssymv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const float alpha, const float
          * A, const int lda, const float * x, const int incx, const
          float beta, float * y, const int incy)

 -- Function: void cblas_ssbmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const int K, const float alpha,
          const float * A, const int lda, const float * x, const
          int incx, const float beta, float * y, const int incy)

 -- Function: void cblas_sspmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const float alpha, const float
          * Ap, const float * x, const int incx, const float beta, float
          * y, const int incy)

 -- Function: void cblas_sger (const enum CBLAS_ORDER order, const
          int M, const int N, const float alpha, const float * x, const
          int incx, const float * y, const int incy, float * A, const
          int lda)

 -- Function: void cblas_ssyr (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const float alpha, const float
          * x, const int incx, float * A, const int lda)

 -- Function: void cblas_sspr (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const float alpha, const float
          * x, const int incx, float * Ap)

 -- Function: void cblas_ssyr2 (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const float alpha, const float
          * x, const int incx, const float * y, const int incy, float
          * A, const int lda)

 -- Function: void cblas_sspr2 (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const float alpha, const float
          * x, const int incx, const float * y, const int incy, float
          * A)

 -- Function: void cblas_dsymv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const double alpha, const double
          * A, const int lda, const double * x, const int incx, const
          double beta, double * y, const int incy)

 -- Function: void cblas_dsbmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const int K, const double alpha,
          const double * A, const int lda, const double * x, const
          int incx, const double beta, double * y, const int incy)

 -- Function: void cblas_dspmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const double alpha, const double
          * Ap, const double * x, const int incx, const double beta,
          double * y, const int incy)

 -- Function: void cblas_dger (const enum CBLAS_ORDER order, const
          int M, const int N, const double alpha, const double * x,
          const int incx, const double * y, const int incy, double * A,
          const int lda)

 -- Function: void cblas_dsyr (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const double alpha, const double
          * x, const int incx, double * A, const int lda)

 -- Function: void cblas_dspr (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const double alpha, const double
          * x, const int incx, double * Ap)

 -- Function: void cblas_dsyr2 (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const double alpha, const double
          * x, const int incx, const double * y, const int incy, double
          * A, const int lda)

 -- Function: void cblas_dspr2 (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const double alpha, const double
          * x, const int incx, const double * y, const int incy, double
          * A)

 -- Function: void cblas_chemv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const void * alpha, const void
          * A, const int lda, const void * x, const int incx, const void
          * beta, void * y, const int incy)

 -- Function: void cblas_chbmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const int K, const void * alpha,
          const void * A, const int lda, const void * x, const int incx,
          const void * beta, void * y, const int incy)

 -- Function: void cblas_chpmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const void * alpha, const void
          * Ap, const void * x, const int incx, const void * beta, void
          * y, const int incy)

 -- Function: void cblas_cgeru (const enum CBLAS_ORDER order, const
          int M, const int N, const void * alpha, const void * x, const
          int incx, const void * y, const int incy, void * A, const
          int lda)

 -- Function: void cblas_cgerc (const enum CBLAS_ORDER order, const
          int M, const int N, const void * alpha, const void * x, const
          int incx, const void * y, const int incy, void * A, const
          int lda)

 -- Function: void cblas_cher (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const float alpha, const void
          * x, const int incx, void * A, const int lda)

 -- Function: void cblas_chpr (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const float alpha, const void
          * x, const int incx, void * A)

 -- Function: void cblas_cher2 (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const void * alpha, const void
          * x, const int incx, const void * y, const int incy, void * A,
          const int lda)

 -- Function: void cblas_chpr2 (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const void * alpha, const void
          * x, const int incx, const void * y, const int incy, void
          * Ap)

 -- Function: void cblas_zhemv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const void * alpha, const void
          * A, const int lda, const void * x, const int incx, const void
          * beta, void * y, const int incy)

 -- Function: void cblas_zhbmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const int K, const void * alpha,
          const void * A, const int lda, const void * x, const int incx,
          const void * beta, void * y, const int incy)

 -- Function: void cblas_zhpmv (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const void * alpha, const void
          * Ap, const void * x, const int incx, const void * beta, void
          * y, const int incy)

 -- Function: void cblas_zgeru (const enum CBLAS_ORDER order, const
          int M, const int N, const void * alpha, const void * x, const
          int incx, const void * y, const int incy, void * A, const
          int lda)

 -- Function: void cblas_zgerc (const enum CBLAS_ORDER order, const
          int M, const int N, const void * alpha, const void * x, const
          int incx, const void * y, const int incy, void * A, const
          int lda)

 -- Function: void cblas_zher (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const double alpha, const void
          * x, const int incx, void * A, const int lda)

 -- Function: void cblas_zhpr (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const double alpha, const void
          * x, const int incx, void * A)

 -- Function: void cblas_zher2 (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const void * alpha, const void
          * x, const int incx, const void * y, const int incy, void * A,
          const int lda)

 -- Function: void cblas_zhpr2 (const enum CBLAS_ORDER order, const enum
          CBLAS_UPLO Uplo, const int N, const void * alpha, const void
          * x, const int incx, const void * y, const int incy, void
          * Ap)


File: gsl-ref.info,  Node: Level 3<2>,  Next: Examples<37>,  Prev: Level 2<2>,  Up: GSL CBLAS Library

51.3 Level 3
============

 -- Function: void cblas_sgemm (const enum CBLAS_ORDER Order, const enum
          CBLAS_TRANSPOSE TransA, const enum CBLAS_TRANSPOSE TransB,
          const int M, const int N, const int K, const float alpha,
          const float * A, const int lda, const float * B, const
          int ldb, const float beta, float * C, const int ldc)

 -- Function: void cblas_ssymm (const enum CBLAS_ORDER Order, const enum
          CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const int M,
          const int N, const float alpha, const float * A, const
          int lda, const float * B, const int ldb, const float beta,
          float * C, const int ldc)

 -- Function: void cblas_ssyrk (const enum CBLAS_ORDER Order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const
          int N, const int K, const float alpha, const float * A, const
          int lda, const float beta, float * C, const int ldc)

 -- Function: void cblas_ssyr2k (const enum CBLAS_ORDER Order, const
          enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const
          int N, const int K, const float alpha, const float * A, const
          int lda, const float * B, const int ldb, const float beta,
          float * C, const int ldc)

 -- Function: void cblas_strmm (const enum CBLAS_ORDER Order, const enum
          CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const enum
          CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const
          int M, const int N, const float alpha, const float * A, const
          int lda, float * B, const int ldb)

 -- Function: void cblas_strsm (const enum CBLAS_ORDER Order, const enum
          CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const enum
          CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const
          int M, const int N, const float alpha, const float * A, const
          int lda, float * B, const int ldb)

 -- Function: void cblas_dgemm (const enum CBLAS_ORDER Order, const enum
          CBLAS_TRANSPOSE TransA, const enum CBLAS_TRANSPOSE TransB,
          const int M, const int N, const int K, const double alpha,
          const double * A, const int lda, const double * B, const
          int ldb, const double beta, double * C, const int ldc)

 -- Function: void cblas_dsymm (const enum CBLAS_ORDER Order, const enum
          CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const int M,
          const int N, const double alpha, const double * A, const
          int lda, const double * B, const int ldb, const double beta,
          double * C, const int ldc)

 -- Function: void cblas_dsyrk (const enum CBLAS_ORDER Order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const
          int N, const int K, const double alpha, const double * A,
          const int lda, const double beta, double * C, const int ldc)

 -- Function: void cblas_dsyr2k (const enum CBLAS_ORDER Order, const
          enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const
          int N, const int K, const double alpha, const double * A,
          const int lda, const double * B, const int ldb, const
          double beta, double * C, const int ldc)

 -- Function: void cblas_dtrmm (const enum CBLAS_ORDER Order, const enum
          CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const enum
          CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const
          int M, const int N, const double alpha, const double * A,
          const int lda, double * B, const int ldb)

 -- Function: void cblas_dtrsm (const enum CBLAS_ORDER Order, const enum
          CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const enum
          CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const
          int M, const int N, const double alpha, const double * A,
          const int lda, double * B, const int ldb)

 -- Function: void cblas_cgemm (const enum CBLAS_ORDER Order, const enum
          CBLAS_TRANSPOSE TransA, const enum CBLAS_TRANSPOSE TransB,
          const int M, const int N, const int K, const void * alpha,
          const void * A, const int lda, const void * B, const int ldb,
          const void * beta, void * C, const int ldc)

 -- Function: void cblas_csymm (const enum CBLAS_ORDER Order, const enum
          CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const int M,
          const int N, const void * alpha, const void * A, const
          int lda, const void * B, const int ldb, const void * beta,
          void * C, const int ldc)

 -- Function: void cblas_csyrk (const enum CBLAS_ORDER Order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const
          int N, const int K, const void * alpha, const void * A, const
          int lda, const void * beta, void * C, const int ldc)

 -- Function: void cblas_csyr2k (const enum CBLAS_ORDER Order, const
          enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const
          int N, const int K, const void * alpha, const void * A, const
          int lda, const void * B, const int ldb, const void * beta,
          void * C, const int ldc)

 -- Function: void cblas_ctrmm (const enum CBLAS_ORDER Order, const enum
          CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const enum
          CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const
          int M, const int N, const void * alpha, const void * A, const
          int lda, void * B, const int ldb)

 -- Function: void cblas_ctrsm (const enum CBLAS_ORDER Order, const enum
          CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const enum
          CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const
          int M, const int N, const void * alpha, const void * A, const
          int lda, void * B, const int ldb)

 -- Function: void cblas_zgemm (const enum CBLAS_ORDER Order, const enum
          CBLAS_TRANSPOSE TransA, const enum CBLAS_TRANSPOSE TransB,
          const int M, const int N, const int K, const void * alpha,
          const void * A, const int lda, const void * B, const int ldb,
          const void * beta, void * C, const int ldc)

 -- Function: void cblas_zsymm (const enum CBLAS_ORDER Order, const enum
          CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const int M,
          const int N, const void * alpha, const void * A, const
          int lda, const void * B, const int ldb, const void * beta,
          void * C, const int ldc)

 -- Function: void cblas_zsyrk (const enum CBLAS_ORDER Order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const
          int N, const int K, const void * alpha, const void * A, const
          int lda, const void * beta, void * C, const int ldc)

 -- Function: void cblas_zsyr2k (const enum CBLAS_ORDER Order, const
          enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const
          int N, const int K, const void * alpha, const void * A, const
          int lda, const void * B, const int ldb, const void * beta,
          void * C, const int ldc)

 -- Function: void cblas_ztrmm (const enum CBLAS_ORDER Order, const enum
          CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const enum
          CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const
          int M, const int N, const void * alpha, const void * A, const
          int lda, void * B, const int ldb)

 -- Function: void cblas_ztrsm (const enum CBLAS_ORDER Order, const enum
          CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const enum
          CBLAS_TRANSPOSE TransA, const enum CBLAS_DIAG Diag, const
          int M, const int N, const void * alpha, const void * A, const
          int lda, void * B, const int ldb)

 -- Function: void cblas_chemm (const enum CBLAS_ORDER Order, const enum
          CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const int M,
          const int N, const void * alpha, const void * A, const
          int lda, const void * B, const int ldb, const void * beta,
          void * C, const int ldc)

 -- Function: void cblas_cherk (const enum CBLAS_ORDER Order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const
          int N, const int K, const float alpha, const void * A, const
          int lda, const float beta, void * C, const int ldc)

 -- Function: void cblas_cher2k (const enum CBLAS_ORDER Order, const
          enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const
          int N, const int K, const void * alpha, const void * A, const
          int lda, const void * B, const int ldb, const float beta, void
          * C, const int ldc)

 -- Function: void cblas_zhemm (const enum CBLAS_ORDER Order, const enum
          CBLAS_SIDE Side, const enum CBLAS_UPLO Uplo, const int M,
          const int N, const void * alpha, const void * A, const
          int lda, const void * B, const int ldb, const void * beta,
          void * C, const int ldc)

 -- Function: void cblas_zherk (const enum CBLAS_ORDER Order, const enum
          CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const
          int N, const int K, const double alpha, const void * A, const
          int lda, const double beta, void * C, const int ldc)

 -- Function: void cblas_zher2k (const enum CBLAS_ORDER Order, const
          enum CBLAS_UPLO Uplo, const enum CBLAS_TRANSPOSE Trans, const
          int N, const int K, const void * alpha, const void * A, const
          int lda, const void * B, const int ldb, const double beta,
          void * C, const int ldc)

 -- Function: void cblas_xerbla (int p, const char * rout, const char
          * form, ...)


File: gsl-ref.info,  Node: Examples<37>,  Prev: Level 3<2>,  Up: GSL CBLAS Library

51.4 Examples
=============

The following program computes the product of two matrices using the
Level-3 BLAS function SGEMM,

     [ 0.11 0.12 0.13 ]  [ 1011 1012 ]     [ 367.76 368.12 ]
     [ 0.21 0.22 0.23 ]  [ 1021 1022 ]  =  [ 674.06 674.72 ]
                         [ 1031 1032 ]

The matrices are stored in row major order but could be stored in column
major order if the first argument of the call to *note cblas_sgemm():
d4d. was changed to ‘CblasColMajor’.

     #include <stdio.h>
     #include <gsl/gsl_cblas.h>

     int
     main (void)
     {
       int lda = 3;

       float A[] = { 0.11, 0.12, 0.13,
                     0.21, 0.22, 0.23 };

       int ldb = 2;

       float B[] = { 1011, 1012,
                     1021, 1022,
                     1031, 1032 };

       int ldc = 2;

       float C[] = { 0.00, 0.00,
                     0.00, 0.00 };

       /* Compute C = A B */

       cblas_sgemm (CblasRowMajor,
                    CblasNoTrans, CblasNoTrans, 2, 2, 3,
                    1.0, A, lda, B, ldb, 0.0, C, ldc);

       printf ("[ %g, %g\n", C[0], C[1]);
       printf ("  %g, %g ]\n", C[2], C[3]);

       return 0;
     }

To compile the program use the following command line:

     $ gcc -Wall demo.c -lgslcblas

There is no need to link with the main library ‘-lgsl’ in this case as
the CBLAS library is an independent unit.  Here is the output from the
program,

     [ 367.76, 368.12
       674.06, 674.72 ]


File: gsl-ref.info,  Node: GNU General Public License,  Next: GNU Free Documentation License,  Prev: GSL CBLAS Library,  Up: Top

52 GNU General Public License
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     produce it from the Program, in the form of source code under the
     terms of section 4, provided that you also meet all of these conditions:

         a) The work must carry prominent notices stating that you modified
         it, and giving a relevant date.

         b) The work must carry prominent notices stating that it is
         released under this License and any conditions added under section
         7.  This requirement modifies the requirement in section 4 to
         "keep intact all notices".

         c) You must license the entire work, as a whole, under this
         License to anyone who comes into possession of a copy.  This
         License will therefore apply, along with any applicable section 7
         additional terms, to the whole of the work, and all its parts,
         regardless of how they are packaged.  This License gives no
         permission to license the work in any other way, but it does not
         invalidate such permission if you have separately received it.

         d) If the work has interactive user interfaces, each must display
         Appropriate Legal Notices; however, if the Program has interactive
         interfaces that do not display Appropriate Legal Notices, your
         work need not make them do so.

       A compilation of a covered work with other separate and independent
     works, which are not by their nature extensions of the covered work,
     and which are not combined with it such as to form a larger program,
     in or on a volume of a storage or distribution medium, is called an
     "aggregate" if the compilation and its resulting copyright are not
     used to limit the access or legal rights of the compilation's users
     beyond what the individual works permit.  Inclusion of a covered work
     in an aggregate does not cause this License to apply to the other
     parts of the aggregate.

       6. Conveying Non-Source Forms.

       You may convey a covered work in object code form under the terms
     of sections 4 and 5, provided that you also convey the
     machine-readable Corresponding Source under the terms of this License,
     in one of these ways:

         a) Convey the object code in, or embodied in, a physical product
         (including a physical distribution medium), accompanied by the
         Corresponding Source fixed on a durable physical medium
         customarily used for software interchange.

         b) Convey the object code in, or embodied in, a physical product
         (including a physical distribution medium), accompanied by a
         written offer, valid for at least three years and valid for as
         long as you offer spare parts or customer support for that product
         model, to give anyone who possesses the object code either (1) a
         copy of the Corresponding Source for all the software in the
         product that is covered by this License, on a durable physical
         medium customarily used for software interchange, for a price no
         more than your reasonable cost of physically performing this
         conveying of source, or (2) access to copy the
         Corresponding Source from a network server at no charge.

         c) Convey individual copies of the object code with a copy of the
         written offer to provide the Corresponding Source.  This
         alternative is allowed only occasionally and noncommercially, and
         only if you received the object code with such an offer, in accord
         with subsection 6b.

         d) Convey the object code by offering access from a designated
         place (gratis or for a charge), and offer equivalent access to the
         Corresponding Source in the same way through the same place at no
         further charge.  You need not require recipients to copy the
         Corresponding Source along with the object code.  If the place to
         copy the object code is a network server, the Corresponding Source
         may be on a different server (operated by you or a third party)
         that supports equivalent copying facilities, provided you maintain
         clear directions next to the object code saying where to find the
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         Corresponding Source, you remain obligated to ensure that it is
         available for as long as needed to satisfy these requirements.

         e) Convey the object code using peer-to-peer transmission, provided
         you inform other peers where the object code and Corresponding
         Source of the work are being offered to the general public at no
         charge under subsection 6d.

       A separable portion of the object code, whose source code is excluded
     from the Corresponding Source as a System Library, need not be
     included in conveying the object code work.

       A "User Product" is either (1) a "consumer product", which means any
     tangible personal property which is normally used for personal, family,
     or household purposes, or (2) anything designed or sold for incorporation
     into a dwelling.  In determining whether a product is a consumer product,
     doubtful cases shall be resolved in favor of coverage.  For a particular
     product received by a particular user, "normally used" refers to a
     typical or common use of that class of product, regardless of the status
     of the particular user or of the way in which the particular user
     actually uses, or expects or is expected to use, the product.  A product
     is a consumer product regardless of whether the product has substantial
     commercial, industrial or non-consumer uses, unless such uses represent
     the only significant mode of use of the product.

       "Installation Information" for a User Product means any methods,
     procedures, authorization keys, or other information required to install
     and execute modified versions of a covered work in that User Product from
     a modified version of its Corresponding Source.  The information must
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     code is in no case prevented or interfered with solely because
     modification has been made.

       If you convey an object code work under this section in, or with, or
     specifically for use in, a User Product, and the conveying occurs as
     part of a transaction in which the right of possession and use of the
     User Product is transferred to the recipient in perpetuity or for a
     fixed term (regardless of how the transaction is characterized), the
     Corresponding Source conveyed under this section must be accompanied
     by the Installation Information.  But this requirement does not apply
     if neither you nor any third party retains the ability to install
     modified object code on the User Product (for example, the work has
     been installed in ROM).

       The requirement to provide Installation Information does not include a
     requirement to continue to provide support service, warranty, or updates
     for a work that has been modified or installed by the recipient, or for
     the User Product in which it has been modified or installed.  Access to a
     network may be denied when the modification itself materially and
     adversely affects the operation of the network or violates the rules and
     protocols for communication across the network.

       Corresponding Source conveyed, and Installation Information provided,
     in accord with this section must be in a format that is publicly
     documented (and with an implementation available to the public in
     source code form), and must require no special password or key for
     unpacking, reading or copying.

       7. Additional Terms.

       "Additional permissions" are terms that supplement the terms of this
     License by making exceptions from one or more of its conditions.
     Additional permissions that are applicable to the entire Program shall
     be treated as though they were included in this License, to the extent
     that they are valid under applicable law.  If additional permissions
     apply only to part of the Program, that part may be used separately
     under those permissions, but the entire Program remains governed by
     this License without regard to the additional permissions.

       When you convey a copy of a covered work, you may at your option
     remove any additional permissions from that copy, or from any part of
     it.  (Additional permissions may be written to require their own
     removal in certain cases when you modify the work.)  You may place
     additional permissions on material, added by you to a covered work,
     for which you have or can give appropriate copyright permission.

       Notwithstanding any other provision of this License, for material you
     add to a covered work, you may (if authorized by the copyright holders of
     that material) supplement the terms of this License with terms:

         a) Disclaiming warranty or limiting liability differently from the
         terms of sections 15 and 16 of this License; or

         b) Requiring preservation of specified reasonable legal notices or
         author attributions in that material or in the Appropriate Legal
         Notices displayed by works containing it; or

         c) Prohibiting misrepresentation of the origin of that material, or
         requiring that modified versions of such material be marked in
         reasonable ways as different from the original version; or

         d) Limiting the use for publicity purposes of names of licensors or
         authors of the material; or

         e) Declining to grant rights under trademark law for use of some
         trade names, trademarks, or service marks; or

         f) Requiring indemnification of licensors and authors of that
         material by anyone who conveys the material (or modified versions of
         it) with contractual assumptions of liability to the recipient, for
         any liability that these contractual assumptions directly impose on
         those licensors and authors.

       All other non-permissive additional terms are considered "further
     restrictions" within the meaning of section 10.  If the Program as you
     received it, or any part of it, contains a notice stating that it is
     governed by this License along with a term that is a further
     restriction, you may remove that term.  If a license document contains
     a further restriction but permits relicensing or conveying under this
     License, you may add to a covered work material governed by the terms
     of that license document, provided that the further restriction does
     not survive such relicensing or conveying.

       If you add terms to a covered work in accord with this section, you
     must place, in the relevant source files, a statement of the
     additional terms that apply to those files, or a notice indicating
     where to find the applicable terms.

       Additional terms, permissive or non-permissive, may be stated in the
     form of a separately written license, or stated as exceptions;
     the above requirements apply either way.

       8. Termination.

       You may not propagate or modify a covered work except as expressly
     provided under this License.  Any attempt otherwise to propagate or
     modify it is void, and will automatically terminate your rights under
     this License (including any patent licenses granted under the third
     paragraph of section 11).

       However, if you cease all violation of this License, then your
     license from a particular copyright holder is reinstated (a)
     provisionally, unless and until the copyright holder explicitly and
     finally terminates your license, and (b) permanently, if the copyright
     holder fails to notify you of the violation by some reasonable means
     prior to 60 days after the cessation.

       Moreover, your license from a particular copyright holder is
     reinstated permanently if the copyright holder notifies you of the
     violation by some reasonable means, this is the first time you have
     received notice of violation of this License (for any work) from that
     copyright holder, and you cure the violation prior to 30 days after
     your receipt of the notice.

       Termination of your rights under this section does not terminate the
     licenses of parties who have received copies or rights from you under
     this License.  If your rights have been terminated and not permanently
     reinstated, you do not qualify to receive new licenses for the same
     material under section 10.

       9. Acceptance Not Required for Having Copies.

       You are not required to accept this License in order to receive or
     run a copy of the Program.  Ancillary propagation of a covered work
     occurring solely as a consequence of using peer-to-peer transmission
     to receive a copy likewise does not require acceptance.  However,
     nothing other than this License grants you permission to propagate or
     modify any covered work.  These actions infringe copyright if you do
     not accept this License.  Therefore, by modifying or propagating a
     covered work, you indicate your acceptance of this License to do so.

       10. Automatic Licensing of Downstream Recipients.

       Each time you convey a covered work, the recipient automatically
     receives a license from the original licensors, to run, modify and
     propagate that work, subject to this License.  You are not responsible
     for enforcing compliance by third parties with this License.

       An "entity transaction" is a transaction transferring control of an
     organization, or substantially all assets of one, or subdividing an
     organization, or merging organizations.  If propagation of a covered
     work results from an entity transaction, each party to that
     transaction who receives a copy of the work also receives whatever
     licenses to the work the party's predecessor in interest had or could
     give under the previous paragraph, plus a right to possession of the
     Corresponding Source of the work from the predecessor in interest, if
     the predecessor has it or can get it with reasonable efforts.

       You may not impose any further restrictions on the exercise of the
     rights granted or affirmed under this License.  For example, you may
     not impose a license fee, royalty, or other charge for exercise of
     rights granted under this License, and you may not initiate litigation
     (including a cross-claim or counterclaim in a lawsuit) alleging that
     any patent claim is infringed by making, using, selling, offering for
     sale, or importing the Program or any portion of it.

       11. Patents.

       A "contributor" is a copyright holder who authorizes use under this
     License of the Program or a work on which the Program is based.  The
     work thus licensed is called the contributor's "contributor version".

       A contributor's "essential patent claims" are all patent claims
     owned or controlled by the contributor, whether already acquired or
     hereafter acquired, that would be infringed by some manner, permitted
     by this License, of making, using, or selling its contributor version,
     but do not include claims that would be infringed only as a
     consequence of further modification of the contributor version.  For
     purposes of this definition, "control" includes the right to grant
     patent sublicenses in a manner consistent with the requirements of
     this License.

       Each contributor grants you a non-exclusive, worldwide, royalty-free
     patent license under the contributor's essential patent claims, to
     make, use, sell, offer for sale, import and otherwise run, modify and
     propagate the contents of its contributor version.

       In the following three paragraphs, a "patent license" is any express
     agreement or commitment, however denominated, not to enforce a patent
     (such as an express permission to practice a patent or covenant not to
     sue for patent infringement).  To "grant" such a patent license to a
     party means to make such an agreement or commitment not to enforce a
     patent against the party.

       If you convey a covered work, knowingly relying on a patent license,
     and the Corresponding Source of the work is not available for anyone
     to copy, free of charge and under the terms of this License, through a
     publicly available network server or other readily accessible means,
     then you must either (1) cause the Corresponding Source to be so
     available, or (2) arrange to deprive yourself of the benefit of the
     patent license for this particular work, or (3) arrange, in a manner
     consistent with the requirements of this License, to extend the patent
     license to downstream recipients.  "Knowingly relying" means you have
     actual knowledge that, but for the patent license, your conveying the
     covered work in a country, or your recipient's use of the covered work
     in a country, would infringe one or more identifiable patents in that
     country that you have reason to believe are valid.

       If, pursuant to or in connection with a single transaction or
     arrangement, you convey, or propagate by procuring conveyance of, a
     covered work, and grant a patent license to some of the parties
     receiving the covered work authorizing them to use, propagate, modify
     or convey a specific copy of the covered work, then the patent license
     you grant is automatically extended to all recipients of the covered
     work and works based on it.

       A patent license is "discriminatory" if it does not include within
     the scope of its coverage, prohibits the exercise of, or is
     conditioned on the non-exercise of one or more of the rights that are
     specifically granted under this License.  You may not convey a covered
     work if you are a party to an arrangement with a third party that is
     in the business of distributing software, under which you make payment
     to the third party based on the extent of your activity of conveying
     the work, and under which the third party grants, to any of the
     parties who would receive the covered work from you, a discriminatory
     patent license (a) in connection with copies of the covered work
     conveyed by you (or copies made from those copies), or (b) primarily
     for and in connection with specific products or compilations that
     contain the covered work, unless you entered into that arrangement,
     or that patent license was granted, prior to 28 March 2007.

       Nothing in this License shall be construed as excluding or limiting
     any implied license or other defenses to infringement that may
     otherwise be available to you under applicable patent law.

       12. No Surrender of Others' Freedom.

       If conditions are imposed on you (whether by court order, agreement or
     otherwise) that contradict the conditions of this License, they do not
     excuse you from the conditions of this License.  If you cannot convey a
     covered work so as to satisfy simultaneously your obligations under this
     License and any other pertinent obligations, then as a consequence you may
     not convey it at all.  For example, if you agree to terms that obligate you
     to collect a royalty for further conveying from those to whom you convey
     the Program, the only way you could satisfy both those terms and this
     License would be to refrain entirely from conveying the Program.

       13. Use with the GNU Affero General Public License.

       Notwithstanding any other provision of this License, you have
     permission to link or combine any covered work with a work licensed
     under version 3 of the GNU Affero General Public License into a single
     combined work, and to convey the resulting work.  The terms of this
     License will continue to apply to the part which is the covered work,
     but the special requirements of the GNU Affero General Public License,
     section 13, concerning interaction through a network will apply to the
     combination as such.

       14. Revised Versions of this License.

       The Free Software Foundation may publish revised and/or new versions of
     the GNU General Public License from time to time.  Such new versions will
     be similar in spirit to the present version, but may differ in detail to
     address new problems or concerns.

       Each version is given a distinguishing version number.  If the
     Program specifies that a certain numbered version of the GNU General
     Public License "or any later version" applies to it, you have the
     option of following the terms and conditions either of that numbered
     version or of any later version published by the Free Software
     Foundation.  If the Program does not specify a version number of the
     GNU General Public License, you may choose any version ever published
     by the Free Software Foundation.

       If the Program specifies that a proxy can decide which future
     versions of the GNU General Public License can be used, that proxy's
     public statement of acceptance of a version permanently authorizes you
     to choose that version for the Program.

       Later license versions may give you additional or different
     permissions.  However, no additional obligations are imposed on any
     author or copyright holder as a result of your choosing to follow a
     later version.

       15. Disclaimer of Warranty.

       THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY
     APPLICABLE LAW.  EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT
     HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY
     OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO,
     THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
     PURPOSE.  THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM
     IS WITH YOU.  SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF
     ALL NECESSARY SERVICING, REPAIR OR CORRECTION.

       16. Limitation of Liability.

       IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING
     WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS
     THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY
     GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE
     USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF
     DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD
     PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS),
     EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF
     SUCH DAMAGES.

       17. Interpretation of Sections 15 and 16.

       If the disclaimer of warranty and limitation of liability provided
     above cannot be given local legal effect according to their terms,
     reviewing courts shall apply local law that most closely approximates
     an absolute waiver of all civil liability in connection with the
     Program, unless a warranty or assumption of liability accompanies a
     copy of the Program in return for a fee.

                          END OF TERMS AND CONDITIONS

                 How to Apply These Terms to Your New Programs

       If you develop a new program, and you want it to be of the greatest
     possible use to the public, the best way to achieve this is to make it
     free software which everyone can redistribute and change under these terms.

       To do so, attach the following notices to the program.  It is safest
     to attach them to the start of each source file to most effectively
     state the exclusion of warranty; and each file should have at least
     the "copyright" line and a pointer to where the full notice is found.

         <one line to give the program's name and a brief idea of what it does.>
         Copyright (C) <year>  <name of author>

         This program is free software: you can redistribute it and/or modify
         it under the terms of the GNU General Public License as published by
         the Free Software Foundation, either version 3 of the License, or
         (at your option) any later version.

         This program is distributed in the hope that it will be useful,
         but WITHOUT ANY WARRANTY; without even the implied warranty of
         MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
         GNU General Public License for more details.

         You should have received a copy of the GNU General Public License
         along with this program.  If not, see <http://www.gnu.org/licenses/>.

     Also add information on how to contact you by electronic and paper mail.

       If the program does terminal interaction, make it output a short
     notice like this when it starts in an interactive mode:

         <program>  Copyright (C) <year>  <name of author>
         This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
         This is free software, and you are welcome to redistribute it
         under certain conditions; type `show c' for details.

     The hypothetical commands `show w' and `show c' should show the appropriate
     parts of the General Public License.  Of course, your program's commands
     might be different; for a GUI interface, you would use an "about box".

       You should also get your employer (if you work as a programmer) or school,
     if any, to sign a "copyright disclaimer" for the program, if necessary.
     For more information on this, and how to apply and follow the GNU GPL, see
     <http://www.gnu.org/licenses/>.

       The GNU General Public License does not permit incorporating your program
     into proprietary programs.  If your program is a subroutine library, you
     may consider it more useful to permit linking proprietary applications with
     the library.  If this is what you want to do, use the GNU Lesser General
     Public License instead of this License.  But first, please read
     <http://www.gnu.org/philosophy/why-not-lgpl.html>.


File: gsl-ref.info,  Node: GNU Free Documentation License,  Next: Index,  Prev: GNU General Public License,  Up: Top

53 GNU Free Documentation License
*********************************


                     GNU Free Documentation License
                      Version 1.3, 3 November 2008


      Copyright (C) 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc.
          <http://fsf.org/>
      Everyone is permitted to copy and distribute verbatim copies
      of this license document, but changing it is not allowed.

     0. PREAMBLE

     The purpose of this License is to make a manual, textbook, or other
     functional and useful document "free" in the sense of freedom: to
     assure everyone the effective freedom to copy and redistribute it,
     with or without modifying it, either commercially or noncommercially.
     Secondarily, this License preserves for the author and publisher a way
     to get credit for their work, while not being considered responsible
     for modifications made by others.

     This License is a kind of "copyleft", which means that derivative
     works of the document must themselves be free in the same sense.  It
     complements the GNU General Public License, which is a copyleft
     license designed for free software.

     We have designed this License in order to use it for manuals for free
     software, because free software needs free documentation: a free
     program should come with manuals providing the same freedoms that the
     software does.  But this License is not limited to software manuals;
     it can be used for any textual work, regardless of subject matter or
     whether it is published as a printed book.  We recommend this License
     principally for works whose purpose is instruction or reference.


     1. APPLICABILITY AND DEFINITIONS

     This License applies to any manual or other work, in any medium, that
     contains a notice placed by the copyright holder saying it can be
     distributed under the terms of this License.  Such a notice grants a
     world-wide, royalty-free license, unlimited in duration, to use that
     work under the conditions stated herein.  The "Document", below,
     refers to any such manual or work.  Any member of the public is a
     licensee, and is addressed as "you".  You accept the license if you
     copy, modify or distribute the work in a way requiring permission
     under copyright law.

     A "Modified Version" of the Document means any work containing the
     Document or a portion of it, either copied verbatim, or with
     modifications and/or translated into another language.

     A "Secondary Section" is a named appendix or a front-matter section of
     the Document that deals exclusively with the relationship of the
     publishers or authors of the Document to the Document's overall
     subject (or to related matters) and contains nothing that could fall
     directly within that overall subject.  (Thus, if the Document is in
     part a textbook of mathematics, a Secondary Section may not explain
     any mathematics.)  The relationship could be a matter of historical
     connection with the subject or with related matters, or of legal,
     commercial, philosophical, ethical or political position regarding
     them.

     The "Invariant Sections" are certain Secondary Sections whose titles
     are designated, as being those of Invariant Sections, in the notice
     that says that the Document is released under this License.  If a
     section does not fit the above definition of Secondary then it is not
     allowed to be designated as Invariant.  The Document may contain zero
     Invariant Sections.  If the Document does not identify any Invariant
     Sections then there are none.

     The "Cover Texts" are certain short passages of text that are listed,
     as Front-Cover Texts or Back-Cover Texts, in the notice that says that
     the Document is released under this License.  A Front-Cover Text may
     be at most 5 words, and a Back-Cover Text may be at most 25 words.

     A "Transparent" copy of the Document means a machine-readable copy,
     represented in a format whose specification is available to the
     general public, that is suitable for revising the document
     straightforwardly with generic text editors or (for images composed of
     pixels) generic paint programs or (for drawings) some widely available
     drawing editor, and that is suitable for input to text formatters or
     for automatic translation to a variety of formats suitable for input
     to text formatters.  A copy made in an otherwise Transparent file
     format whose markup, or absence of markup, has been arranged to thwart
     or discourage subsequent modification by readers is not Transparent.
     An image format is not Transparent if used for any substantial amount
     of text.  A copy that is not "Transparent" is called "Opaque".

     Examples of suitable formats for Transparent copies include plain
     ASCII without markup, Texinfo input format, LaTeX input format, SGML
     or XML using a publicly available DTD, and standard-conforming simple
     HTML, PostScript or PDF designed for human modification.  Examples of
     transparent image formats include PNG, XCF and JPG.  Opaque formats
     include proprietary formats that can be read and edited only by
     proprietary word processors, SGML or XML for which the DTD and/or
     processing tools are not generally available, and the
     machine-generated HTML, PostScript or PDF produced by some word
     processors for output purposes only.

     The "Title Page" means, for a printed book, the title page itself,
     plus such following pages as are needed to hold, legibly, the material
     this License requires to appear in the title page.  For works in
     formats which do not have any title page as such, "Title Page" means
     the text near the most prominent appearance of the work's title,
     preceding the beginning of the body of the text.

     The "publisher" means any person or entity that distributes copies of
     the Document to the public.

     A section "Entitled XYZ" means a named subunit of the Document whose
     title either is precisely XYZ or contains XYZ in parentheses following
     text that translates XYZ in another language.  (Here XYZ stands for a
     specific section name mentioned below, such as "Acknowledgements",
     "Dedications", "Endorsements", or "History".)  To "Preserve the Title"
     of such a section when you modify the Document means that it remains a
     section "Entitled XYZ" according to this definition.

     The Document may include Warranty Disclaimers next to the notice which
     states that this License applies to the Document.  These Warranty
     Disclaimers are considered to be included by reference in this
     License, but only as regards disclaiming warranties: any other
     implication that these Warranty Disclaimers may have is void and has
     no effect on the meaning of this License.

     2. VERBATIM COPYING

     You may copy and distribute the Document in any medium, either
     commercially or noncommercially, provided that this License, the
     copyright notices, and the license notice saying this License applies
     to the Document are reproduced in all copies, and that you add no
     other conditions whatsoever to those of this License.  You may not use
     technical measures to obstruct or control the reading or further
     copying of the copies you make or distribute.  However, you may accept
     compensation in exchange for copies.  If you distribute a large enough
     number of copies you must also follow the conditions in section 3.

     You may also lend copies, under the same conditions stated above, and
     you may publicly display copies.


     3. COPYING IN QUANTITY

     If you publish printed copies (or copies in media that commonly have
     printed covers) of the Document, numbering more than 100, and the
     Document's license notice requires Cover Texts, you must enclose the
     copies in covers that carry, clearly and legibly, all these Cover
     Texts: Front-Cover Texts on the front cover, and Back-Cover Texts on
     the back cover.  Both covers must also clearly and legibly identify
     you as the publisher of these copies.  The front cover must present
     the full title with all words of the title equally prominent and
     visible.  You may add other material on the covers in addition.
     Copying with changes limited to the covers, as long as they preserve
     the title of the Document and satisfy these conditions, can be treated
     as verbatim copying in other respects.

     If the required texts for either cover are too voluminous to fit
     legibly, you should put the first ones listed (as many as fit
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     It is requested, but not required, that you contact the authors of the
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     Document.


     4. MODIFICATIONS

     You may copy and distribute a Modified Version of the Document under
     the conditions of sections 2 and 3 above, provided that you release
     the Modified Version under precisely this License, with the Modified
     Version filling the role of the Document, thus licensing distribution
     and modification of the Modified Version to whoever possesses a copy
     of it.  In addition, you must do these things in the Modified Version:

     A. Use in the Title Page (and on the covers, if any) a title distinct
        from that of the Document, and from those of previous versions
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        if the original publisher of that version gives permission.
     B. List on the Title Page, as authors, one or more persons or entities
        responsible for authorship of the modifications in the Modified
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     C. State on the Title page the name of the publisher of the
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     D. Preserve all the copyright notices of the Document.
     E. Add an appropriate copyright notice for your modifications
        adjacent to the other copyright notices.
     F. Include, immediately after the copyright notices, a license notice
        giving the public permission to use the Modified Version under the
        terms of this License, in the form shown in the Addendum below.
     G. Preserve in that license notice the full lists of Invariant Sections
        and required Cover Texts given in the Document's license notice.
     H. Include an unaltered copy of this License.
     I. Preserve the section Entitled "History", Preserve its Title, and add
        to it an item stating at least the title, year, new authors, and
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     J. Preserve the network location, if any, given in the Document for
        public access to a Transparent copy of the Document, and likewise
        the network locations given in the Document for previous versions
        it was based on.  These may be placed in the "History" section.
        You may omit a network location for a work that was published at
        least four years before the Document itself, or if the original
        publisher of the version it refers to gives permission.
     K. For any section Entitled "Acknowledgements" or "Dedications",
        Preserve the Title of the section, and preserve in the section all
        the substance and tone of each of the contributor acknowledgements
        and/or dedications given therein.
     L. Preserve all the Invariant Sections of the Document,
        unaltered in their text and in their titles.  Section numbers
        or the equivalent are not considered part of the section titles.
     M. Delete any section Entitled "Endorsements".  Such a section
        may not be included in the Modified Version.
     N. Do not retitle any existing section to be Entitled "Endorsements"
        or to conflict in title with any Invariant Section.
     O. Preserve any Warranty Disclaimers.

     If the Modified Version includes new front-matter sections or
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     You may add a section Entitled "Endorsements", provided it contains
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     You may add a passage of up to five words as a Front-Cover Text, and a
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     Front-Cover Text and one of Back-Cover Text may be added by (or
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     includes a cover text for the same cover, previously added by you or
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     you may not add another; but you may replace the old one, on explicit
     permission from the previous publisher that added the old one.

     The author(s) and publisher(s) of the Document do not by this License
     give permission to use their names for publicity for or to assert or
     imply endorsement of any Modified Version.


     5. COMBINING DOCUMENTS

     You may combine the Document with other documents released under this
     License, under the terms defined in section 4 above for modified
     versions, provided that you include in the combination all of the
     Invariant Sections of all of the original documents, unmodified, and
     list them all as Invariant Sections of your combined work in its
     license notice, and that you preserve all their Warranty Disclaimers.

     The combined work need only contain one copy of this License, and
     multiple identical Invariant Sections may be replaced with a single
     copy.  If there are multiple Invariant Sections with the same name but
     different contents, make the title of each such section unique by
     adding at the end of it, in parentheses, the name of the original
     author or publisher of that section if known, or else a unique number.
     Make the same adjustment to the section titles in the list of
     Invariant Sections in the license notice of the combined work.

     In the combination, you must combine any sections Entitled "History"
     in the various original documents, forming one section Entitled
     "History"; likewise combine any sections Entitled "Acknowledgements",
     and any sections Entitled "Dedications".  You must delete all sections
     Entitled "Endorsements".


     6. COLLECTIONS OF DOCUMENTS

     You may make a collection consisting of the Document and other
     documents released under this License, and replace the individual
     copies of this License in the various documents with a single copy
     that is included in the collection, provided that you follow the rules
     of this License for verbatim copying of each of the documents in all
     other respects.

     You may extract a single document from such a collection, and
     distribute it individually under this License, provided you insert a
     copy of this License into the extracted document, and follow this
     License in all other respects regarding verbatim copying of that
     document.


     7. AGGREGATION WITH INDEPENDENT WORKS

     A compilation of the Document or its derivatives with other separate
     and independent documents or works, in or on a volume of a storage or
     distribution medium, is called an "aggregate" if the copyright
     resulting from the compilation is not used to limit the legal rights
     of the compilation's users beyond what the individual works permit.
     When the Document is included in an aggregate, this License does not
     apply to the other works in the aggregate which are not themselves
     derivative works of the Document.

     If the Cover Text requirement of section 3 is applicable to these
     copies of the Document, then if the Document is less than one half of
     the entire aggregate, the Document's Cover Texts may be placed on
     covers that bracket the Document within the aggregate, or the
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     8. TRANSLATION

     Translation is considered a kind of modification, so you may
     distribute translations of the Document under the terms of section 4.
     Replacing Invariant Sections with translations requires special
     permission from their copyright holders, but you may include
     translations of some or all Invariant Sections in addition to the
     original versions of these Invariant Sections.  You may include a
     translation of this License, and all the license notices in the
     Document, and any Warranty Disclaimers, provided that you also include
     the original English version of this License and the original versions
     of those notices and disclaimers.  In case of a disagreement between
     the translation and the original version of this License or a notice
     or disclaimer, the original version will prevail.

     If a section in the Document is Entitled "Acknowledgements",
     "Dedications", or "History", the requirement (section 4) to Preserve
     its Title (section 1) will typically require changing the actual
     title.


     9. TERMINATION

     You may not copy, modify, sublicense, or distribute the Document
     except as expressly provided under this License.  Any attempt
     otherwise to copy, modify, sublicense, or distribute it is void, and
     will automatically terminate your rights under this License.

     However, if you cease all violation of this License, then your license
     from a particular copyright holder is reinstated (a) provisionally,
     unless and until the copyright holder explicitly and finally
     terminates your license, and (b) permanently, if the copyright holder
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     Moreover, your license from a particular copyright holder is
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     received notice of violation of this License (for any work) from that
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     your receipt of the notice.

     Termination of your rights under this section does not terminate the
     licenses of parties who have received copies or rights from you under
     this License.  If your rights have been terminated and not permanently
     reinstated, receipt of a copy of some or all of the same material does
     not give you any rights to use it.


     10. FUTURE REVISIONS OF THIS LICENSE

     The Free Software Foundation may publish new, revised versions of the
     GNU Free Documentation License from time to time.  Such new versions
     will be similar in spirit to the present version, but may differ in
     detail to address new problems or concerns.  See
     http://www.gnu.org/copyleft/.

     Each version of the License is given a distinguishing version number.
     If the Document specifies that a particular numbered version of this
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     following the terms and conditions either of that specified version or
     of any later version that has been published (not as a draft) by the
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     Document.

     11. RELICENSING

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     "CC-BY-SA" means the Creative Commons Attribution-Share Alike 3.0
     license published by Creative Commons Corporation, a not-for-profit
     corporation with a principal place of business in San Francisco,
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     "Incorporate" means to publish or republish a Document, in whole or in
     part, as part of another Document.

     An MMC is "eligible for relicensing" if it is licensed under this
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     somewhere other than this MMC, and subsequently incorporated in whole or
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     The operator of an MMC Site may republish an MMC contained in the site
     under CC-BY-SA on the same site at any time before August 1, 2009,
     provided the MMC is eligible for relicensing.


     ADDENDUM: How to use this License for your documents

     To use this License in a document you have written, include a copy of
     the License in the document and put the following copyright and
     license notices just after the title page:

         Copyright (c)  YEAR  YOUR NAME.
         Permission is granted to copy, distribute and/or modify this document
         under the terms of the GNU Free Documentation License, Version 1.3
         or any later version published by the Free Software Foundation;
         with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.
         A copy of the license is included in the section entitled "GNU
         Free Documentation License".

     If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts,
     replace the "with...Texts." line with this:

         with the Invariant Sections being LIST THEIR TITLES, with the
         Front-Cover Texts being LIST, and with the Back-Cover Texts being LIST.

     If you have Invariant Sections without Cover Texts, or some other
     combination of the three, merge those two alternatives to suit the
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     If your document contains nontrivial examples of program code, we
     recommend releasing these examples in parallel under your choice of
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     to permit their use in free software.

   * genindex


File: gsl-ref.info,  Node: Index,  Prev: GNU Free Documentation License,  Up: Top

Index
*****

[index]
* Menu:

* $, shell prompt:                       Further Information. (line  26)
* 2D histograms:                         Example programs for histograms.
                                                              (line  64)
* 2D random direction vector:            Spherical Vector Distributions.
                                                              (line  10)
* 3-j symbols:                           Coupling Coefficients.
                                                              (line   6)
* 3D random direction vector:            Spherical Vector Distributions.
                                                              (line  32)
* 6-j symbols:                           Coupling Coefficients.
                                                              (line   6)
* 9-j symbols:                           Coupling Coefficients.
                                                              (line   6)
* acceleration of series:                References and Further Reading<25>.
                                                              (line  12)
* acosh:                                 Elementary Functions.
                                                              (line  37)
* Adams method:                          Stepping Functions.  (line 156)
* Adaptive step-size control, differential equations: Stepping Functions.
                                                              (line 177)
* Ai(x):                                 Airy Functions and Derivatives.
                                                              (line   6)
* Airy functions:                        Airy Functions and Derivatives.
                                                              (line   6)
* Akima splines:                         1D Interpolation Types.
                                                              (line  41)
* aliasing of arrays:                    Compatibility with C++.
                                                              (line  13)
* alpha (C variable):                    MISER.               (line 131)
* alpha (C variable) <1>:                VEGAS.               (line 166)
* alternative optimized functions:       Portability functions.
                                                              (line  34)
* AMAX, Level-1 BLAS:                    Level 1.             (line  67)
* Angular Mathieu Functions:             Angular Mathieu Functions.
                                                              (line   6)
* angular reduction:                     Restriction Functions.
                                                              (line   6)
* ANSI C, use of:                        Conventions used in this manual.
                                                              (line  20)
* Apell symbol, see Pochhammer symbol:   Pochhammer Symbol.   (line   6)
* approximate comparison of floating point numbers: Approximate Comparison of Floating Point Numbers.
                                                              (line  12)
* arctangent integral:                   Arctangent Integral. (line   6)
* argument of complex number:            Properties of complex numbers.
                                                              (line   6)
* arithmetic exceptions:                 Representation of floating point numbers.
                                                              (line 136)
* asinh:                                 Elementary Functions.
                                                              (line  42)
* astronomical constants:                Fundamental Constants.
                                                              (line  55)
* ASUM, Level-1 BLAS:                    Level 1.             (line  54)
* atanh:                                 Elementary Functions.
                                                              (line  47)
* atomic physics, constants:             Astronomy and Astrophysics.
                                                              (line  30)
* autoconf, using with GSL:              Contributors to GSL. (line 145)
* AXPY, Level-1 BLAS:                    Level 1.             (line 102)
* B-spline wavelets:                     Initialization.      (line  36)
* Bader and Deuflhard, Bulirsch-Stoer method.: Stepping Functions.
                                                              (line 150)
* balancing matrices:                    Banded LDLT Decomposition.
                                                              (line  60)
* banded Cholesky Decomposition:         Symmetric Banded Format.
                                                              (line  26)
* banded general matrices:               Banded Systems.      (line  14)
* banded LDLT decomposition:             Banded Cholesky Decomposition.
                                                              (line  75)
* banded matrices:                       Triangular Systems.  (line  45)
* banded symmetric matrices:             General Banded Format.
                                                              (line  27)
* Basic Linear Algebra Subroutines (BLAS): References and Further Reading<7>.
                                                              (line  14)
* Basic Linear Algebra Subroutines (BLAS) <1>: Autoconf Macros.
                                                              (line 106)
* basis splines, B-splines:              References and Further Reading<34>.
                                                              (line  38)
* basis splines, derivatives:            Evaluation of B-splines.
                                                              (line  35)
* basis splines, evaluation:             Constructing the knots vector.
                                                              (line  19)
* basis splines, examples:               Working with the Greville abscissae.
                                                              (line  20)
* basis splines, Greville abscissae:     Evaluation of B-spline derivatives.
                                                              (line  36)
* basis splines, initializing:           Overview<7>.         (line  34)
* basis splines, Marsden-Schoenberg points: Evaluation of B-spline derivatives.
                                                              (line  35)
* basis splines, overview:               Basis Splines.       (line  15)
* BDF method:                            Stepping Functions.  (line 166)
* Bernoulli trial, random variates:      The Bernoulli Distribution.
                                                              (line   6)
* Bessel functions:                      Bessel Functions.    (line   6)
* Bessel Functions, Fractional Order:    Regular Bessel Function—Fractional Order.
                                                              (line   6)
* best-fit parameters, covariance:       High Level Driver.   (line  39)
* Beta distribution:                     The Beta Distribution.
                                                              (line   6)
* Beta function:                         Beta Functions.      (line   6)
* Beta function, incomplete normalized:  Incomplete Beta Function.
                                                              (line   6)
* BFGS algorithm, minimization:          Algorithms with Derivatives.
                                                              (line  43)
* Bi(x):                                 Airy Functions and Derivatives.
                                                              (line   6)
* bias, IEEE format:                     IEEE floating-point arithmetic.
                                                              (line  11)
* bicubic interpolation:                 2D Interpolation Types.
                                                              (line  16)
* bidiagonalization of real matrices:    Hessenberg-Triangular Decomposition of Real Matrices.
                                                              (line  26)
* bilinear interpolation:                2D Interpolation Types.
                                                              (line  11)
* binning data:                          References and Further Reading<18>.
                                                              (line  18)
* Binomial random variates:              The Binomial Distribution.
                                                              (line   6)
* biorthogonal wavelets:                 Initialization.      (line  36)
* bisection algorithm for finding roots: Root Bracketing Algorithms.
                                                              (line  18)
* Bivariate Gaussian distribution:       The Bivariate Gaussian Distribution.
                                                              (line   6)
* Bivariate Gaussian distribution <1>:   The Multivariate Gaussian Distribution.
                                                              (line   6)
* BLAS:                                  References and Further Reading<7>.
                                                              (line  15)
* BLAS, Low-level C interface:           Autoconf Macros.     (line 107)
* BLAS, sparse:                          References and Further Reading<36>.
                                                              (line  13)
* blocks:                                Data types.          (line  60)
* bounds checking, extension to GCC:     Vector allocation.   (line  36)
* breakpoints:                           Debugging Numerical Programs.
                                                              (line   9)
* Brent’s method for finding minima:     Minimization Algorithms.
                                                              (line  33)
* Brent’s method for finding roots:      Root Bracketing Algorithms.
                                                              (line  52)
* Broyden algorithm for multidimensional roots: Algorithms without Derivatives.
                                                              (line  54)
* BSD random number generator:           Unix random number generators.
                                                              (line  17)
* bug-gsl:                               No Warranty.         (line  11)
* bug-gsl mailing list:                  No Warranty.         (line  12)
* bugs:                                  No Warranty.         (line  12)
* Bulirsch-Stoer method:                 Stepping Functions.  (line 150)
* C extensions, compatible use of:       Conventions used in this manual.
                                                              (line  20)
* C++, compatibility:                    Support for different numeric types.
                                                              (line  67)
* C99, inline keyword:                   ANSI C Compliance.   (line  25)
* Carlson forms of Elliptic integrals:   Definition of Carlson Forms.
                                                              (line   6)
* Cash-Karp, Runge-Kutta method:         Stepping Functions.  (line 119)
* Cauchy distribution:                   The Cauchy Distribution.
                                                              (line   6)
* Cauchy principal value, by numerical quadrature: QAWC adaptive integration for Cauchy principal values.
                                                              (line   6)
* CBLAS:                                 References and Further Reading<7>.
                                                              (line  15)
* CBLAS, Low-level interface:            Autoconf Macros.     (line 107)
* cblas_caxpy (C function):              Level 1<2>.          (line 105)
* cblas_ccopy (C function):              Level 1<2>.          (line 102)
* cblas_cdotc_sub (C function):          Level 1<2>.          (line  24)
* cblas_cdotu_sub (C function):          Level 1<2>.          (line  20)
* cblas_cgbmv (C function):              Level 2<2>.          (line  97)
* cblas_cgemm (C function):              Level 3<2>.          (line  86)
* cblas_cgemv (C function):              Level 2<2>.          (line  90)
* cblas_cgerc (C function):              Level 2<2>.          (line 292)
* cblas_cgeru (C function):              Level 2<2>.          (line 286)
* cblas_chbmv (C function):              Level 2<2>.          (line 274)
* cblas_chemm (C function):              Level 3<2>.          (line 170)
* cblas_chemv (C function):              Level 2<2>.          (line 268)
* cblas_cher (C function):               Level 2<2>.          (line 298)
* cblas_cher2 (C function):              Level 2<2>.          (line 308)
* cblas_cher2k (C function):             Level 3<2>.          (line 182)
* cblas_cherk (C function):              Level 3<2>.          (line 176)
* cblas_chpmv (C function):              Level 2<2>.          (line 279)
* cblas_chpr (C function):               Level 2<2>.          (line 303)
* cblas_chpr2 (C function):              Level 2<2>.          (line 314)
* cblas_cscal (C function):              Level 1<2>.          (line 149)
* cblas_csscal (C function):             Level 1<2>.          (line 155)
* cblas_cswap (C function):              Level 1<2>.          (line  99)
* cblas_csymm (C function):              Level 3<2>.          (line  92)
* cblas_csyr2k (C function):             Level 3<2>.          (line 104)
* cblas_csyrk (C function):              Level 3<2>.          (line  98)
* cblas_ctbmv (C function):              Level 2<2>.          (line 108)
* cblas_ctbsv (C function):              Level 2<2>.          (line 123)
* cblas_ctpmv (C function):              Level 2<2>.          (line 113)
* cblas_ctpsv (C function):              Level 2<2>.          (line 128)
* cblas_ctrmm (C function):              Level 3<2>.          (line 112)
* cblas_ctrmv (C function):              Level 2<2>.          (line 103)
* cblas_ctrsm (C function):              Level 3<2>.          (line 120)
* cblas_ctrsv (C function):              Level 2<2>.          (line 118)
* cblas_dasum (C function):              Level 1<2>.          (line  48)
* cblas_daxpy (C function):              Level 1<2>.          (line  96)
* cblas_dcopy (C function):              Level 1<2>.          (line  92)
* cblas_ddot (C function):               Level 1<2>.          (line  16)
* cblas_dgbmv (C function):              Level 2<2>.          (line  54)
* cblas_dgemm (C function):              Level 3<2>.          (line  48)
* cblas_dgemv (C function):              Level 2<2>.          (line  47)
* cblas_dger (C function):               Level 2<2>.          (line 241)
* cblas_dnrm2 (C function):              Level 1<2>.          (line  44)
* cblas_drot (C function):               Level 1<2>.          (line 136)
* cblas_drotg (C function):              Level 1<2>.          (line 130)
* cblas_drotm (C function):              Level 1<2>.          (line 139)
* cblas_drotmg (C function):             Level 1<2>.          (line 133)
* cblas_dsbmv (C function):              Level 2<2>.          (line 231)
* cblas_dscal (C function):              Level 1<2>.          (line 145)
* cblas_dsdot (C function):              Level 1<2>.          (line   9)
* cblas_dspmv (C function):              Level 2<2>.          (line 236)
* cblas_dspr (C function):               Level 2<2>.          (line 251)
* cblas_dspr2 (C function):              Level 2<2>.          (line 262)
* cblas_dswap (C function):              Level 1<2>.          (line  89)
* cblas_dsymm (C function):              Level 3<2>.          (line  54)
* cblas_dsymv (C function):              Level 2<2>.          (line 225)
* cblas_dsyr (C function):               Level 2<2>.          (line 246)
* cblas_dsyr2 (C function):              Level 2<2>.          (line 256)
* cblas_dsyr2k (C function):             Level 3<2>.          (line  65)
* cblas_dsyrk (C function):              Level 3<2>.          (line  60)
* cblas_dtbmv (C function):              Level 2<2>.          (line  65)
* cblas_dtbsv (C function):              Level 2<2>.          (line  80)
* cblas_dtpmv (C function):              Level 2<2>.          (line  70)
* cblas_dtpsv (C function):              Level 2<2>.          (line  85)
* cblas_dtrmm (C function):              Level 3<2>.          (line  72)
* cblas_dtrmv (C function):              Level 2<2>.          (line  60)
* cblas_dtrsm (C function):              Level 3<2>.          (line  79)
* cblas_dtrsv (C function):              Level 2<2>.          (line  75)
* cblas_dzasum (C function):             Level 1<2>.          (line  64)
* cblas_dznrm2 (C function):             Level 1<2>.          (line  60)
* cblas_icamax (C function):             Level 1<2>.          (line  74)
* cblas_idamax (C function):             Level 1<2>.          (line  71)
* cblas_isamax (C function):             Level 1<2>.          (line  68)
* cblas_izamax (C function):             Level 1<2>.          (line  77)
* cblas_sasum (C function):              Level 1<2>.          (line  40)
* cblas_saxpy (C function):              Level 1<2>.          (line  86)
* cblas_scasum (C function):             Level 1<2>.          (line  56)
* cblas_scnrm2 (C function):             Level 1<2>.          (line  52)
* cblas_scopy (C function):              Level 1<2>.          (line  83)
* cblas_sdot (C function):               Level 1<2>.          (line  13)
* cblas_sdsdot (C function):             Level 1<2>.          (line   6)
* cblas_sgbmv (C function):              Level 2<2>.          (line  11)
* cblas_sgemm (C function):              Level 3<2>.          (line   6)
* cblas_sgemv (C function):              Level 2<2>.          (line   6)
* cblas_sger (C function):               Level 2<2>.          (line 194)
* cblas_snrm2 (C function):              Level 1<2>.          (line  36)
* cblas_srot (C function):               Level 1<2>.          (line 124)
* cblas_srotg (C function):              Level 1<2>.          (line 117)
* cblas_srotm (C function):              Level 1<2>.          (line 127)
* cblas_srotmg (C function):             Level 1<2>.          (line 121)
* cblas_ssbmv (C function):              Level 2<2>.          (line 183)
* cblas_sscal (C function):              Level 1<2>.          (line 142)
* cblas_sspmv (C function):              Level 2<2>.          (line 188)
* cblas_sspr (C function):               Level 2<2>.          (line 207)
* cblas_sspr2 (C function):              Level 2<2>.          (line 219)
* cblas_sswap (C function):              Level 1<2>.          (line  80)
* cblas_ssymm (C function):              Level 3<2>.          (line  12)
* cblas_ssymv (C function):              Level 2<2>.          (line 176)
* cblas_ssyr (C function):               Level 2<2>.          (line 201)
* cblas_ssyr2 (C function):              Level 2<2>.          (line 213)
* cblas_ssyr2k (C function):             Level 3<2>.          (line  24)
* cblas_ssyrk (C function):              Level 3<2>.          (line  18)
* cblas_stbmv (C function):              Level 2<2>.          (line  22)
* cblas_stbsv (C function):              Level 2<2>.          (line  37)
* cblas_stpmv (C function):              Level 2<2>.          (line  27)
* cblas_stpsv (C function):              Level 2<2>.          (line  42)
* cblas_strmm (C function):              Level 3<2>.          (line  32)
* cblas_strmv (C function):              Level 2<2>.          (line  17)
* cblas_strsm (C function):              Level 3<2>.          (line  40)
* cblas_strsv (C function):              Level 2<2>.          (line  32)
* cblas_xerbla (C function):             Level 3<2>.          (line 210)
* cblas_zaxpy (C function):              Level 1<2>.          (line 114)
* cblas_zcopy (C function):              Level 1<2>.          (line 111)
* cblas_zdotc_sub (C function):          Level 1<2>.          (line  32)
* cblas_zdotu_sub (C function):          Level 1<2>.          (line  28)
* cblas_zdscal (C function):             Level 1<2>.          (line 158)
* cblas_zgbmv (C function):              Level 2<2>.          (line 140)
* cblas_zgemm (C function):              Level 3<2>.          (line 128)
* cblas_zgemv (C function):              Level 2<2>.          (line 133)
* cblas_zgerc (C function):              Level 2<2>.          (line 344)
* cblas_zgeru (C function):              Level 2<2>.          (line 338)
* cblas_zhbmv (C function):              Level 2<2>.          (line 326)
* cblas_zhemm (C function):              Level 3<2>.          (line 190)
* cblas_zhemv (C function):              Level 2<2>.          (line 320)
* cblas_zher (C function):               Level 2<2>.          (line 350)
* cblas_zher2 (C function):              Level 2<2>.          (line 362)
* cblas_zher2k (C function):             Level 3<2>.          (line 202)
* cblas_zherk (C function):              Level 3<2>.          (line 196)
* cblas_zhpmv (C function):              Level 2<2>.          (line 331)
* cblas_zhpr (C function):               Level 2<2>.          (line 356)
* cblas_zhpr2 (C function):              Level 2<2>.          (line 368)
* cblas_zscal (C function):              Level 1<2>.          (line 152)
* cblas_zswap (C function):              Level 1<2>.          (line 108)
* cblas_zsymm (C function):              Level 3<2>.          (line 134)
* cblas_zsyr2k (C function):             Level 3<2>.          (line 146)
* cblas_zsyrk (C function):              Level 3<2>.          (line 140)
* cblas_ztbmv (C function):              Level 2<2>.          (line 151)
* cblas_ztbsv (C function):              Level 2<2>.          (line 166)
* cblas_ztpmv (C function):              Level 2<2>.          (line 156)
* cblas_ztpsv (C function):              Level 2<2>.          (line 171)
* cblas_ztrmm (C function):              Level 3<2>.          (line 154)
* cblas_ztrmv (C function):              Level 2<2>.          (line 146)
* cblas_ztrsm (C function):              Level 3<2>.          (line 162)
* cblas_ztrsv (C function):              Level 2<2>.          (line 161)
* CDFs, cumulative distribution functions: References.        (line  14)
* ce(q,x), Mathieu function:             Angular Mathieu Functions.
                                                              (line   6)
* Chebyshev series:                      References and Further Reading<24>.
                                                              (line  15)
* checking combination for validity:     Combination properties.
                                                              (line  20)
* checking multiset for validity:        Multiset properties. (line  20)
* checking permutation for validity:     Permutation properties.
                                                              (line  15)
* Chi(x):                                Hyperbolic Integrals.
                                                              (line   6)
* Chi-squared distribution:              The Chi-squared Distribution.
                                                              (line  14)
* Cholesky decomposition:                Singular Value Decomposition.
                                                              (line  95)
* Cholesky decomposition, banded:        Symmetric Banded Format.
                                                              (line  27)
* Cholesky decomposition, modified:      Pivoted Cholesky Decomposition.
                                                              (line 107)
* Cholesky decomposition, pivoted:       Cholesky Decomposition.
                                                              (line 150)
* Cholesky decomposition, square root free: Modified Cholesky Decomposition.
                                                              (line  68)
* Ci(x):                                 Trigonometric Integrals.
                                                              (line   6)
* Clausen functions:                     Clausen Functions.   (line   6)
* Clenshaw-Curtis quadrature:            Integrands with weight functions.
                                                              (line   6)
* CMRG, combined multiple recursive random number generator: Random number generator algorithms.
                                                              (line 103)
* code reuse in applications:            Deprecated Functions.
                                                              (line  14)
* combinations:                          References and Further Reading<5>.
                                                              (line  16)
* combinatorial factor C(m:              Factorials.          (line  44)
* combinatorial optimization:            References and Further Reading<20>.
                                                              (line  21)
* comparison functions, definition:      Sorting objects.     (line  16)
* compatibility:                         Conventions used in this manual.
                                                              (line  19)
* compiling programs, include paths:     An Example Program.  (line  29)
* compiling programs, library paths:     Compiling and Linking.
                                                              (line  24)
* complementary incomplete Gamma function: Incomplete Gamma Functions.
                                                              (line  23)
* complete Fermi-Dirac integrals:        Complete Fermi-Dirac Integrals.
                                                              (line   6)
* complete orthogonal decomposition:     LQ Decomposition.    (line  64)
* complex arithmetic:                    Properties of complex numbers.
                                                              (line  27)
* complex cosine function, special functions: Trigonometric Functions for Complex Arguments.
                                                              (line  12)
* Complex Gamma function:                Gamma Functions.     (line  60)
* complex hermitian matrix, eigensystem: Complex Hermitian Matrices.
                                                              (line   9)
* complex log sine function, special functions: Trigonometric Functions for Complex Arguments.
                                                              (line  18)
* complex numbers:                       Approximate Comparison of Floating Point Numbers.
                                                              (line  30)
* complex sinc function, special functions: Circular Trigonometric Functions.
                                                              (line  23)
* complex sine function, special functions: Trigonometric Functions for Complex Arguments.
                                                              (line   6)
* confluent hypergeometric function:     Laguerre Functions.  (line   6)
* confluent hypergeometric functions:    Hypergeometric Functions.
                                                              (line   6)
* conical functions:                     Legendre Functions and Spherical Harmonics.
                                                              (line   6)
* Conjugate gradient algorithm, minimization: Algorithms with Derivatives.
                                                              (line  15)
* conjugate of complex number:           Complex arithmetic operators.
                                                              (line  66)
* constant matrix:                       Accessing matrix elements.
                                                              (line  49)
* constants, fundamental:                Physical Constants.  (line  22)
* constants, mathematical (defined as macros): Mathematical Functions.
                                                              (line  14)
* constants, physical:                   References and Further Reading<38>.
                                                              (line  15)
* constants, prefixes:                   Force and Energy.    (line  21)
* contacting the GSL developers:         Reporting Bugs.      (line  30)
* conventions, used in manual:           Further Information. (line  27)
* convergence, accelerating a series:    References and Further Reading<25>.
                                                              (line  11)
* conversion of units:                   References and Further Reading<38>.
                                                              (line  15)
* cooling schedule:                      Simulated Annealing algorithm.
                                                              (line  20)
* COPY, Level-1 BLAS:                    Level 1.             (line  91)
* correlation, of two datasets:          Covariance.          (line  24)
* cosine function, special functions:    Circular Trigonometric Functions.
                                                              (line  11)
* cosine of complex number:              Complex Trigonometric Functions.
                                                              (line  11)
* cost function:                         References and Further Reading<20>.
                                                              (line  20)
* Coulomb wave functions:                Coulomb Functions.   (line   6)
* coupling coefficients:                 Coupling Coefficients.
                                                              (line   6)
* covariance matrix, from linear regression: Linear regression with a constant term.
                                                              (line   9)
* covariance matrix, linear fits:        Overview<5>.         (line  18)
* covariance matrix, nonlinear fits:     High Level Driver.   (line  38)
* covariance, of two datasets:           Autocorrelation.     (line  23)
* cquad, doubly-adaptive integration:    QAWF adaptive integration for Fourier integrals.
                                                              (line  59)
* CRAY random number generator, RANF:    Other random number generators.
                                                              (line  22)
* cubic equation, solving:               Quadratic Equations. (line  43)
* cubic splines:                         1D Interpolation Types.
                                                              (line  23)
* cumulative distribution functions (CDFs): References.       (line  14)
* Cylindrical Bessel Functions:          Regular Cylindrical Bessel Functions.
                                                              (line   6)
* Daubechies wavelets:                   Initialization.      (line  23)
* Dawson function:                       Dawson Function.     (line   6)
* DAXPY, Level-1 BLAS:                   Level 1.             (line 102)
* debugging numerical programs:          Debugging Numerical Programs.
                                                              (line   9)
* Debye functions:                       Debye Functions.     (line   6)
* denormalized form, IEEE format:        Representation of floating point numbers.
                                                              (line  14)
* deprecated functions:                  Thread-safety.       (line  21)
* derivatives, calculating numerically:  References and Further Reading<23>.
                                                              (line  18)
* determinant of a matrix, by LU decomposition: LU Decomposition.
                                                              (line 111)
* Deuflhard and Bader, Bulirsch-Stoer method.: Stepping Functions.
                                                              (line 150)
* DFTs, see FFT:                         References and Further Reading<10>.
                                                              (line  27)
* diagonal, of a matrix:                 Creating row and column views.
                                                              (line  67)
* differential equations, initial value problems: References and Further Reading<21>.
                                                              (line  11)
* differentiation of functions, numeric: References and Further Reading<23>.
                                                              (line  18)
* digamma function:                      Psi Digamma Function.
                                                              (line   6)
* dilogarithm:                           Dilogarithm.         (line   6)
* direction vector, random 2D:           Spherical Vector Distributions.
                                                              (line  10)
* direction vector, random 3D:           Spherical Vector Distributions.
                                                              (line  32)
* direction vector, random N-dimensional: Spherical Vector Distributions.
                                                              (line  42)
* Dirichlet distribution:                The Dirichlet Distribution.
                                                              (line   6)
* discontinuities, in ODE systems:       Evolution.           (line  90)
* Discrete Fourier Transforms, see FFT:  References and Further Reading<10>.
                                                              (line  28)
* discrete Hankel transforms:            References and Further Reading<27>.
                                                              (line  54)
* Discrete Newton algorithm for multidimensional roots: Algorithms without Derivatives.
                                                              (line  32)
* Discrete random numbers:               General Discrete Distributions.
                                                              (line  49)
* Discrete random numbers <1>:           General Discrete Distributions.
                                                              (line  65)
* Discrete random numbers <2>:           General Discrete Distributions.
                                                              (line  71)
* Discrete random numbers <3>:           General Discrete Distributions.
                                                              (line  80)
* Discrete random numbers, preprocessing: General Discrete Distributions.
                                                              (line  49)
* dither (C variable):                   MISER.               (line 149)
* divided differences, polynomials:      Polynomial Evaluation.
                                                              (line  41)
* division by zero, IEEE exceptions:     Representation of floating point numbers.
                                                              (line 136)
* Dogleg algorithm:                      Levenberg-Marquardt with Geodesic Acceleration.
                                                              (line  30)
* Dogleg algorithm, double:              Dogleg.              (line  27)
* dollar sign $, shell prompt:           Conventions used in this manual.
                                                              (line   6)
* DOT, Level-1 BLAS:                     Level 1.             (line   6)
* double Dogleg algorithm:               Dogleg.              (line  27)
* double factorial:                      Factorials.          (line  19)
* double precision, IEEE format:         Representation of floating point numbers.
                                                              (line  43)
* downloading GSL:                       GSL is Free Software.
                                                              (line  46)
* DWT initialization:                    Definitions<2>.      (line  27)
* DWT, mathematical definition:          Wavelet Transforms.  (line  11)
* DWT, one dimensional:                  Transform Functions. (line  12)
* DWT, see wavelet transforms:           References and Further Reading<26>.
                                                              (line  25)
* DWT, two dimensional:                  Wavelet transforms in one dimension.
                                                              (line  43)
* e, defined as a macro:                 Mathematical Constants.
                                                              (line   9)
* E1(x):                                 Exponential Integral.
                                                              (line   6)
* E2(x):                                 Exponential Integral.
                                                              (line   6)
* Ei(x):                                 Exponential Integral.
                                                              (line   6)
* eigenvalues and eigenvectors:          References and Further Reading<9>.
                                                              (line  64)
* elementary functions:                  Examples.            (line  39)
* elementary operations:                 Elementary Operations.
                                                              (line   6)
* elliptic functions (Jacobi):           Elliptic Functions Jacobi.
                                                              (line   6)
* elliptic integrals:                    Elliptic Integrals.  (line   6)
* energy function:                       References and Further Reading<20>.
                                                              (line  21)
* energy, units of:                      Mass and Weight.     (line  50)
* erf(x):                                Error Functions.     (line   6)
* erfc(x):                               Error Functions.     (line   6)
* Erlang distribution:                   The Gamma Distribution.
                                                              (line   6)
* error codes:                           Error Codes.         (line  13)
* error codes, reserved:                 Error Reporting.     (line  44)
* error function:                        Error Functions.     (line   6)
* error handlers:                        Error Codes.         (line  50)
* error handling:                        Code Reuse.          (line  14)
* error handling macros:                 Error Handlers.      (line  81)
* estimated standard deviation:          References and Further Reading<14>.
                                                              (line  52)
* estimated variance:                    References and Further Reading<14>.
                                                              (line  52)
* estimate_frac (C variable):            MISER.               (line 104)
* estimation, location:                  Order Statistics.    (line  21)
* estimation, scale:                     Gastwirth Estimator. (line  23)
* Eta Function:                          Eta Function.        (line   6)
* euclidean distance function, hypot:    Elementary Functions.
                                                              (line  24)
* euclidean distance function, hypot3:   Elementary Functions.
                                                              (line  30)
* Euler’s constant, defined as a macro:  Mathematical Constants.
                                                              (line   9)
* evaluation of polynomials:             Polynomials.         (line  13)
* evaluation of polynomials, in divided difference form: Polynomial Evaluation.
                                                              (line  40)
* examples, conventions used in:         Further Information. (line  27)
* exceptions, C++:                       Support for different numeric types.
                                                              (line  66)
* exceptions, floating point:            Examining floating point registers.
                                                              (line  32)
* exceptions, IEEE arithmetic:           Representation of floating point numbers.
                                                              (line 136)
* exchanging permutation elements:       Accessing permutation elements.
                                                              (line  17)
* exp:                                   Exponential Functions.
                                                              (line   6)
* expm1:                                 Elementary Functions.
                                                              (line  18)
* exponent, IEEE format:                 IEEE floating-point arithmetic.
                                                              (line  11)
* Exponential distribution:              The Exponential Distribution.
                                                              (line   6)
* exponential function:                  Exponential Functions.
                                                              (line   6)
* exponential integrals:                 Exponential Integrals.
                                                              (line   6)
* Exponential power distribution:        The Exponential Power Distribution.
                                                              (line   6)
* exponential, difference from 1 computed accurately: Elementary Functions.
                                                              (line  18)
* exponentiation of complex number:      Elementary Complex Functions.
                                                              (line  17)
* extern inline:                         ANSI C Compliance.   (line  24)
* F-distribution:                        The F-distribution.  (line  13)
* factorial:                             Factorials.          (line   6)
* factorial <1>:                         Factorials.          (line  11)
* factorization of matrices:             References and Further Reading<8>.
                                                              (line  32)
* false position algorithm for finding roots: Root Bracketing Algorithms.
                                                              (line  34)
* Fast Fourier Transforms, see FFT:      References and Further Reading<10>.
                                                              (line  28)
* Fehlberg method, differential equations: Stepping Functions.
                                                              (line 114)
* Fermi-Dirac function:                  Fermi-Dirac Function.
                                                              (line   6)
* FFT:                                   References and Further Reading<10>.
                                                              (line  28)
* FFT mathematical definition:           Fast Fourier Transforms FFTs.
                                                              (line  18)
* FFT of complex data, mixed-radix algorithm: Radix-2 FFT routines for complex data.
                                                              (line 129)
* FFT of complex data, radix-2 algorithm: Overview of complex data FFTs.
                                                              (line  73)
* FFT of real data:                      Mixed-radix FFT routines for complex data.
                                                              (line 226)
* FFT of real data, mixed-radix algorithm: Radix-2 FFT routines for real data.
                                                              (line 102)
* FFT of real data, radix-2 algorithm:   Overview of real data FFTs.
                                                              (line  50)
* FFT, complex data:                     Mathematical Definitions.
                                                              (line  54)
* finding minima:                        References and Further Reading<29>.
                                                              (line  17)
* finding roots:                         References and Further Reading<28>.
                                                              (line  15)
* finding zeros:                         References and Further Reading<28>.
                                                              (line  15)
* fits, multi-parameter linear:          Linear regression without a constant term.
                                                              (line  47)
* fitting:                               References and Further Reading<32>.
                                                              (line  24)
* fitting, using Chebyshev polynomials:  References and Further Reading<24>.
                                                              (line  15)
* Fj(x), Fermi-Dirac integral:           Complete Fermi-Dirac Integrals.
                                                              (line   6)
* Fj(x,b), incomplete Fermi-Dirac integral: Incomplete Fermi-Dirac Integrals.
                                                              (line   6)
* flat distribution:                     The Flat Uniform Distribution.
                                                              (line   6)
* Fletcher-Reeves conjugate gradient algorithm, minimization: Algorithms with Derivatives.
                                                              (line  15)
* floating point exceptions:             Examining floating point registers.
                                                              (line  31)
* floating point numbers, approximate comparison: Approximate Comparison of Floating Point Numbers.
                                                              (line  12)
* floating point registers:              Using gdb.           (line  89)
* force and energy:                      Radioactivity.       (line  18)
* Fortran range checking, equivalent in gcc: Vector allocation.
                                                              (line  35)
* Four-tap Generalized Feedback Shift Register: Random number generator algorithms.
                                                              (line 181)
* Fourier integrals, numerical:          QAWF adaptive integration for Fourier integrals.
                                                              (line   6)
* Fourier Transforms, see FFT:           References and Further Reading<10>.
                                                              (line  28)
* Fractional Order Bessel Functions:     Regular Bessel Function—Fractional Order.
                                                              (line   6)
* free software, explanation of:         Routines available in GSL.
                                                              (line  53)
* frexp:                                 Elementary Functions.
                                                              (line  57)
* functions, numerical differentiation:  References and Further Reading<23>.
                                                              (line  18)
* fundamental constants:                 Physical Constants.  (line  22)
* Gamma distribution:                    The Gamma Distribution.
                                                              (line   6)
* gamma functions:                       Gamma Functions.     (line   6)
* Gastwirth estimator:                   Trimmed Mean.        (line  25)
* Gauss-Kronrod quadrature:              Introduction<2>.     (line  57)
* Gaussian distribution:                 The Gaussian Distribution.
                                                              (line   6)
* Gaussian distribution, bivariate:      The Bivariate Gaussian Distribution.
                                                              (line   6)
* Gaussian distribution, bivariate <1>:  The Multivariate Gaussian Distribution.
                                                              (line   6)
* Gaussian Tail distribution:            The Gaussian Tail Distribution.
                                                              (line   6)
* gcc extensions, range-checking:        Vector allocation.   (line  36)
* gcc warning options:                   Handling floating point exceptions.
                                                              (line  30)
* gdb:                                   Debugging Numerical Programs.
                                                              (line   9)
* Gegenbauer functions:                  Gegenbauer Functions.
                                                              (line   6)
* GEMM, Level-3 BLAS:                    Level 3.             (line   6)
* GEMV, Level-2 BLAS:                    Level 2.             (line   6)
* general polynomial equations, solving: Cubic Equations.     (line  38)
* generalized eigensystems:              Real Generalized Nonsymmetric Eigensystems.
                                                              (line   6)
* generalized hermitian definite eigensystems: Complex Generalized Hermitian-Definite Eigensystems.
                                                              (line   6)
* generalized symmetric eigensystems:    Real Generalized Symmetric-Definite Eigensystems.
                                                              (line   6)
* Geometric random variates:             The Geometric Distribution.
                                                              (line   6)
* Geometric random variates <1>:         The Hypergeometric Distribution.
                                                              (line   6)
* GER, Level-2 BLAS:                     Level 2.             (line 107)
* GERC, Level-2 BLAS:                    Level 2.             (line 122)
* GERU, Level-2 BLAS:                    Level 2.             (line 107)
* Givens rotation:                       Bidiagonalization.   (line  56)
* Givens Rotation, BLAS:                 Level 1.             (line 128)
* Givens Rotation, Modified, BLAS:       Level 1.             (line 153)
* gmres:                                 Types of Sparse Iterative Solvers.
                                                              (line  11)
* GNU General Public License:            Top.                 (line  12)
* golden section algorithm for finding minima: Minimization Algorithms.
                                                              (line  15)
* gsl_acosh (C function):                Elementary Functions.
                                                              (line  37)
* gsl_asinh (C function):                Elementary Functions.
                                                              (line  42)
* gsl_atanh (C function):                Elementary Functions.
                                                              (line  47)
* gsl_blas_caxpy (C function):           Level 1.             (line 102)
* gsl_blas_ccopy (C function):           Level 1.             (line  91)
* gsl_blas_cdotc (C function):           Level 1.             (line  32)
* gsl_blas_cdotu (C function):           Level 1.             (line  24)
* gsl_blas_cgemm (C function):           Level 3.             (line   6)
* gsl_blas_cgemv (C function):           Level 2.             (line   6)
* gsl_blas_cgerc (C function):           Level 2.             (line 122)
* gsl_blas_cgeru (C function):           Level 2.             (line 107)
* gsl_blas_chemm (C function):           Level 3.             (line  63)
* gsl_blas_chemv (C function):           Level 2.             (line  88)
* gsl_blas_cher (C function):            Level 2.             (line 145)
* gsl_blas_cher2 (C function):           Level 2.             (line 171)
* gsl_blas_cher2k (C function):          Level 3.             (line 217)
* gsl_blas_cherk (C function):           Level 3.             (line 170)
* gsl_blas_cscal (C function):           Level 1.             (line 114)
* gsl_blas_csscal (C function):          Level 1.             (line 114)
* gsl_blas_cswap (C function):           Level 1.             (line  81)
* gsl_blas_csymm (C function):           Level 3.             (line  36)
* gsl_blas_csyr2k (C function):          Level 3.             (line 187)
* gsl_blas_csyrk (C function):           Level 3.             (line 145)
* gsl_blas_ctrmm (C function):           Level 3.             (line  82)
* gsl_blas_ctrmv (C function):           Level 2.             (line  28)
* gsl_blas_ctrsm (C function):           Level 3.             (line 113)
* gsl_blas_ctrsv (C function):           Level 2.             (line  51)
* gsl_blas_dasum (C function):           Level 1.             (line  54)
* gsl_blas_daxpy (C function):           Level 1.             (line 102)
* gsl_blas_dcopy (C function):           Level 1.             (line  91)
* gsl_blas_ddot (C function):            Level 1.             (line  13)
* gsl_blas_dgemm (C function):           Level 3.             (line   6)
* gsl_blas_dgemv (C function):           Level 2.             (line   6)
* gsl_blas_dger (C function):            Level 2.             (line 107)
* gsl_blas_dnrm2 (C function):           Level 1.             (line  40)
* gsl_blas_drot (C function):            Level 1.             (line 143)
* gsl_blas_drotg (C function):           Level 1.             (line 128)
* gsl_blas_drotm (C function):           Level 1.             (line 163)
* gsl_blas_drotmg (C function):          Level 1.             (line 153)
* gsl_blas_dscal (C function):           Level 1.             (line 114)
* gsl_blas_dsdot (C function):           Level 1.             (line  13)
* gsl_blas_dswap (C function):           Level 1.             (line  81)
* gsl_blas_dsymm (C function):           Level 3.             (line  36)
* gsl_blas_dsymv (C function):           Level 2.             (line  73)
* gsl_blas_dsyr (C function):            Level 2.             (line 133)
* gsl_blas_dsyr2 (C function):           Level 2.             (line 158)
* gsl_blas_dsyr2k (C function):          Level 3.             (line 187)
* gsl_blas_dsyrk (C function):           Level 3.             (line 145)
* gsl_blas_dtrmm (C function):           Level 3.             (line  82)
* gsl_blas_dtrmv (C function):           Level 2.             (line  28)
* gsl_blas_dtrsm (C function):           Level 3.             (line 113)
* gsl_blas_dtrsv (C function):           Level 2.             (line  51)
* gsl_blas_dzasum (C function):          Level 1.             (line  60)
* gsl_blas_dznrm2 (C function):          Level 1.             (line  46)
* gsl_blas_icamax (C function):          Level 1.             (line  67)
* gsl_blas_idamax (C function):          Level 1.             (line  67)
* gsl_blas_isamax (C function):          Level 1.             (line  67)
* gsl_blas_izamax (C function):          Level 1.             (line  67)
* gsl_blas_sasum (C function):           Level 1.             (line  54)
* gsl_blas_saxpy (C function):           Level 1.             (line 102)
* gsl_blas_scasum (C function):          Level 1.             (line  60)
* gsl_blas_scnrm2 (C function):          Level 1.             (line  46)
* gsl_blas_scopy (C function):           Level 1.             (line  91)
* gsl_blas_sdot (C function):            Level 1.             (line  13)
* gsl_blas_sdsdot (C function):          Level 1.             (line   6)
* gsl_blas_sgemm (C function):           Level 3.             (line   6)
* gsl_blas_sgemv (C function):           Level 2.             (line   6)
* gsl_blas_sger (C function):            Level 2.             (line 107)
* gsl_blas_snrm2 (C function):           Level 1.             (line  40)
* gsl_blas_srot (C function):            Level 1.             (line 143)
* gsl_blas_srotg (C function):           Level 1.             (line 128)
* gsl_blas_srotm (C function):           Level 1.             (line 163)
* gsl_blas_srotmg (C function):          Level 1.             (line 153)
* gsl_blas_sscal (C function):           Level 1.             (line 114)
* gsl_blas_sswap (C function):           Level 1.             (line  81)
* gsl_blas_ssymm (C function):           Level 3.             (line  36)
* gsl_blas_ssymv (C function):           Level 2.             (line  73)
* gsl_blas_ssyr (C function):            Level 2.             (line 133)
* gsl_blas_ssyr2 (C function):           Level 2.             (line 158)
* gsl_blas_ssyr2k (C function):          Level 3.             (line 187)
* gsl_blas_ssyrk (C function):           Level 3.             (line 145)
* gsl_blas_strmm (C function):           Level 3.             (line  82)
* gsl_blas_strmv (C function):           Level 2.             (line  28)
* gsl_blas_strsm (C function):           Level 3.             (line 113)
* gsl_blas_strsv (C function):           Level 2.             (line  51)
* gsl_blas_zaxpy (C function):           Level 1.             (line 102)
* gsl_blas_zcopy (C function):           Level 1.             (line  91)
* gsl_blas_zdotc (C function):           Level 1.             (line  32)
* gsl_blas_zdotu (C function):           Level 1.             (line  24)
* gsl_blas_zdscal (C function):          Level 1.             (line 114)
* gsl_blas_zgemm (C function):           Level 3.             (line   6)
* gsl_blas_zgemv (C function):           Level 2.             (line   6)
* gsl_blas_zgerc (C function):           Level 2.             (line 122)
* gsl_blas_zgeru (C function):           Level 2.             (line 107)
* gsl_blas_zhemm (C function):           Level 3.             (line  63)
* gsl_blas_zhemv (C function):           Level 2.             (line  88)
* gsl_blas_zher (C function):            Level 2.             (line 145)
* gsl_blas_zher2 (C function):           Level 2.             (line 171)
* gsl_blas_zher2k (C function):          Level 3.             (line 217)
* gsl_blas_zherk (C function):           Level 3.             (line 170)
* gsl_blas_zscal (C function):           Level 1.             (line 114)
* gsl_blas_zswap (C function):           Level 1.             (line  81)
* gsl_blas_zsymm (C function):           Level 3.             (line  36)
* gsl_blas_zsyr2k (C function):          Level 3.             (line 187)
* gsl_blas_zsyrk (C function):           Level 3.             (line 145)
* gsl_blas_ztrmm (C function):           Level 3.             (line  82)
* gsl_blas_ztrmv (C function):           Level 2.             (line  28)
* gsl_blas_ztrsm (C function):           Level 3.             (line 113)
* gsl_blas_ztrsv (C function):           Level 2.             (line  51)
* gsl_block (C type):                    Blocks.              (line  10)
* gsl_block_alloc (C function):          Block allocation.    (line  14)
* gsl_block_calloc (C function):         Block allocation.    (line  26)
* gsl_block_fprintf (C function):        Reading and writing blocks.
                                                              (line  27)
* gsl_block_fread (C function):          Reading and writing blocks.
                                                              (line  17)
* gsl_block_free (C function):           Block allocation.    (line  31)
* gsl_block_fscanf (C function):         Reading and writing blocks.
                                                              (line  36)
* gsl_block_fwrite (C function):         Reading and writing blocks.
                                                              (line   9)
* gsl_bspline_alloc (C function):        Initializing the B-splines solver.
                                                              (line  11)
* gsl_bspline_deriv_eval (C function):   Evaluation of B-spline derivatives.
                                                              (line   6)
* gsl_bspline_deriv_eval_nonzero (C function): Evaluation of B-spline derivatives.
                                                              (line  20)
* gsl_bspline_eval (C function):         Evaluation of B-splines.
                                                              (line   6)
* gsl_bspline_eval_nonzero (C function): Evaluation of B-splines.
                                                              (line  17)
* gsl_bspline_free (C function):         Initializing the B-splines solver.
                                                              (line  20)
* gsl_bspline_greville_abscissa (C function): Working with the Greville abscissae.
                                                              (line  14)
* gsl_bspline_knots (C function):        Constructing the knots vector.
                                                              (line   6)
* gsl_bspline_knots_uniform (C function): Constructing the knots vector.
                                                              (line  12)
* gsl_bspline_ncoeffs (C function):      Evaluation of B-splines.
                                                              (line  31)
* gsl_bspline_workspace (C type):        Initializing the B-splines solver.
                                                              (line   6)
* GSL_C99_INLINE:                        ANSI C Compliance.   (line  25)
* GSL_C99_INLINE (C macro):              Accessing vector elements.
                                                              (line  29)
* gsl_cdf_beta_P (C function):           The Beta Distribution.
                                                              (line  23)
* gsl_cdf_beta_Pinv (C function):        The Beta Distribution.
                                                              (line  23)
* gsl_cdf_beta_Q (C function):           The Beta Distribution.
                                                              (line  23)
* gsl_cdf_beta_Qinv (C function):        The Beta Distribution.
                                                              (line  23)
* gsl_cdf_binomial_P (C function):       The Binomial Distribution.
                                                              (line  26)
* gsl_cdf_binomial_Q (C function):       The Binomial Distribution.
                                                              (line  26)
* gsl_cdf_cauchy_P (C function):         The Cauchy Distribution.
                                                              (line  24)
* gsl_cdf_cauchy_Pinv (C function):      The Cauchy Distribution.
                                                              (line  24)
* gsl_cdf_cauchy_Q (C function):         The Cauchy Distribution.
                                                              (line  24)
* gsl_cdf_cauchy_Qinv (C function):      The Cauchy Distribution.
                                                              (line  24)
* gsl_cdf_chisq_P (C function):          The Chi-squared Distribution.
                                                              (line  31)
* gsl_cdf_chisq_Pinv (C function):       The Chi-squared Distribution.
                                                              (line  31)
* gsl_cdf_chisq_Q (C function):          The Chi-squared Distribution.
                                                              (line  31)
* gsl_cdf_chisq_Qinv (C function):       The Chi-squared Distribution.
                                                              (line  31)
* gsl_cdf_exponential_P (C function):    The Exponential Distribution.
                                                              (line  22)
* gsl_cdf_exponential_Pinv (C function): The Exponential Distribution.
                                                              (line  22)
* gsl_cdf_exponential_Q (C function):    The Exponential Distribution.
                                                              (line  22)
* gsl_cdf_exponential_Qinv (C function): The Exponential Distribution.
                                                              (line  22)
* gsl_cdf_exppow_P (C function):         The Exponential Power Distribution.
                                                              (line  27)
* gsl_cdf_exppow_Q (C function):         The Exponential Power Distribution.
                                                              (line  27)
* gsl_cdf_fdist_P (C function):          The F-distribution.  (line  36)
* gsl_cdf_fdist_Pinv (C function):       The F-distribution.  (line  36)
* gsl_cdf_fdist_Q (C function):          The F-distribution.  (line  36)
* gsl_cdf_fdist_Qinv (C function):       The F-distribution.  (line  36)
* gsl_cdf_flat_P (C function):           The Flat Uniform Distribution.
                                                              (line  23)
* gsl_cdf_flat_Pinv (C function):        The Flat Uniform Distribution.
                                                              (line  23)
* gsl_cdf_flat_Q (C function):           The Flat Uniform Distribution.
                                                              (line  23)
* gsl_cdf_flat_Qinv (C function):        The Flat Uniform Distribution.
                                                              (line  23)
* gsl_cdf_gamma_P (C function):          The Gamma Distribution.
                                                              (line  37)
* gsl_cdf_gamma_Pinv (C function):       The Gamma Distribution.
                                                              (line  37)
* gsl_cdf_gamma_Q (C function):          The Gamma Distribution.
                                                              (line  37)
* gsl_cdf_gamma_Qinv (C function):       The Gamma Distribution.
                                                              (line  37)
* gsl_cdf_gaussian_P (C function):       The Gaussian Distribution.
                                                              (line  47)
* gsl_cdf_gaussian_Pinv (C function):    The Gaussian Distribution.
                                                              (line  47)
* gsl_cdf_gaussian_Q (C function):       The Gaussian Distribution.
                                                              (line  47)
* gsl_cdf_gaussian_Qinv (C function):    The Gaussian Distribution.
                                                              (line  47)
* gsl_cdf_geometric_P (C function):      The Geometric Distribution.
                                                              (line  27)
* gsl_cdf_geometric_Q (C function):      The Geometric Distribution.
                                                              (line  27)
* gsl_cdf_gumbel1_P (C function):        The Type-1 Gumbel Distribution.
                                                              (line  24)
* gsl_cdf_gumbel1_Pinv (C function):     The Type-1 Gumbel Distribution.
                                                              (line  24)
* gsl_cdf_gumbel1_Q (C function):        The Type-1 Gumbel Distribution.
                                                              (line  24)
* gsl_cdf_gumbel1_Qinv (C function):     The Type-1 Gumbel Distribution.
                                                              (line  24)
* gsl_cdf_gumbel2_P (C function):        The Type-2 Gumbel Distribution.
                                                              (line  24)
* gsl_cdf_gumbel2_Pinv (C function):     The Type-2 Gumbel Distribution.
                                                              (line  24)
* gsl_cdf_gumbel2_Q (C function):        The Type-2 Gumbel Distribution.
                                                              (line  24)
* gsl_cdf_gumbel2_Qinv (C function):     The Type-2 Gumbel Distribution.
                                                              (line  24)
* gsl_cdf_hypergeometric_P (C function): The Hypergeometric Distribution.
                                                              (line  31)
* gsl_cdf_hypergeometric_Q (C function): The Hypergeometric Distribution.
                                                              (line  31)
* gsl_cdf_laplace_P (C function):        The Laplace Distribution.
                                                              (line  21)
* gsl_cdf_laplace_Pinv (C function):     The Laplace Distribution.
                                                              (line  21)
* gsl_cdf_laplace_Q (C function):        The Laplace Distribution.
                                                              (line  21)
* gsl_cdf_laplace_Qinv (C function):     The Laplace Distribution.
                                                              (line  21)
* gsl_cdf_logistic_P (C function):       The Logistic Distribution.
                                                              (line  22)
* gsl_cdf_logistic_Pinv (C function):    The Logistic Distribution.
                                                              (line  22)
* gsl_cdf_logistic_Q (C function):       The Logistic Distribution.
                                                              (line  22)
* gsl_cdf_logistic_Qinv (C function):    The Logistic Distribution.
                                                              (line  22)
* gsl_cdf_lognormal_P (C function):      The Lognormal Distribution.
                                                              (line  26)
* gsl_cdf_lognormal_Pinv (C function):   The Lognormal Distribution.
                                                              (line  26)
* gsl_cdf_lognormal_Q (C function):      The Lognormal Distribution.
                                                              (line  26)
* gsl_cdf_lognormal_Qinv (C function):   The Lognormal Distribution.
                                                              (line  26)
* gsl_cdf_negative_binomial_P (C function): The Negative Binomial Distribution.
                                                              (line  27)
* gsl_cdf_negative_binomial_Q (C function): The Negative Binomial Distribution.
                                                              (line  27)
* gsl_cdf_pareto_P (C function):         The Pareto Distribution.
                                                              (line  24)
* gsl_cdf_pareto_Pinv (C function):      The Pareto Distribution.
                                                              (line  24)
* gsl_cdf_pareto_Q (C function):         The Pareto Distribution.
                                                              (line  24)
* gsl_cdf_pareto_Qinv (C function):      The Pareto Distribution.
                                                              (line  24)
* gsl_cdf_pascal_P (C function):         The Pascal Distribution.
                                                              (line  25)
* gsl_cdf_pascal_Q (C function):         The Pascal Distribution.
                                                              (line  25)
* gsl_cdf_poisson_P (C function):        The Poisson Distribution.
                                                              (line  23)
* gsl_cdf_poisson_Q (C function):        The Poisson Distribution.
                                                              (line  23)
* gsl_cdf_rayleigh_P (C function):       The Rayleigh Distribution.
                                                              (line  22)
* gsl_cdf_rayleigh_Pinv (C function):    The Rayleigh Distribution.
                                                              (line  22)
* gsl_cdf_rayleigh_Q (C function):       The Rayleigh Distribution.
                                                              (line  22)
* gsl_cdf_rayleigh_Qinv (C function):    The Rayleigh Distribution.
                                                              (line  22)
* gsl_cdf_tdist_P (C function):          The t-distribution.  (line  30)
* gsl_cdf_tdist_Pinv (C function):       The t-distribution.  (line  30)
* gsl_cdf_tdist_Q (C function):          The t-distribution.  (line  30)
* gsl_cdf_tdist_Qinv (C function):       The t-distribution.  (line  30)
* gsl_cdf_ugaussian_P (C function):      The Gaussian Distribution.
                                                              (line  56)
* gsl_cdf_ugaussian_Pinv (C function):   The Gaussian Distribution.
                                                              (line  56)
* gsl_cdf_ugaussian_Q (C function):      The Gaussian Distribution.
                                                              (line  56)
* gsl_cdf_ugaussian_Qinv (C function):   The Gaussian Distribution.
                                                              (line  56)
* gsl_cdf_weibull_P (C function):        The Weibull Distribution.
                                                              (line  24)
* gsl_cdf_weibull_Pinv (C function):     The Weibull Distribution.
                                                              (line  24)
* gsl_cdf_weibull_Q (C function):        The Weibull Distribution.
                                                              (line  24)
* gsl_cdf_weibull_Qinv (C function):     The Weibull Distribution.
                                                              (line  24)
* gsl_cheb_alloc (C function):           Creation and Calculation of Chebyshev Series.
                                                              (line   6)
* gsl_cheb_calc_deriv (C function):      Derivatives and Integrals.
                                                              (line  12)
* gsl_cheb_calc_integ (C function):      Derivatives and Integrals.
                                                              (line  20)
* gsl_cheb_coeffs (C function):          Auxiliary Functions. (line  13)
* gsl_cheb_eval (C function):            Chebyshev Series Evaluation.
                                                              (line   6)
* gsl_cheb_eval_err (C function):        Chebyshev Series Evaluation.
                                                              (line  12)
* gsl_cheb_eval_n (C function):          Chebyshev Series Evaluation.
                                                              (line  21)
* gsl_cheb_eval_n_err (C function):      Chebyshev Series Evaluation.
                                                              (line  28)
* gsl_cheb_free (C function):            Creation and Calculation of Chebyshev Series.
                                                              (line  11)
* gsl_cheb_init (C function):            Creation and Calculation of Chebyshev Series.
                                                              (line  15)
* gsl_cheb_order (C function):           Auxiliary Functions. (line   9)
* gsl_cheb_series (C type):              Definitions.         (line   6)
* gsl_cheb_size (C function):            Auxiliary Functions. (line  13)
* gsl_check_range (C variable):          Accessing vector elements.
                                                              (line  37)
* gsl_combination (C type):              The Combination struct.
                                                              (line   6)
* gsl_combination_alloc (C function):    Combination allocation.
                                                              (line   6)
* gsl_combination_calloc (C function):   Combination allocation.
                                                              (line  17)
* gsl_combination_data (C function):     Combination properties.
                                                              (line  15)
* gsl_combination_fprintf (C function):  Reading and writing combinations.
                                                              (line  30)
* gsl_combination_fread (C function):    Reading and writing combinations.
                                                              (line  18)
* gsl_combination_free (C function):     Combination allocation.
                                                              (line  37)
* gsl_combination_fscanf (C function):   Reading and writing combinations.
                                                              (line  40)
* gsl_combination_fwrite (C function):   Reading and writing combinations.
                                                              (line   9)
* gsl_combination_get (C function):      Accessing combination elements.
                                                              (line   9)
* gsl_combination_init_first (C function): Combination allocation.
                                                              (line  26)
* gsl_combination_init_last (C function): Combination allocation.
                                                              (line  31)
* gsl_combination_k (C function):        Combination properties.
                                                              (line  10)
* gsl_combination_memcpy (C function):   Combination allocation.
                                                              (line  41)
* gsl_combination_n (C function):        Combination properties.
                                                              (line   6)
* gsl_combination_next (C function):     Combination functions.
                                                              (line   6)
* gsl_combination_prev (C function):     Combination functions.
                                                              (line  15)
* gsl_combination_valid (C function):    Combination properties.
                                                              (line  20)
* gsl_comparison_fn_t (C type):          Sorting objects.     (line  23)
* gsl_complex:                           Complex Numbers.     (line  21)
* gsl_complex_abs (C function):          Properties of complex numbers.
                                                              (line  11)
* gsl_complex_abs2 (C function):         Properties of complex numbers.
                                                              (line  15)
* gsl_complex_add (C function):          Complex arithmetic operators.
                                                              (line   6)
* gsl_complex_add_imag (C function):     Complex arithmetic operators.
                                                              (line  46)
* gsl_complex_add_real (C function):     Complex arithmetic operators.
                                                              (line  26)
* gsl_complex_arccos (C function):       Inverse Complex Trigonometric Functions.
                                                              (line  21)
* gsl_complex_arccosh (C function):      Inverse Complex Hyperbolic Functions.
                                                              (line  12)
* gsl_complex_arccosh_real (C function): Inverse Complex Hyperbolic Functions.
                                                              (line  20)
* gsl_complex_arccos_real (C function):  Inverse Complex Trigonometric Functions.
                                                              (line  27)
* gsl_complex_arccot (C function):       Inverse Complex Trigonometric Functions.
                                                              (line  61)
* gsl_complex_arccoth (C function):      Inverse Complex Hyperbolic Functions.
                                                              (line  46)
* gsl_complex_arccsc (C function):       Inverse Complex Trigonometric Functions.
                                                              (line  51)
* gsl_complex_arccsch (C function):      Inverse Complex Hyperbolic Functions.
                                                              (line  41)
* gsl_complex_arccsc_real (C function):  Inverse Complex Trigonometric Functions.
                                                              (line  56)
* gsl_complex_arcsec (C function):       Inverse Complex Trigonometric Functions.
                                                              (line  41)
* gsl_complex_arcsech (C function):      Inverse Complex Hyperbolic Functions.
                                                              (line  36)
* gsl_complex_arcsec_real (C function):  Inverse Complex Trigonometric Functions.
                                                              (line  46)
* gsl_complex_arcsin (C function):       Inverse Complex Trigonometric Functions.
                                                              (line   6)
* gsl_complex_arcsinh (C function):      Inverse Complex Hyperbolic Functions.
                                                              (line   6)
* gsl_complex_arcsin_real (C function):  Inverse Complex Trigonometric Functions.
                                                              (line  12)
* gsl_complex_arctan (C function):       Inverse Complex Trigonometric Functions.
                                                              (line  35)
* gsl_complex_arctanh (C function):      Inverse Complex Hyperbolic Functions.
                                                              (line  25)
* gsl_complex_arctanh_real (C function): Inverse Complex Hyperbolic Functions.
                                                              (line  31)
* gsl_complex_arg (C function):          Properties of complex numbers.
                                                              (line   6)
* gsl_complex_conjugate (C function):    Complex arithmetic operators.
                                                              (line  66)
* gsl_complex_cos (C function):          Complex Trigonometric Functions.
                                                              (line  11)
* gsl_complex_cosh (C function):         Complex Hyperbolic Functions.
                                                              (line  11)
* gsl_complex_cot (C function):          Complex Trigonometric Functions.
                                                              (line  31)
* gsl_complex_coth (C function):         Complex Hyperbolic Functions.
                                                              (line  31)
* gsl_complex_csc (C function):          Complex Trigonometric Functions.
                                                              (line  26)
* gsl_complex_csch (C function):         Complex Hyperbolic Functions.
                                                              (line  26)
* gsl_complex_div (C function):          Complex arithmetic operators.
                                                              (line  21)
* gsl_complex_div_imag (C function):     Complex arithmetic operators.
                                                              (line  61)
* gsl_complex_div_real (C function):     Complex arithmetic operators.
                                                              (line  41)
* gsl_complex_exp (C function):          Elementary Complex Functions.
                                                              (line  28)
* gsl_complex_inverse (C function):      Complex arithmetic operators.
                                                              (line  71)
* gsl_complex_log (C function):          Elementary Complex Functions.
                                                              (line  33)
* gsl_complex_log10 (C function):        Elementary Complex Functions.
                                                              (line  39)
* gsl_complex_logabs (C function):       Properties of complex numbers.
                                                              (line  20)
* gsl_complex_log_b (C function):        Elementary Complex Functions.
                                                              (line  44)
* gsl_complex_mul (C function):          Complex arithmetic operators.
                                                              (line  16)
* gsl_complex_mul_imag (C function):     Complex arithmetic operators.
                                                              (line  56)
* gsl_complex_mul_real (C function):     Complex arithmetic operators.
                                                              (line  36)
* gsl_complex_negative (C function):     Complex arithmetic operators.
                                                              (line  76)
* gsl_complex_polar (C function):        Representation of complex numbers.
                                                              (line  30)
* gsl_complex_poly_complex_eval (C function): Polynomial Evaluation.
                                                              (line  25)
* gsl_complex_pow (C function):          Elementary Complex Functions.
                                                              (line  17)
* gsl_complex_pow_real (C function):     Elementary Complex Functions.
                                                              (line  23)
* gsl_complex_rect (C function):         Representation of complex numbers.
                                                              (line  24)
* gsl_complex_sec (C function):          Complex Trigonometric Functions.
                                                              (line  21)
* gsl_complex_sech (C function):         Complex Hyperbolic Functions.
                                                              (line  21)
* gsl_complex_sin (C function):          Complex Trigonometric Functions.
                                                              (line   6)
* gsl_complex_sinh (C function):         Complex Hyperbolic Functions.
                                                              (line   6)
* gsl_complex_sqrt (C function):         Elementary Complex Functions.
                                                              (line   6)
* gsl_complex_sqrt_real (C function):    Elementary Complex Functions.
                                                              (line  12)
* gsl_complex_sub (C function):          Complex arithmetic operators.
                                                              (line  11)
* gsl_complex_sub_imag (C function):     Complex arithmetic operators.
                                                              (line  51)
* gsl_complex_sub_real (C function):     Complex arithmetic operators.
                                                              (line  31)
* gsl_complex_tan (C function):          Complex Trigonometric Functions.
                                                              (line  16)
* gsl_complex_tanh (C function):         Complex Hyperbolic Functions.
                                                              (line  16)
* GSL_CONST_MKSA_ACRE (C macro):         Volume Area and Length.
                                                              (line  14)
* GSL_CONST_MKSA_ANGSTROM (C macro):     Atomic and Nuclear Physics.
                                                              (line  47)
* GSL_CONST_MKSA_ASTRONOMICAL_UNIT (C macro): Astronomy and Astrophysics.
                                                              (line   6)
* GSL_CONST_MKSA_BAR (C macro):          Pressure.            (line   6)
* GSL_CONST_MKSA_BARN (C macro):         Atomic and Nuclear Physics.
                                                              (line  51)
* GSL_CONST_MKSA_BOHR_MAGNETON (C macro): Atomic and Nuclear Physics.
                                                              (line  55)
* GSL_CONST_MKSA_BOHR_RADIUS (C macro):  Atomic and Nuclear Physics.
                                                              (line  43)
* GSL_CONST_MKSA_BOLTZMANN (C macro):    Fundamental Constants.
                                                              (line  36)
* GSL_CONST_MKSA_BTU (C macro):          Thermal Energy and Power.
                                                              (line  10)
* GSL_CONST_MKSA_CALORIE (C macro):      Thermal Energy and Power.
                                                              (line   6)
* GSL_CONST_MKSA_CANADIAN_GALLON (C macro): Volume Area and Length.
                                                              (line  26)
* GSL_CONST_MKSA_CARAT (C macro):        Mass and Weight.     (line  30)
* GSL_CONST_MKSA_CURIE (C macro):        Radioactivity.       (line   6)
* GSL_CONST_MKSA_DAY (C macro):          Measurement of Time. (line  14)
* GSL_CONST_MKSA_DEBYE (C macro):        Atomic and Nuclear Physics.
                                                              (line  76)
* GSL_CONST_MKSA_DYNE (C macro):         Force and Energy.    (line  10)
* GSL_CONST_MKSA_ELECTRON_CHARGE (C macro): Atomic and Nuclear Physics.
                                                              (line   6)
* GSL_CONST_MKSA_ELECTRON_MAGNETIC_MOMENT (C macro): Atomic and Nuclear Physics.
                                                              (line  63)
* GSL_CONST_MKSA_ELECTRON_VOLT (C macro): Atomic and Nuclear Physics.
                                                              (line  10)
* GSL_CONST_MKSA_ERG (C macro):          Force and Energy.    (line  18)
* GSL_CONST_MKSA_FARADAY (C macro):      Fundamental Constants.
                                                              (line  32)
* GSL_CONST_MKSA_FATHOM (C macro):       Speed and Nautical Units.
                                                              (line  18)
* GSL_CONST_MKSA_FOOT (C macro):         Imperial Units.      (line  10)
* GSL_CONST_MKSA_FOOTCANDLE (C macro):   Light and Illumination.
                                                              (line  22)
* GSL_CONST_MKSA_FOOTLAMBERT (C macro):  Light and Illumination.
                                                              (line  30)
* GSL_CONST_MKSA_GAUSS (C macro):        Fundamental Constants.
                                                              (line  52)
* GSL_CONST_MKSA_GRAM_FORCE (C macro):   Mass and Weight.     (line  34)
* GSL_CONST_MKSA_GRAVITATIONAL_CONSTANT (C macro): Astronomy and Astrophysics.
                                                              (line  10)
* GSL_CONST_MKSA_GRAV_ACCEL (C macro):   Astronomy and Astrophysics.
                                                              (line  22)
* GSL_CONST_MKSA_HECTARE (C macro):      Volume Area and Length.
                                                              (line  10)
* GSL_CONST_MKSA_HORSEPOWER (C macro):   Thermal Energy and Power.
                                                              (line  18)
* GSL_CONST_MKSA_HOUR (C macro):         Measurement of Time. (line  10)
* GSL_CONST_MKSA_INCH (C macro):         Imperial Units.      (line   6)
* GSL_CONST_MKSA_INCH_OF_MERCURY (C macro): Pressure.         (line  22)
* GSL_CONST_MKSA_INCH_OF_WATER (C macro): Pressure.           (line  26)
* GSL_CONST_MKSA_JOULE (C macro):        Force and Energy.    (line  14)
* GSL_CONST_MKSA_KILOMETERS_PER_HOUR (C macro): Speed and Nautical Units.
                                                              (line   6)
* GSL_CONST_MKSA_KILOPOUND_FORCE (C macro): Mass and Weight.  (line  42)
* GSL_CONST_MKSA_KNOT (C macro):         Speed and Nautical Units.
                                                              (line  22)
* GSL_CONST_MKSA_LAMBERT (C macro):      Light and Illumination.
                                                              (line  26)
* GSL_CONST_MKSA_LIGHT_YEAR (C macro):   Astronomy and Astrophysics.
                                                              (line  14)
* GSL_CONST_MKSA_LITER (C macro):        Volume Area and Length.
                                                              (line  18)
* GSL_CONST_MKSA_LUMEN (C macro):        Light and Illumination.
                                                              (line  10)
* GSL_CONST_MKSA_LUX (C macro):          Light and Illumination.
                                                              (line  14)
* GSL_CONST_MKSA_MASS_ELECTRON (C macro): Atomic and Nuclear Physics.
                                                              (line  18)
* GSL_CONST_MKSA_MASS_MUON (C macro):    Atomic and Nuclear Physics.
                                                              (line  22)
* GSL_CONST_MKSA_MASS_NEUTRON (C macro): Atomic and Nuclear Physics.
                                                              (line  30)
* GSL_CONST_MKSA_MASS_PROTON (C macro):  Atomic and Nuclear Physics.
                                                              (line  26)
* GSL_CONST_MKSA_METER_OF_MERCURY (C macro): Pressure.        (line  18)
* GSL_CONST_MKSA_METRIC_TON (C macro):   Mass and Weight.     (line  18)
* GSL_CONST_MKSA_MICRON (C macro):       Volume Area and Length.
                                                              (line   6)
* GSL_CONST_MKSA_MIL (C macro):          Imperial Units.      (line  22)
* GSL_CONST_MKSA_MILE (C macro):         Imperial Units.      (line  18)
* GSL_CONST_MKSA_MILES_PER_HOUR (C macro): Speed and Nautical Units.
                                                              (line  10)
* GSL_CONST_MKSA_MINUTE (C macro):       Measurement of Time. (line   6)
* GSL_CONST_MKSA_MOLAR_GAS (C macro):    Fundamental Constants.
                                                              (line  40)
* GSL_CONST_MKSA_NAUTICAL_MILE (C macro): Speed and Nautical Units.
                                                              (line  14)
* GSL_CONST_MKSA_NEWTON (C macro):       Force and Energy.    (line   6)
* GSL_CONST_MKSA_NUCLEAR_MAGNETON (C macro): Atomic and Nuclear Physics.
                                                              (line  59)
* GSL_CONST_MKSA_OUNCE_MASS (C macro):   Mass and Weight.     (line  10)
* GSL_CONST_MKSA_PARSEC (C macro):       Astronomy and Astrophysics.
                                                              (line  18)
* GSL_CONST_MKSA_PHOT (C macro):         Light and Illumination.
                                                              (line  18)
* GSL_CONST_MKSA_PINT (C macro):         Volume Area and Length.
                                                              (line  38)
* GSL_CONST_MKSA_PLANCKS_CONSTANT_H (C macro): Fundamental Constants.
                                                              (line  20)
* GSL_CONST_MKSA_PLANCKS_CONSTANT_HBAR (C macro): Fundamental Constants.
                                                              (line  24)
* GSL_CONST_MKSA_POINT (C macro):        Printers Units.      (line   6)
* GSL_CONST_MKSA_POISE (C macro):        Viscosity.           (line   6)
* GSL_CONST_MKSA_POUNDAL (C macro):      Mass and Weight.     (line  46)
* GSL_CONST_MKSA_POUND_FORCE (C macro):  Mass and Weight.     (line  38)
* GSL_CONST_MKSA_POUND_MASS (C macro):   Mass and Weight.     (line   6)
* GSL_CONST_MKSA_PROTON_MAGNETIC_MOMENT (C macro): Atomic and Nuclear Physics.
                                                              (line  68)
* GSL_CONST_MKSA_PSI (C macro):          Pressure.            (line  30)
* GSL_CONST_MKSA_QUART (C macro):        Volume Area and Length.
                                                              (line  34)
* GSL_CONST_MKSA_RAD (C macro):          Radioactivity.       (line  14)
* GSL_CONST_MKSA_ROENTGEN (C macro):     Radioactivity.       (line  10)
* GSL_CONST_MKSA_RYDBERG (C macro):      Atomic and Nuclear Physics.
                                                              (line  38)
* GSL_CONST_MKSA_SOLAR_MASS (C macro):   Astronomy and Astrophysics.
                                                              (line  26)
* GSL_CONST_MKSA_SPEED_OF_LIGHT (C macro): Fundamental Constants.
                                                              (line   6)
* GSL_CONST_MKSA_STANDARD_GAS_VOLUME (C macro): Fundamental Constants.
                                                              (line  44)
* GSL_CONST_MKSA_STD_ATMOSPHERE (C macro): Pressure.          (line  10)
* GSL_CONST_MKSA_STEFAN_BOLTZMANN_CONSTANT (C macro): Fundamental Constants.
                                                              (line  48)
* GSL_CONST_MKSA_STILB (C macro):        Light and Illumination.
                                                              (line   6)
* GSL_CONST_MKSA_STOKES (C macro):       Viscosity.           (line  10)
* GSL_CONST_MKSA_TEXPOINT (C macro):     Printers Units.      (line  10)
* GSL_CONST_MKSA_THERM (C macro):        Thermal Energy and Power.
                                                              (line  14)
* GSL_CONST_MKSA_THOMSON_CROSS_SECTION (C macro): Atomic and Nuclear Physics.
                                                              (line  72)
* GSL_CONST_MKSA_TON (C macro):          Mass and Weight.     (line  14)
* GSL_CONST_MKSA_TORR (C macro):         Pressure.            (line  14)
* GSL_CONST_MKSA_TROY_OUNCE (C macro):   Mass and Weight.     (line  26)
* GSL_CONST_MKSA_UK_GALLON (C macro):    Volume Area and Length.
                                                              (line  30)
* GSL_CONST_MKSA_UK_TON (C macro):       Mass and Weight.     (line  22)
* GSL_CONST_MKSA_UNIFIED_ATOMIC_MASS (C macro): Atomic and Nuclear Physics.
                                                              (line  14)
* GSL_CONST_MKSA_US_GALLON (C macro):    Volume Area and Length.
                                                              (line  22)
* GSL_CONST_MKSA_VACUUM_PERMEABILITY (C macro): Fundamental Constants.
                                                              (line  10)
* GSL_CONST_MKSA_VACUUM_PERMITTIVITY (C macro): Fundamental Constants.
                                                              (line  15)
* GSL_CONST_MKSA_WEEK (C macro):         Measurement of Time. (line  18)
* GSL_CONST_MKSA_YARD (C macro):         Imperial Units.      (line  14)
* GSL_CONST_NUM_ATTO (C macro):          Prefixes.            (line  60)
* GSL_CONST_NUM_AVOGADRO (C macro):      Fundamental Constants.
                                                              (line  28)
* GSL_CONST_NUM_EXA (C macro):           Prefixes.            (line  16)
* GSL_CONST_NUM_FEMTO (C macro):         Prefixes.            (line  56)
* GSL_CONST_NUM_FINE_STRUCTURE (C macro): Atomic and Nuclear Physics.
                                                              (line  34)
* GSL_CONST_NUM_GIGA (C macro):          Prefixes.            (line  28)
* GSL_CONST_NUM_KILO (C macro):          Prefixes.            (line  36)
* GSL_CONST_NUM_MEGA (C macro):          Prefixes.            (line  32)
* GSL_CONST_NUM_MICRO (C macro):         Prefixes.            (line  44)
* GSL_CONST_NUM_MILLI (C macro):         Prefixes.            (line  40)
* GSL_CONST_NUM_NANO (C macro):          Prefixes.            (line  48)
* GSL_CONST_NUM_PETA (C macro):          Prefixes.            (line  20)
* GSL_CONST_NUM_PICO (C macro):          Prefixes.            (line  52)
* GSL_CONST_NUM_TERA (C macro):          Prefixes.            (line  24)
* GSL_CONST_NUM_YOCTO (C macro):         Prefixes.            (line  68)
* GSL_CONST_NUM_YOTTA (C macro):         Prefixes.            (line   8)
* GSL_CONST_NUM_ZEPTO (C macro):         Prefixes.            (line  64)
* GSL_CONST_NUM_ZETTA (C macro):         Prefixes.            (line  12)
* gsl_deriv_backward (C function):       Functions.           (line  44)
* gsl_deriv_central (C function):        Functions.           (line   6)
* gsl_deriv_forward (C function):        Functions.           (line  24)
* gsl_dht (C type):                      Functions<2>.        (line   6)
* gsl_dht_alloc (C function):            Functions<2>.        (line  10)
* gsl_dht_apply (C function):            Functions<2>.        (line  30)
* gsl_dht_free (C function):             Functions<2>.        (line  26)
* gsl_dht_init (C function):             Functions<2>.        (line  15)
* gsl_dht_k_sample (C function):         Functions<2>.        (line  46)
* gsl_dht_new (C function):              Functions<2>.        (line  20)
* gsl_dht_x_sample (C function):         Functions<2>.        (line  40)
* GSL_EDOM (C variable):                 Error Codes.         (line  13)
* gsl_eigen_gen (C function):            Real Generalized Nonsymmetric Eigensystems.
                                                              (line  89)
* gsl_eigen_genherm (C function):        Complex Generalized Hermitian-Definite Eigensystems.
                                                              (line  37)
* gsl_eigen_genhermv (C function):       Complex Generalized Hermitian-Definite Eigensystems.
                                                              (line  63)
* gsl_eigen_genhermv_alloc (C function): Complex Generalized Hermitian-Definite Eigensystems.
                                                              (line  51)
* gsl_eigen_genhermv_free (C function):  Complex Generalized Hermitian-Definite Eigensystems.
                                                              (line  58)
* gsl_eigen_genhermv_sort (C function):  Sorting Eigenvalues and Eigenvectors.
                                                              (line  54)
* gsl_eigen_genhermv_workspace (C type): Complex Generalized Hermitian-Definite Eigensystems.
                                                              (line  46)
* gsl_eigen_genherm_alloc (C function):  Complex Generalized Hermitian-Definite Eigensystems.
                                                              (line  24)
* gsl_eigen_genherm_free (C function):   Complex Generalized Hermitian-Definite Eigensystems.
                                                              (line  31)
* gsl_eigen_genherm_workspace (C type):  Complex Generalized Hermitian-Definite Eigensystems.
                                                              (line  19)
* gsl_eigen_gensymm (C function):        Real Generalized Symmetric-Definite Eigensystems.
                                                              (line  44)
* gsl_eigen_gensymmv (C function):       Real Generalized Symmetric-Definite Eigensystems.
                                                              (line  69)
* gsl_eigen_gensymmv_alloc (C function): Real Generalized Symmetric-Definite Eigensystems.
                                                              (line  57)
* gsl_eigen_gensymmv_free (C function):  Real Generalized Symmetric-Definite Eigensystems.
                                                              (line  64)
* gsl_eigen_gensymmv_sort (C function):  Sorting Eigenvalues and Eigenvectors.
                                                              (line  46)
* gsl_eigen_gensymmv_workspace (C type): Real Generalized Symmetric-Definite Eigensystems.
                                                              (line  52)
* gsl_eigen_gensymm_alloc (C function):  Real Generalized Symmetric-Definite Eigensystems.
                                                              (line  31)
* gsl_eigen_gensymm_free (C function):   Real Generalized Symmetric-Definite Eigensystems.
                                                              (line  38)
* gsl_eigen_gensymm_workspace (C type):  Real Generalized Symmetric-Definite Eigensystems.
                                                              (line  26)
* gsl_eigen_genv (C function):           Real Generalized Nonsymmetric Eigensystems.
                                                              (line 134)
* gsl_eigen_genv_alloc (C function):     Real Generalized Nonsymmetric Eigensystems.
                                                              (line 122)
* gsl_eigen_genv_free (C function):      Real Generalized Nonsymmetric Eigensystems.
                                                              (line 130)
* gsl_eigen_genv_QZ (C function):        Real Generalized Nonsymmetric Eigensystems.
                                                              (line 151)
* gsl_eigen_genv_sort (C function):      Sorting Eigenvalues and Eigenvectors.
                                                              (line  62)
* gsl_eigen_genv_workspace (C type):     Real Generalized Nonsymmetric Eigensystems.
                                                              (line 117)
* gsl_eigen_gen_alloc (C function):      Real Generalized Nonsymmetric Eigensystems.
                                                              (line  54)
* gsl_eigen_gen_free (C function):       Real Generalized Nonsymmetric Eigensystems.
                                                              (line  62)
* gsl_eigen_gen_params (C function):     Real Generalized Nonsymmetric Eigensystems.
                                                              (line  66)
* gsl_eigen_gen_QZ (C function):         Real Generalized Nonsymmetric Eigensystems.
                                                              (line 109)
* gsl_eigen_gen_workspace (C type):      Real Generalized Nonsymmetric Eigensystems.
                                                              (line  49)
* gsl_eigen_herm (C function):           Complex Hermitian Matrices.
                                                              (line  26)
* gsl_eigen_hermv (C function):          Complex Hermitian Matrices.
                                                              (line  54)
* gsl_eigen_hermv_alloc (C function):    Complex Hermitian Matrices.
                                                              (line  42)
* gsl_eigen_hermv_free (C function):     Complex Hermitian Matrices.
                                                              (line  50)
* gsl_eigen_hermv_sort (C function):     Sorting Eigenvalues and Eigenvectors.
                                                              (line  28)
* gsl_eigen_hermv_workspace (C type):    Complex Hermitian Matrices.
                                                              (line  37)
* gsl_eigen_herm_alloc (C function):     Complex Hermitian Matrices.
                                                              (line  14)
* gsl_eigen_herm_free (C function):      Complex Hermitian Matrices.
                                                              (line  22)
* gsl_eigen_herm_workspace (C type):     Complex Hermitian Matrices.
                                                              (line   9)
* gsl_eigen_nonsymm (C function):        Real Nonsymmetric Matrices.
                                                              (line  72)
* gsl_eigen_nonsymmv (C function):       Real Nonsymmetric Matrices.
                                                              (line 123)
* gsl_eigen_nonsymmv_alloc (C function): Real Nonsymmetric Matrices.
                                                              (line 100)
* gsl_eigen_nonsymmv_free (C function):  Real Nonsymmetric Matrices.
                                                              (line 107)
* gsl_eigen_nonsymmv_params (C function): Real Nonsymmetric Matrices.
                                                              (line 112)
* gsl_eigen_nonsymmv_sort (C function):  Sorting Eigenvalues and Eigenvectors.
                                                              (line  36)
* gsl_eigen_nonsymmv_workspace (C type): Real Nonsymmetric Matrices.
                                                              (line  95)
* gsl_eigen_nonsymmv_Z (C function):     Real Nonsymmetric Matrices.
                                                              (line 138)
* gsl_eigen_nonsymm_alloc (C function):  Real Nonsymmetric Matrices.
                                                              (line  22)
* gsl_eigen_nonsymm_free (C function):   Real Nonsymmetric Matrices.
                                                              (line  29)
* gsl_eigen_nonsymm_params (C function): Real Nonsymmetric Matrices.
                                                              (line  35)
* gsl_eigen_nonsymm_workspace (C type):  Real Nonsymmetric Matrices.
                                                              (line  17)
* gsl_eigen_nonsymm_Z (C function):      Real Nonsymmetric Matrices.
                                                              (line  87)
* gsl_eigen_sort_t (C type):             Sorting Eigenvalues and Eigenvectors.
                                                              (line  14)
* gsl_eigen_symm (C function):           Real Symmetric Matrices.
                                                              (line  29)
* gsl_eigen_symmv (C function):          Real Symmetric Matrices.
                                                              (line  56)
* gsl_eigen_symmv_alloc (C function):    Real Symmetric Matrices.
                                                              (line  44)
* gsl_eigen_symmv_free (C function):     Real Symmetric Matrices.
                                                              (line  52)
* gsl_eigen_symmv_sort (C function):     Sorting Eigenvalues and Eigenvectors.
                                                              (line   6)
* gsl_eigen_symmv_workspace (C type):    Real Symmetric Matrices.
                                                              (line  39)
* gsl_eigen_symm_alloc (C function):     Real Symmetric Matrices.
                                                              (line  17)
* gsl_eigen_symm_free (C function):      Real Symmetric Matrices.
                                                              (line  25)
* gsl_eigen_symm_workspace (C type):     Real Symmetric Matrices.
                                                              (line  12)
* GSL_EINVAL (C variable):               Error Codes.         (line  32)
* GSL_ENOMEM (C variable):               Error Codes.         (line  25)
* GSL_ERANGE (C variable):               Error Codes.         (line  19)
* GSL_ERROR (C macro):                   Using GSL error reporting in your own functions.
                                                              (line  15)
* gsl_error_handler_t (C type):          Error Handlers.      (line  23)
* GSL_ERROR_VAL (C macro):               Using GSL error reporting in your own functions.
                                                              (line  36)
* gsl_expm1 (C function):                Elementary Functions.
                                                              (line  18)
* gsl_fcmp (C function):                 Approximate Comparison of Floating Point Numbers.
                                                              (line  12)
* gsl_fft_complex_backward (C function): Mixed-radix FFT routines for complex data.
                                                              (line 120)
* gsl_fft_complex_forward (C function):  Mixed-radix FFT routines for complex data.
                                                              (line 120)
* gsl_fft_complex_inverse (C function):  Mixed-radix FFT routines for complex data.
                                                              (line 120)
* gsl_fft_complex_radix2_backward (C function): Radix-2 FFT routines for complex data.
                                                              (line  16)
* gsl_fft_complex_radix2_dif_backward (C function): Radix-2 FFT routines for complex data.
                                                              (line  42)
* gsl_fft_complex_radix2_dif_forward (C function): Radix-2 FFT routines for complex data.
                                                              (line  42)
* gsl_fft_complex_radix2_dif_inverse (C function): Radix-2 FFT routines for complex data.
                                                              (line  42)
* gsl_fft_complex_radix2_dif_transform (C function): Radix-2 FFT routines for complex data.
                                                              (line  42)
* gsl_fft_complex_radix2_forward (C function): Radix-2 FFT routines for complex data.
                                                              (line  16)
* gsl_fft_complex_radix2_inverse (C function): Radix-2 FFT routines for complex data.
                                                              (line  16)
* gsl_fft_complex_radix2_transform (C function): Radix-2 FFT routines for complex data.
                                                              (line  16)
* gsl_fft_complex_transform (C function): Mixed-radix FFT routines for complex data.
                                                              (line 120)
* gsl_fft_complex_wavetable (C type):    Mixed-radix FFT routines for complex data.
                                                              (line  79)
* gsl_fft_complex_wavetable_alloc (C function): Mixed-radix FFT routines for complex data.
                                                              (line  44)
* gsl_fft_complex_wavetable_free (C function): Mixed-radix FFT routines for complex data.
                                                              (line  64)
* gsl_fft_complex_workspace (C type):    Mixed-radix FFT routines for complex data.
                                                              (line 100)
* gsl_fft_complex_workspace_alloc (C function): Mixed-radix FFT routines for complex data.
                                                              (line 105)
* gsl_fft_complex_workspace_free (C function): Mixed-radix FFT routines for complex data.
                                                              (line 111)
* gsl_fft_halfcomplex_radix2_backward (C function): Radix-2 FFT routines for real data.
                                                              (line  57)
* gsl_fft_halfcomplex_radix2_inverse (C function): Radix-2 FFT routines for real data.
                                                              (line  57)
* gsl_fft_halfcomplex_radix2_unpack (C function): Radix-2 FFT routines for real data.
                                                              (line  70)
* gsl_fft_halfcomplex_transform (C function): Mixed-radix FFT routines for real data.
                                                              (line 137)
* gsl_fft_halfcomplex_unpack (C function): Mixed-radix FFT routines for real data.
                                                              (line 175)
* gsl_fft_halfcomplex_wavetable (C type): Mixed-radix FFT routines for real data.
                                                              (line  72)
* gsl_fft_halfcomplex_wavetable_alloc (C function): Mixed-radix FFT routines for real data.
                                                              (line  78)
* gsl_fft_halfcomplex_wavetable_free (C function): Mixed-radix FFT routines for real data.
                                                              (line 101)
* gsl_fft_real_radix2_transform (C function): Radix-2 FFT routines for real data.
                                                              (line  13)
* gsl_fft_real_transform (C function):   Mixed-radix FFT routines for real data.
                                                              (line 137)
* gsl_fft_real_unpack (C function):      Mixed-radix FFT routines for real data.
                                                              (line 158)
* gsl_fft_real_wavetable (C type):       Mixed-radix FFT routines for real data.
                                                              (line  72)
* gsl_fft_real_wavetable_alloc (C function): Mixed-radix FFT routines for real data.
                                                              (line  78)
* gsl_fft_real_wavetable_free (C function): Mixed-radix FFT routines for real data.
                                                              (line 101)
* gsl_fft_real_workspace (C type):       Mixed-radix FFT routines for real data.
                                                              (line 114)
* gsl_fft_real_workspace_alloc (C function): Mixed-radix FFT routines for real data.
                                                              (line 118)
* gsl_fft_real_workspace_free (C function): Mixed-radix FFT routines for real data.
                                                              (line 126)
* GSL_FILTER_END_PADVALUE (C macro):     Handling Endpoints<2>.
                                                              (line  27)
* GSL_FILTER_END_PADZERO (C macro):      Handling Endpoints<2>.
                                                              (line  17)
* gsl_filter_end_t (C type):             Handling Endpoints<2>.
                                                              (line  12)
* GSL_FILTER_END_TRUNCATE (C macro):     Handling Endpoints<2>.
                                                              (line  36)
* gsl_filter_gaussian (C function):      Gaussian Filter.     (line  50)
* gsl_filter_gaussian_alloc (C function): Gaussian Filter.    (line  37)
* gsl_filter_gaussian_free (C function): Gaussian Filter.     (line  45)
* gsl_filter_gaussian_kernel (C function): Gaussian Filter.   (line  62)
* gsl_filter_impulse (C function):       Impulse Detection Filter.
                                                              (line 114)
* gsl_filter_impulse_alloc (C function): Impulse Detection Filter.
                                                              (line 101)
* gsl_filter_impulse_free (C function):  Impulse Detection Filter.
                                                              (line 109)
* gsl_filter_median (C function):        Standard Median Filter.
                                                              (line  25)
* gsl_filter_median_alloc (C function):  Standard Median Filter.
                                                              (line  11)
* gsl_filter_median_free (C function):   Standard Median Filter.
                                                              (line  19)
* gsl_filter_rmedian (C function):       Recursive Median Filter.
                                                              (line  33)
* gsl_filter_rmedian_alloc (C function): Recursive Median Filter.
                                                              (line  19)
* gsl_filter_rmedian_free (C function):  Recursive Median Filter.
                                                              (line  28)
* GSL_FILTER_SCALE_IQR (C macro):        Impulse Detection Filter.
                                                              (line  61)
* GSL_FILTER_SCALE_MAD (C macro):        Impulse Detection Filter.
                                                              (line  50)
* GSL_FILTER_SCALE_QN (C macro):         Impulse Detection Filter.
                                                              (line  81)
* GSL_FILTER_SCALE_SN (C macro):         Impulse Detection Filter.
                                                              (line  75)
* gsl_filter_scale_t (C type):           Impulse Detection Filter.
                                                              (line  45)
* gsl_finite (C function):               Infinities and Not-a-number.
                                                              (line  30)
* gsl_fit_linear (C function):           Linear regression with a constant term.
                                                              (line   9)
* gsl_fit_linear_est (C function):       Linear regression with a constant term.
                                                              (line  45)
* gsl_fit_mul (C function):              Linear regression without a constant term.
                                                              (line  10)
* gsl_fit_mul_est (C function):          Linear regression without a constant term.
                                                              (line  40)
* gsl_fit_wlinear (C function):          Linear regression with a constant term.
                                                              (line  27)
* gsl_fit_wmul (C function):             Linear regression without a constant term.
                                                              (line  23)
* gsl_frexp (C function):                Elementary Functions.
                                                              (line  57)
* gsl_function (C type):                 Providing the function to solve.
                                                              (line  11)
* gsl_function_fdf (C type):             Providing the function to solve.
                                                              (line  54)
* gsl_heapsort (C function):             Sorting objects.     (line  16)
* gsl_heapsort_index (C function):       Sorting objects.     (line  57)
* gsl_histogram (C type):                The histogram struct.
                                                              (line   8)
* gsl_histogram2d (C type):              The 2D histogram struct.
                                                              (line   8)
* gsl_histogram2d_accumulate (C function): Updating and accessing 2D histogram elements.
                                                              (line  27)
* gsl_histogram2d_add (C function):      2D Histogram Operations.
                                                              (line  12)
* gsl_histogram2d_alloc (C function):    2D Histogram allocation.
                                                              (line  14)
* gsl_histogram2d_clone (C function):    Copying 2D Histograms.
                                                              (line  13)
* gsl_histogram2d_cov (C function):      2D Histogram Statistics.
                                                              (line  60)
* gsl_histogram2d_div (C function):      2D Histogram Operations.
                                                              (line  36)
* gsl_histogram2d_equal_bins_p (C function): 2D Histogram Operations.
                                                              (line   6)
* gsl_histogram2d_find (C function):     Searching 2D histogram ranges.
                                                              (line   9)
* gsl_histogram2d_fprintf (C function):  Reading and writing 2D histograms.
                                                              (line  30)
* gsl_histogram2d_fread (C function):    Reading and writing 2D histograms.
                                                              (line  18)
* gsl_histogram2d_free (C function):     2D Histogram allocation.
                                                              (line  42)
* gsl_histogram2d_fscanf (C function):   Reading and writing 2D histograms.
                                                              (line  68)
* gsl_histogram2d_fwrite (C function):   Reading and writing 2D histograms.
                                                              (line   9)
* gsl_histogram2d_get (C function):      Updating and accessing 2D histogram elements.
                                                              (line  35)
* gsl_histogram2d_get_xrange (C function): Updating and accessing 2D histogram elements.
                                                              (line  44)
* gsl_histogram2d_get_yrange (C function): Updating and accessing 2D histogram elements.
                                                              (line  44)
* gsl_histogram2d_increment (C function): Updating and accessing 2D histogram elements.
                                                              (line  12)
* gsl_histogram2d_max_bin (C function):  2D Histogram Statistics.
                                                              (line  11)
* gsl_histogram2d_max_val (C function):  2D Histogram Statistics.
                                                              (line   6)
* gsl_histogram2d_memcpy (C function):   Copying 2D Histograms.
                                                              (line   6)
* gsl_histogram2d_min_bin (C function):  2D Histogram Statistics.
                                                              (line  24)
* gsl_histogram2d_min_val (C function):  2D Histogram Statistics.
                                                              (line  19)
* gsl_histogram2d_mul (C function):      2D Histogram Operations.
                                                              (line  28)
* gsl_histogram2d_nx (C function):       Updating and accessing 2D histogram elements.
                                                              (line  60)
* gsl_histogram2d_ny (C function):       Updating and accessing 2D histogram elements.
                                                              (line  60)
* gsl_histogram2d_pdf (C type):          Resampling from 2D histograms.
                                                              (line  20)
* gsl_histogram2d_pdf_alloc (C function): Resampling from 2D histograms.
                                                              (line  42)
* gsl_histogram2d_pdf_free (C function): Resampling from 2D histograms.
                                                              (line  61)
* gsl_histogram2d_pdf_init (C function): Resampling from 2D histograms.
                                                              (line  52)
* gsl_histogram2d_pdf_sample (C function): Resampling from 2D histograms.
                                                              (line  66)
* gsl_histogram2d_reset (C function):    Updating and accessing 2D histogram elements.
                                                              (line  72)
* gsl_histogram2d_scale (C function):    2D Histogram Operations.
                                                              (line  44)
* gsl_histogram2d_set_ranges (C function): 2D Histogram allocation.
                                                              (line  27)
* gsl_histogram2d_set_ranges_uniform (C function): 2D Histogram allocation.
                                                              (line  35)
* gsl_histogram2d_shift (C function):    2D Histogram Operations.
                                                              (line  52)
* gsl_histogram2d_sub (C function):      2D Histogram Operations.
                                                              (line  20)
* gsl_histogram2d_sum (C function):      2D Histogram Statistics.
                                                              (line  67)
* gsl_histogram2d_xmax (C function):     Updating and accessing 2D histogram elements.
                                                              (line  60)
* gsl_histogram2d_xmean (C function):    2D Histogram Statistics.
                                                              (line  32)
* gsl_histogram2d_xmin (C function):     Updating and accessing 2D histogram elements.
                                                              (line  60)
* gsl_histogram2d_xsigma (C function):   2D Histogram Statistics.
                                                              (line  46)
* gsl_histogram2d_ymax (C function):     Updating and accessing 2D histogram elements.
                                                              (line  60)
* gsl_histogram2d_ymean (C function):    2D Histogram Statistics.
                                                              (line  39)
* gsl_histogram2d_ymin (C function):     Updating and accessing 2D histogram elements.
                                                              (line  60)
* gsl_histogram2d_ysigma (C function):   2D Histogram Statistics.
                                                              (line  53)
* gsl_histogram_accumulate (C function): Updating and accessing histogram elements.
                                                              (line  26)
* gsl_histogram_add (C function):        Histogram Operations.
                                                              (line  12)
* gsl_histogram_alloc (C function):      Histogram allocation.
                                                              (line  14)
* gsl_histogram_bins (C function):       Updating and accessing histogram elements.
                                                              (line  56)
* gsl_histogram_clone (C function):      Copying Histograms.  (line  13)
* gsl_histogram_div (C function):        Histogram Operations.
                                                              (line  35)
* gsl_histogram_equal_bins_p (C function): Histogram Operations.
                                                              (line   6)
* gsl_histogram_find (C function):       Searching histogram ranges.
                                                              (line   9)
* gsl_histogram_fprintf (C function):    Reading and writing histograms.
                                                              (line  28)
* gsl_histogram_fread (C function):      Reading and writing histograms.
                                                              (line  18)
* gsl_histogram_free (C function):       Histogram allocation.
                                                              (line  67)
* gsl_histogram_fscanf (C function):     Reading and writing histograms.
                                                              (line  54)
* gsl_histogram_fwrite (C function):     Reading and writing histograms.
                                                              (line   9)
* gsl_histogram_get (C function):        Updating and accessing histogram elements.
                                                              (line  33)
* gsl_histogram_get_range (C function):  Updating and accessing histogram elements.
                                                              (line  41)
* gsl_histogram_increment (C function):  Updating and accessing histogram elements.
                                                              (line  11)
* gsl_histogram_max (C function):        Updating and accessing histogram elements.
                                                              (line  56)
* gsl_histogram_max_bin (C function):    Histogram Statistics.
                                                              (line  11)
* gsl_histogram_max_val (C function):    Histogram Statistics.
                                                              (line   6)
* gsl_histogram_mean (C function):       Histogram Statistics.
                                                              (line  28)
* gsl_histogram_memcpy (C function):     Copying Histograms.  (line   6)
* gsl_histogram_min (C function):        Updating and accessing histogram elements.
                                                              (line  56)
* gsl_histogram_min_bin (C function):    Histogram Statistics.
                                                              (line  22)
* gsl_histogram_min_val (C function):    Histogram Statistics.
                                                              (line  17)
* gsl_histogram_mul (C function):        Histogram Operations.
                                                              (line  27)
* gsl_histogram_pdf (C type):            The histogram probability distribution struct.
                                                              (line  18)
* gsl_histogram_pdf_alloc (C function):  The histogram probability distribution struct.
                                                              (line  36)
* gsl_histogram_pdf_free (C function):   The histogram probability distribution struct.
                                                              (line  53)
* gsl_histogram_pdf_init (C function):   The histogram probability distribution struct.
                                                              (line  44)
* gsl_histogram_pdf_sample (C function): The histogram probability distribution struct.
                                                              (line  58)
* gsl_histogram_reset (C function):      Updating and accessing histogram elements.
                                                              (line  65)
* gsl_histogram_scale (C function):      Histogram Operations.
                                                              (line  43)
* gsl_histogram_set_ranges (C function): Histogram allocation.
                                                              (line  24)
* gsl_histogram_set_ranges_uniform (C function): Histogram allocation.
                                                              (line  51)
* gsl_histogram_shift (C function):      Histogram Operations.
                                                              (line  50)
* gsl_histogram_sigma (C function):      Histogram Statistics.
                                                              (line  35)
* gsl_histogram_sub (C function):        Histogram Operations.
                                                              (line  19)
* gsl_histogram_sum (C function):        Histogram Statistics.
                                                              (line  43)
* gsl_hypot (C function):                Elementary Functions.
                                                              (line  24)
* gsl_hypot3 (C function):               Elementary Functions.
                                                              (line  30)
* gsl_ieee_env_setup (C function):       Setting up your IEEE environment.
                                                              (line  27)
* gsl_ieee_fprintf_double (C function):  Representation of floating point numbers.
                                                              (line  56)
* gsl_ieee_fprintf_float (C function):   Representation of floating point numbers.
                                                              (line  56)
* GSL_IEEE_MODE (C macro):               Setting up your IEEE environment.
                                                              (line  23)
* gsl_ieee_printf_double (C function):   Representation of floating point numbers.
                                                              (line  91)
* gsl_ieee_printf_float (C function):    Representation of floating point numbers.
                                                              (line  91)
* GSL_IMAG (C macro):                    Representation of complex numbers.
                                                              (line  36)
* gsl_integration_cquad (C function):    CQUAD doubly-adaptive integration.
                                                              (line  35)
* gsl_integration_cquad_workspace:       CQUAD doubly-adaptive integration.
                                                              (line  20)
* gsl_integration_cquad_workspace_alloc (C function): CQUAD doubly-adaptive integration.
                                                              (line  20)
* gsl_integration_cquad_workspace_free (C function): CQUAD doubly-adaptive integration.
                                                              (line  30)
* gsl_integration_fixed (C function):    Fixed point quadratures.
                                                              (line 161)
* gsl_integration_fixed_alloc (C function): Fixed point quadratures.
                                                              (line  74)
* gsl_integration_fixed_chebyshev (C variable): Fixed point quadratures.
                                                              (line  99)
* gsl_integration_fixed_chebyshev2 (C variable): Fixed point quadratures.
                                                              (line 133)
* gsl_integration_fixed_exponential (C variable): Fixed point quadratures.
                                                              (line 124)
* gsl_integration_fixed_free (C function): Fixed point quadratures.
                                                              (line 139)
* gsl_integration_fixed_gegenbauer (C variable): Fixed point quadratures.
                                                              (line 105)
* gsl_integration_fixed_hermite (C variable): Fixed point quadratures.
                                                              (line 119)
* gsl_integration_fixed_jacobi (C variable): Fixed point quadratures.
                                                              (line 110)
* gsl_integration_fixed_laguerre (C variable): Fixed point quadratures.
                                                              (line 114)
* gsl_integration_fixed_legendre (C variable): Fixed point quadratures.
                                                              (line  93)
* gsl_integration_fixed_n (C function):  Fixed point quadratures.
                                                              (line 144)
* gsl_integration_fixed_nodes (C function): Fixed point quadratures.
                                                              (line 149)
* gsl_integration_fixed_rational (C variable): Fixed point quadratures.
                                                              (line 129)
* gsl_integration_fixed_type (C type):   Fixed point quadratures.
                                                              (line  88)
* gsl_integration_fixed_weights (C function): Fixed point quadratures.
                                                              (line 155)
* gsl_integration_fixed_workspace (C type): Fixed point quadratures.
                                                              (line  59)
* gsl_integration_glfixed (C function):  Gauss-Legendre integration.
                                                              (line  23)
* gsl_integration_glfixed_point (C function): Gauss-Legendre integration.
                                                              (line  30)
* gsl_integration_glfixed_table:         Gauss-Legendre integration.
                                                              (line  14)
* gsl_integration_glfixed_table <1>:     Gauss-Legendre integration.
                                                              (line  41)
* gsl_integration_glfixed_table_alloc (C function): Gauss-Legendre integration.
                                                              (line  14)
* gsl_integration_glfixed_table_free (C function): Gauss-Legendre integration.
                                                              (line  41)
* gsl_integration_qag (C function):      QAG adaptive integration.
                                                              (line  32)
* gsl_integration_qagi (C function):     QAGI adaptive integration on infinite intervals.
                                                              (line   6)
* gsl_integration_qagil (C function):    QAGI adaptive integration on infinite intervals.
                                                              (line  36)
* gsl_integration_qagiu (C function):    QAGI adaptive integration on infinite intervals.
                                                              (line  21)
* gsl_integration_qagp (C function):     QAGP adaptive integration with known singular points.
                                                              (line   6)
* gsl_integration_qags (C function):     QAGS adaptive integration with singularities.
                                                              (line  15)
* gsl_integration_qawc (C function):     QAWC adaptive integration for Cauchy principal values.
                                                              (line   6)
* gsl_integration_qawf (C function):     QAWF adaptive integration for Fourier integrals.
                                                              (line   6)
* gsl_integration_qawo (C function):     QAWO adaptive integration for oscillatory functions.
                                                              (line  60)
* gsl_integration_qawo_table:            QAWO adaptive integration for oscillatory functions.
                                                              (line  11)
* gsl_integration_qawo_table_alloc (C function): QAWO adaptive integration for oscillatory functions.
                                                              (line  11)
* gsl_integration_qawo_table_free (C function): QAWO adaptive integration for oscillatory functions.
                                                              (line  54)
* gsl_integration_qawo_table_set (C function): QAWO adaptive integration for oscillatory functions.
                                                              (line  40)
* gsl_integration_qawo_table_set_length (C function): QAWO adaptive integration for oscillatory functions.
                                                              (line  48)
* gsl_integration_qaws (C function):     QAWS adaptive integration for singular functions.
                                                              (line  69)
* gsl_integration_qaws_table (C type):   QAWS adaptive integration for singular functions.
                                                              (line  11)
* gsl_integration_qaws_table_alloc (C function): QAWS adaptive integration for singular functions.
                                                              (line  16)
* gsl_integration_qaws_table_free (C function): QAWS adaptive integration for singular functions.
                                                              (line  63)
* gsl_integration_qaws_table_set (C function): QAWS adaptive integration for singular functions.
                                                              (line  55)
* gsl_integration_qng (C function):      QNG non-adaptive Gauss-Kronrod integration.
                                                              (line  10)
* gsl_integration_romberg (C function):  Romberg integration. (line  35)
* gsl_integration_romberg_alloc (C function): Romberg integration.
                                                              (line  21)
* gsl_integration_romberg_free (C function): Romberg integration.
                                                              (line  30)
* gsl_integration_workspace:             QAG adaptive integration.
                                                              (line  18)
* gsl_integration_workspace (C type):    QAG adaptive integration.
                                                              (line  13)
* gsl_integration_workspace_alloc (C function): QAG adaptive integration.
                                                              (line  18)
* gsl_integration_workspace_free (C function): QAG adaptive integration.
                                                              (line  27)
* GSL_INTEG_COSINE (C macro):            QAWO adaptive integration for oscillatory functions.
                                                              (line  27)
* GSL_INTEG_GAUSS15 (C macro):           QAG adaptive integration.
                                                              (line  45)
* GSL_INTEG_GAUSS21 (C macro):           QAG adaptive integration.
                                                              (line  47)
* GSL_INTEG_GAUSS31 (C macro):           QAG adaptive integration.
                                                              (line  49)
* GSL_INTEG_GAUSS41 (C macro):           QAG adaptive integration.
                                                              (line  51)
* GSL_INTEG_GAUSS51 (C macro):           QAG adaptive integration.
                                                              (line  53)
* GSL_INTEG_GAUSS61 (C macro):           QAG adaptive integration.
                                                              (line  55)
* GSL_INTEG_SINE (C macro):              QAWO adaptive integration for oscillatory functions.
                                                              (line  29)
* gsl_interp (C type):                   1D Interpolation Functions.
                                                              (line   9)
* gsl_interp2d (C type):                 2D Interpolation Functions.
                                                              (line  10)
* gsl_interp2d_alloc (C function):       2D Interpolation Functions.
                                                              (line  14)
* gsl_interp2d_bicubic (C variable):     2D Interpolation Types.
                                                              (line  16)
* gsl_interp2d_bilinear (C variable):    2D Interpolation Types.
                                                              (line  11)
* gsl_interp2d_eval (C function):        2D Evaluation of Interpolating Functions.
                                                              (line   6)
* gsl_interp2d_eval_deriv_x (C function): 2D Evaluation of Interpolating Functions.
                                                              (line  39)
* gsl_interp2d_eval_deriv_xx (C function): 2D Evaluation of Interpolating Functions.
                                                              (line  75)
* gsl_interp2d_eval_deriv_xx_e (C function): 2D Evaluation of Interpolating Functions.
                                                              (line  75)
* gsl_interp2d_eval_deriv_xy (C function): 2D Evaluation of Interpolating Functions.
                                                              (line 111)
* gsl_interp2d_eval_deriv_xy_e (C function): 2D Evaluation of Interpolating Functions.
                                                              (line 111)
* gsl_interp2d_eval_deriv_x_e (C function): 2D Evaluation of Interpolating Functions.
                                                              (line  39)
* gsl_interp2d_eval_deriv_y (C function): 2D Evaluation of Interpolating Functions.
                                                              (line  57)
* gsl_interp2d_eval_deriv_yy (C function): 2D Evaluation of Interpolating Functions.
                                                              (line  93)
* gsl_interp2d_eval_deriv_yy_e (C function): 2D Evaluation of Interpolating Functions.
                                                              (line  93)
* gsl_interp2d_eval_deriv_y_e (C function): 2D Evaluation of Interpolating Functions.
                                                              (line  57)
* gsl_interp2d_eval_e (C function):      2D Evaluation of Interpolating Functions.
                                                              (line   6)
* gsl_interp2d_eval_extrap (C function): 2D Evaluation of Interpolating Functions.
                                                              (line  21)
* gsl_interp2d_eval_extrap_e (C function): 2D Evaluation of Interpolating Functions.
                                                              (line  21)
* gsl_interp2d_free (C function):        2D Interpolation Functions.
                                                              (line  37)
* gsl_interp2d_get (C function):         2D Interpolation Grids.
                                                              (line  21)
* gsl_interp2d_idx (C function):         2D Interpolation Grids.
                                                              (line  27)
* gsl_interp2d_init (C function):        2D Interpolation Functions.
                                                              (line  21)
* gsl_interp2d_min_size (C function):    2D Interpolation Types.
                                                              (line  32)
* gsl_interp2d_name (C function):        2D Interpolation Types.
                                                              (line  20)
* gsl_interp2d_set (C function):         2D Interpolation Grids.
                                                              (line  15)
* gsl_interp2d_type (C type):            2D Interpolation Types.
                                                              (line   6)
* gsl_interp2d_type_min_size (C function): 2D Interpolation Types.
                                                              (line  32)
* gsl_interp_accel (C type):             1D Index Look-up and Acceleration.
                                                              (line   9)
* gsl_interp_accel_alloc (C function):   1D Index Look-up and Acceleration.
                                                              (line  24)
* gsl_interp_accel_find (C function):    1D Index Look-up and Acceleration.
                                                              (line  31)
* gsl_interp_accel_free (C function):    1D Index Look-up and Acceleration.
                                                              (line  48)
* gsl_interp_akima (C variable):         1D Interpolation Types.
                                                              (line  41)
* gsl_interp_akima_periodic (C variable): 1D Interpolation Types.
                                                              (line  46)
* gsl_interp_alloc (C function):         1D Interpolation Functions.
                                                              (line  13)
* gsl_interp_bsearch (C function):       1D Index Look-up and Acceleration.
                                                              (line  16)
* gsl_interp_cspline (C variable):       1D Interpolation Types.
                                                              (line  23)
* gsl_interp_cspline_periodic (C variable): 1D Interpolation Types.
                                                              (line  31)
* gsl_interp_eval (C function):          1D Evaluation of Interpolating Functions.
                                                              (line   6)
* gsl_interp_eval_deriv (C function):    1D Evaluation of Interpolating Functions.
                                                              (line  21)
* gsl_interp_eval_deriv2 (C function):   1D Evaluation of Interpolating Functions.
                                                              (line  32)
* gsl_interp_eval_deriv2_e (C function): 1D Evaluation of Interpolating Functions.
                                                              (line  32)
* gsl_interp_eval_deriv_e (C function):  1D Evaluation of Interpolating Functions.
                                                              (line  21)
* gsl_interp_eval_e (C function):        1D Evaluation of Interpolating Functions.
                                                              (line   6)
* gsl_interp_eval_integ (C function):    1D Evaluation of Interpolating Functions.
                                                              (line  44)
* gsl_interp_eval_integ_e (C function):  1D Evaluation of Interpolating Functions.
                                                              (line  44)
* gsl_interp_free (C function):          1D Interpolation Functions.
                                                              (line  31)
* gsl_interp_init (C function):          1D Interpolation Functions.
                                                              (line  20)
* gsl_interp_linear (C variable):        1D Interpolation Types.
                                                              (line  10)
* gsl_interp_min_size (C function):      1D Interpolation Types.
                                                              (line  75)
* gsl_interp_name (C function):          1D Interpolation Types.
                                                              (line  64)
* gsl_interp_polynomial (C variable):    1D Interpolation Types.
                                                              (line  15)
* gsl_interp_steffen (C variable):       1D Interpolation Types.
                                                              (line  51)
* gsl_interp_type (C type):              1D Interpolation Types.
                                                              (line   8)
* gsl_interp_type_min_size (C function): 1D Interpolation Types.
                                                              (line  75)
* gsl_isinf (C function):                Infinities and Not-a-number.
                                                              (line  25)
* gsl_isnan (C function):                Infinities and Not-a-number.
                                                              (line  21)
* GSL_IS_EVEN (C macro):                 Testing for Odd and Even Numbers.
                                                              (line  11)
* GSL_IS_ODD (C macro):                  Testing for Odd and Even Numbers.
                                                              (line   6)
* gsl_ldexp (C function):                Elementary Functions.
                                                              (line  52)
* gsl_linalg_balance_matrix (C function): Balancing.          (line  17)
* gsl_linalg_bidiag_decomp (C function): Bidiagonalization.   (line  15)
* gsl_linalg_bidiag_unpack (C function): Bidiagonalization.   (line  28)
* gsl_linalg_bidiag_unpack2 (C function): Bidiagonalization.  (line  39)
* gsl_linalg_bidiag_unpack_B (C function): Bidiagonalization. (line  49)
* gsl_linalg_cholesky_band_decomp (C function): Banded Cholesky Decomposition.
                                                              (line  14)
* gsl_linalg_cholesky_band_invert (C function): Banded Cholesky Decomposition.
                                                              (line  49)
* gsl_linalg_cholesky_band_rcond (C function): Banded Cholesky Decomposition.
                                                              (line  65)
* gsl_linalg_cholesky_band_solve (C function): Banded Cholesky Decomposition.
                                                              (line  31)
* gsl_linalg_cholesky_band_svx (C function): Banded Cholesky Decomposition.
                                                              (line  39)
* gsl_linalg_cholesky_band_unpack (C function): Banded Cholesky Decomposition.
                                                              (line  58)
* gsl_linalg_cholesky_decomp (C function): Cholesky Decomposition.
                                                              (line  48)
* gsl_linalg_cholesky_decomp1 (C function): Cholesky Decomposition.
                                                              (line  29)
* gsl_linalg_cholesky_decomp2 (C function): Cholesky Decomposition.
                                                              (line  87)
* gsl_linalg_cholesky_invert (C function): Cholesky Decomposition.
                                                              (line  77)
* gsl_linalg_cholesky_rcond (C function): Cholesky Decomposition.
                                                              (line 140)
* gsl_linalg_cholesky_scale (C function): Cholesky Decomposition.
                                                              (line 123)
* gsl_linalg_cholesky_scale_apply (C function): Cholesky Decomposition.
                                                              (line 134)
* gsl_linalg_cholesky_solve (C function): Cholesky Decomposition.
                                                              (line  53)
* gsl_linalg_cholesky_solve2 (C function): Cholesky Decomposition.
                                                              (line 105)
* gsl_linalg_cholesky_svx (C function):  Cholesky Decomposition.
                                                              (line  64)
* gsl_linalg_cholesky_svx2 (C function): Cholesky Decomposition.
                                                              (line 114)
* gsl_linalg_COD_decomp (C function):    Complete Orthogonal Decomposition.
                                                              (line  49)
* gsl_linalg_COD_decomp_e (C function):  Complete Orthogonal Decomposition.
                                                              (line  49)
* gsl_linalg_COD_lssolve (C function):   Complete Orthogonal Decomposition.
                                                              (line  68)
* gsl_linalg_COD_lssolve2 (C function):  Complete Orthogonal Decomposition.
                                                              (line  83)
* gsl_linalg_COD_matZ (C function):      Complete Orthogonal Decomposition.
                                                              (line 108)
* gsl_linalg_COD_unpack (C function):    Complete Orthogonal Decomposition.
                                                              (line 100)
* gsl_linalg_complex_cholesky_decomp (C function): Cholesky Decomposition.
                                                              (line  29)
* gsl_linalg_complex_cholesky_invert (C function): Cholesky Decomposition.
                                                              (line  77)
* gsl_linalg_complex_cholesky_solve (C function): Cholesky Decomposition.
                                                              (line  53)
* gsl_linalg_complex_cholesky_svx (C function): Cholesky Decomposition.
                                                              (line  64)
* gsl_linalg_complex_householder_hm (C function): Householder Transformations.
                                                              (line  30)
* gsl_linalg_complex_householder_hv (C function): Householder Transformations.
                                                              (line  48)
* gsl_linalg_complex_householder_mh (C function): Householder Transformations.
                                                              (line  39)
* gsl_linalg_complex_householder_transform (C function): Householder Transformations.
                                                              (line  17)
* gsl_linalg_complex_LU_decomp (C function): LU Decomposition.
                                                              (line  19)
* gsl_linalg_complex_LU_det (C function): LU Decomposition.   (line 111)
* gsl_linalg_complex_LU_invert (C function): LU Decomposition.
                                                              (line  79)
* gsl_linalg_complex_LU_invx (C function): LU Decomposition.  (line  95)
* gsl_linalg_complex_LU_lndet (C function): LU Decomposition. (line 121)
* gsl_linalg_complex_LU_refine (C function): LU Decomposition.
                                                              (line  66)
* gsl_linalg_complex_LU_sgndet (C function): LU Decomposition.
                                                              (line 130)
* gsl_linalg_complex_LU_solve (C function): LU Decomposition. (line  44)
* gsl_linalg_complex_LU_svx (C function): LU Decomposition.   (line  56)
* gsl_linalg_complex_tri_invert (C function): Triangular Systems.
                                                              (line   6)
* gsl_linalg_complex_tri_LHL (C function): Triangular Systems.
                                                              (line  19)
* gsl_linalg_complex_tri_UL (C function): Triangular Systems. (line  25)
* gsl_linalg_givens (C function):        Givens Rotations.    (line  21)
* gsl_linalg_givens_gv (C function):     Givens Rotations.    (line  28)
* gsl_linalg_hermtd_decomp (C function): Tridiagonal Decomposition of Hermitian Matrices.
                                                              (line  14)
* gsl_linalg_hermtd_unpack (C function): Tridiagonal Decomposition of Hermitian Matrices.
                                                              (line  27)
* gsl_linalg_hermtd_unpack_T (C function): Tridiagonal Decomposition of Hermitian Matrices.
                                                              (line  36)
* gsl_linalg_hessenberg_decomp (C function): Hessenberg Decomposition of Real Matrices.
                                                              (line  16)
* gsl_linalg_hessenberg_set_zero (C function): Hessenberg Decomposition of Real Matrices.
                                                              (line  47)
* gsl_linalg_hessenberg_unpack (C function): Hessenberg Decomposition of Real Matrices.
                                                              (line  28)
* gsl_linalg_hessenberg_unpack_accum (C function): Hessenberg Decomposition of Real Matrices.
                                                              (line  36)
* gsl_linalg_hesstri_decomp (C function): Hessenberg-Triangular Decomposition of Real Matrices.
                                                              (line  17)
* gsl_linalg_HH_solve (C function):      Householder solver for linear systems.
                                                              (line   6)
* gsl_linalg_HH_svx (C function):        Householder solver for linear systems.
                                                              (line  15)
* gsl_linalg_householder_hm (C function): Householder Transformations.
                                                              (line  30)
* gsl_linalg_householder_hv (C function): Householder Transformations.
                                                              (line  48)
* gsl_linalg_householder_mh (C function): Householder Transformations.
                                                              (line  39)
* gsl_linalg_householder_transform (C function): Householder Transformations.
                                                              (line  17)
* gsl_linalg_ldlt_band_decomp (C function): Banded LDLT Decomposition.
                                                              (line  14)
* gsl_linalg_ldlt_band_rcond (C function): Banded LDLT Decomposition.
                                                              (line  51)
* gsl_linalg_ldlt_band_solve (C function): Banded LDLT Decomposition.
                                                              (line  26)
* gsl_linalg_ldlt_band_svx (C function): Banded LDLT Decomposition.
                                                              (line  34)
* gsl_linalg_ldlt_band_unpack (C function): Banded LDLT Decomposition.
                                                              (line  43)
* gsl_linalg_ldlt_decomp (C function):   LDLT Decomposition.  (line  22)
* gsl_linalg_ldlt_rcond (C function):    LDLT Decomposition.  (line  52)
* gsl_linalg_ldlt_solve (C function):    LDLT Decomposition.  (line  36)
* gsl_linalg_ldlt_svx (C function):      LDLT Decomposition.  (line  43)
* gsl_linalg_LQ_decomp (C function):     LQ Decomposition.    (line  27)
* gsl_linalg_LQ_lssolve (C function):    LQ Decomposition.    (line  41)
* gsl_linalg_LQ_QTvec (C function):      LQ Decomposition.    (line  59)
* gsl_linalg_LQ_unpack (C function):     LQ Decomposition.    (line  52)
* gsl_linalg_LU_decomp (C function):     LU Decomposition.    (line  19)
* gsl_linalg_LU_det (C function):        LU Decomposition.    (line 111)
* gsl_linalg_LU_invert (C function):     LU Decomposition.    (line  79)
* gsl_linalg_LU_invx (C function):       LU Decomposition.    (line  95)
* gsl_linalg_LU_lndet (C function):      LU Decomposition.    (line 121)
* gsl_linalg_LU_refine (C function):     LU Decomposition.    (line  66)
* gsl_linalg_LU_sgndet (C function):     LU Decomposition.    (line 130)
* gsl_linalg_LU_solve (C function):      LU Decomposition.    (line  44)
* gsl_linalg_LU_svx (C function):        LU Decomposition.    (line  56)
* gsl_linalg_mcholesky_decomp (C function): Modified Cholesky Decomposition.
                                                              (line  25)
* gsl_linalg_mcholesky_rcond (C function): Modified Cholesky Decomposition.
                                                              (line  59)
* gsl_linalg_mcholesky_solve (C function): Modified Cholesky Decomposition.
                                                              (line  40)
* gsl_linalg_mcholesky_svx (C function): Modified Cholesky Decomposition.
                                                              (line  49)
* gsl_linalg_pcholesky_decomp (C function): Pivoted Cholesky Decomposition.
                                                              (line  20)
* gsl_linalg_pcholesky_decomp2 (C function): Pivoted Cholesky Decomposition.
                                                              (line  52)
* gsl_linalg_pcholesky_invert (C function): Pivoted Cholesky Decomposition.
                                                              (line  90)
* gsl_linalg_pcholesky_rcond (C function): Pivoted Cholesky Decomposition.
                                                              (line  97)
* gsl_linalg_pcholesky_solve (C function): Pivoted Cholesky Decomposition.
                                                              (line  33)
* gsl_linalg_pcholesky_solve2 (C function): Pivoted Cholesky Decomposition.
                                                              (line  70)
* gsl_linalg_pcholesky_svx (C function): Pivoted Cholesky Decomposition.
                                                              (line  42)
* gsl_linalg_pcholesky_svx2 (C function): Pivoted Cholesky Decomposition.
                                                              (line  79)
* gsl_linalg_QRPT_decomp (C function):   QR Decomposition with Column Pivoting.
                                                              (line  33)
* gsl_linalg_QRPT_decomp2 (C function):  QR Decomposition with Column Pivoting.
                                                              (line  58)
* gsl_linalg_QRPT_lssolve (C function):  QR Decomposition with Column Pivoting.
                                                              (line  83)
* gsl_linalg_QRPT_lssolve2 (C function): QR Decomposition with Column Pivoting.
                                                              (line  98)
* gsl_linalg_QRPT_QRsolve (C function):  QR Decomposition with Column Pivoting.
                                                              (line 115)
* gsl_linalg_QRPT_rank (C function):     QR Decomposition with Column Pivoting.
                                                              (line 146)
* gsl_linalg_QRPT_rcond (C function):    QR Decomposition with Column Pivoting.
                                                              (line 156)
* gsl_linalg_QRPT_Rsolve (C function):   QR Decomposition with Column Pivoting.
                                                              (line 133)
* gsl_linalg_QRPT_Rsvx (C function):     QR Decomposition with Column Pivoting.
                                                              (line 139)
* gsl_linalg_QRPT_solve (C function):    QR Decomposition with Column Pivoting.
                                                              (line  66)
* gsl_linalg_QRPT_svx (C function):      QR Decomposition with Column Pivoting.
                                                              (line  75)
* gsl_linalg_QRPT_update (C function):   QR Decomposition with Column Pivoting.
                                                              (line 123)
* gsl_linalg_QR_decomp (C function):     QR Decomposition.    (line  63)
* gsl_linalg_QR_decomp_r (C function):   QR Decomposition.    (line  82)
* gsl_linalg_QR_lssolve (C function):    QR Decomposition.    (line 118)
* gsl_linalg_QR_lssolve_r (C function):  QR Decomposition.    (line 130)
* gsl_linalg_QR_QRsolve (C function):    QR Decomposition.    (line 216)
* gsl_linalg_QR_QTmat (C function):      QR Decomposition.    (line 169)
* gsl_linalg_QR_QTmat_r (C function):    QR Decomposition.    (line 169)
* gsl_linalg_QR_QTvec (C function):      QR Decomposition.    (line 146)
* gsl_linalg_QR_QTvec_r (C function):    QR Decomposition.    (line 146)
* gsl_linalg_QR_Qvec (C function):       QR Decomposition.    (line 160)
* gsl_linalg_QR_rcond (C function):      QR Decomposition.    (line 244)
* gsl_linalg_QR_Rsolve (C function):     QR Decomposition.    (line 183)
* gsl_linalg_QR_Rsvx (C function):       QR Decomposition.    (line 190)
* gsl_linalg_QR_solve (C function):      QR Decomposition.    (line  96)
* gsl_linalg_QR_solve_r (C function):    QR Decomposition.    (line  96)
* gsl_linalg_QR_svx (C function):        QR Decomposition.    (line 109)
* gsl_linalg_QR_TR_decomp (C function):  Triangle on Top of Rectangle.
                                                              (line  27)
* gsl_linalg_QR_unpack (C function):     QR Decomposition.    (line 200)
* gsl_linalg_QR_unpack_r (C function):   QR Decomposition.    (line 207)
* gsl_linalg_QR_update (C function):     QR Decomposition.    (line 223)
* gsl_linalg_R_solve (C function):       QR Decomposition.    (line 231)
* gsl_linalg_R_svx (C function):         QR Decomposition.    (line 237)
* gsl_linalg_solve_cyc_tridiag (C function): Tridiagonal Systems.
                                                              (line  42)
* gsl_linalg_solve_symm_cyc_tridiag (C function): Tridiagonal Systems.
                                                              (line  58)
* gsl_linalg_solve_symm_tridiag (C function): Tridiagonal Systems.
                                                              (line  28)
* gsl_linalg_solve_tridiag (C function): Tridiagonal Systems. (line  13)
* gsl_linalg_SV_decomp (C function):     Singular Value Decomposition.
                                                              (line  38)
* gsl_linalg_SV_decomp_jacobi (C function): Singular Value Decomposition.
                                                              (line  61)
* gsl_linalg_SV_decomp_mod (C function): Singular Value Decomposition.
                                                              (line  52)
* gsl_linalg_SV_leverage (C function):   Singular Value Decomposition.
                                                              (line  86)
* gsl_linalg_SV_solve (C function):      Singular Value Decomposition.
                                                              (line  69)
* gsl_linalg_symmtd_decomp (C function): Tridiagonal Decomposition of Real Symmetric Matrices.
                                                              (line  13)
* gsl_linalg_symmtd_unpack (C function): Tridiagonal Decomposition of Real Symmetric Matrices.
                                                              (line  26)
* gsl_linalg_symmtd_unpack_T (C function): Tridiagonal Decomposition of Real Symmetric Matrices.
                                                              (line  36)
* gsl_linalg_tri_invert (C function):    Triangular Systems.  (line   6)
* gsl_linalg_tri_LTL (C function):       Triangular Systems.  (line  19)
* gsl_linalg_tri_rcond (C function):     Triangular Systems.  (line  34)
* gsl_linalg_tri_UL (C function):        Triangular Systems.  (line  25)
* gsl_log1p (C function):                Elementary Functions.
                                                              (line  12)
* gsl_matrix (C type):                   Matrices.            (line  11)
* gsl_matrix_add (C function):           Matrix operations.   (line   8)
* gsl_matrix_add_constant (C function):  Matrix operations.   (line  44)
* gsl_matrix_alloc (C function):         Matrix allocation.   (line  14)
* gsl_matrix_calloc (C function):        Matrix allocation.   (line  24)
* gsl_matrix_column (C function):        Creating row and column views.
                                                              (line  25)
* gsl_matrix_const_column (C function):  Creating row and column views.
                                                              (line  25)
* gsl_matrix_const_diagonal (C function): Creating row and column views.
                                                              (line  67)
* gsl_matrix_const_ptr (C function):     Accessing matrix elements.
                                                              (line  38)
* gsl_matrix_const_row (C function):     Creating row and column views.
                                                              (line  13)
* gsl_matrix_const_subcolumn (C function): Creating row and column views.
                                                              (line  52)
* gsl_matrix_const_subdiagonal (C function): Creating row and column views.
                                                              (line  80)
* gsl_matrix_const_submatrix (C function): Matrix views.      (line  21)
* gsl_matrix_const_subrow (C function):  Creating row and column views.
                                                              (line  38)
* gsl_matrix_const_superdiagonal (C function): Creating row and column views.
                                                              (line  94)
* gsl_matrix_const_view (C type):        Matrix views.        (line   6)
* gsl_matrix_const_view_array (C function): Matrix views.     (line  55)
* gsl_matrix_const_view_array_with_tda (C function): Matrix views.
                                                              (line  81)
* gsl_matrix_const_view_vector (C function): Matrix views.    (line 108)
* gsl_matrix_const_view_vector_with_tda (C function): Matrix views.
                                                              (line 134)
* gsl_matrix_diagonal (C function):      Creating row and column views.
                                                              (line  67)
* gsl_matrix_div_elements (C function):  Matrix operations.   (line  30)
* gsl_matrix_equal (C function):         Matrix properties.   (line  20)
* gsl_matrix_fprintf (C function):       Reading and writing matrices.
                                                              (line  28)
* gsl_matrix_fread (C function):         Reading and writing matrices.
                                                              (line  18)
* gsl_matrix_free (C function):          Matrix allocation.   (line  30)
* gsl_matrix_fscanf (C function):        Reading and writing matrices.
                                                              (line  37)
* gsl_matrix_fwrite (C function):        Reading and writing matrices.
                                                              (line   9)
* gsl_matrix_get (C function):           Accessing matrix elements.
                                                              (line  20)
* gsl_matrix_get_col (C function):       Copying rows and columns.
                                                              (line  21)
* gsl_matrix_get_row (C function):       Copying rows and columns.
                                                              (line  13)
* gsl_matrix_isneg (C function):         Matrix properties.   (line  10)
* gsl_matrix_isnonneg (C function):      Matrix properties.   (line  10)
* gsl_matrix_isnull (C function):        Matrix properties.   (line  10)
* gsl_matrix_ispos (C function):         Matrix properties.   (line  10)
* gsl_matrix_max (C function):           Finding maximum and minimum elements of matrices.
                                                              (line   8)
* gsl_matrix_max_index (C function):     Finding maximum and minimum elements of matrices.
                                                              (line  22)
* gsl_matrix_memcpy (C function):        Copying matrices.    (line   6)
* gsl_matrix_min (C function):           Finding maximum and minimum elements of matrices.
                                                              (line  12)
* gsl_matrix_minmax (C function):        Finding maximum and minimum elements of matrices.
                                                              (line  16)
* gsl_matrix_minmax_index (C function):  Finding maximum and minimum elements of matrices.
                                                              (line  38)
* gsl_matrix_min_index (C function):     Finding maximum and minimum elements of matrices.
                                                              (line  30)
* gsl_matrix_mul_elements (C function):  Matrix operations.   (line  22)
* gsl_matrix_ptr (C function):           Accessing matrix elements.
                                                              (line  38)
* gsl_matrix_row (C function):           Creating row and column views.
                                                              (line  13)
* gsl_matrix_scale (C function):         Matrix operations.   (line  38)
* gsl_matrix_set (C function):           Accessing matrix elements.
                                                              (line  29)
* gsl_matrix_set_all (C function):       Initializing matrix elements.
                                                              (line   6)
* gsl_matrix_set_col (C function):       Copying rows and columns.
                                                              (line  36)
* gsl_matrix_set_identity (C function):  Initializing matrix elements.
                                                              (line  15)
* gsl_matrix_set_row (C function):       Copying rows and columns.
                                                              (line  29)
* gsl_matrix_set_zero (C function):      Initializing matrix elements.
                                                              (line  11)
* gsl_matrix_sub (C function):           Matrix operations.   (line  15)
* gsl_matrix_subcolumn (C function):     Creating row and column views.
                                                              (line  52)
* gsl_matrix_subdiagonal (C function):   Creating row and column views.
                                                              (line  80)
* gsl_matrix_submatrix (C function):     Matrix views.        (line  21)
* gsl_matrix_subrow (C function):        Creating row and column views.
                                                              (line  38)
* gsl_matrix_superdiagonal (C function): Creating row and column views.
                                                              (line  94)
* gsl_matrix_swap (C function):          Copying matrices.    (line  12)
* gsl_matrix_swap_columns (C function):  Exchanging rows and columns.
                                                              (line  16)
* gsl_matrix_swap_rowcol (C function):   Exchanging rows and columns.
                                                              (line  23)
* gsl_matrix_swap_rows (C function):     Exchanging rows and columns.
                                                              (line   9)
* gsl_matrix_transpose (C function):     Exchanging rows and columns.
                                                              (line  39)
* gsl_matrix_transpose_memcpy (C function): Exchanging rows and columns.
                                                              (line  31)
* gsl_matrix_view (C type):              Matrix views.        (line   6)
* gsl_matrix_view_array (C function):    Matrix views.        (line  55)
* gsl_matrix_view_array_with_tda (C function): Matrix views.  (line  81)
* gsl_matrix_view_vector (C function):   Matrix views.        (line 108)
* gsl_matrix_view_vector_with_tda (C function): Matrix views. (line 134)
* GSL_MAX (C macro):                     Maximum and Minimum functions.
                                                              (line  10)
* GSL_MAX_DBL (C function):              Maximum and Minimum functions.
                                                              (line  20)
* GSL_MAX_INT (C function):              Maximum and Minimum functions.
                                                              (line  36)
* GSL_MAX_LDBL (C function):             Maximum and Minimum functions.
                                                              (line  44)
* GSL_MIN (C macro):                     Maximum and Minimum functions.
                                                              (line  15)
* GSL_MIN_DBL (C function):              Maximum and Minimum functions.
                                                              (line  28)
* gsl_min_fminimizer (C type):           Initializing the Minimizer.
                                                              (line   6)
* gsl_min_fminimizer_alloc (C function): Initializing the Minimizer.
                                                              (line  10)
* gsl_min_fminimizer_brent (C variable): Minimization Algorithms.
                                                              (line  33)
* gsl_min_fminimizer_free (C function):  Initializing the Minimizer.
                                                              (line  46)
* gsl_min_fminimizer_f_lower (C function): Iteration<2>.      (line  46)
* gsl_min_fminimizer_f_minimum (C function): Iteration<2>.    (line  46)
* gsl_min_fminimizer_f_upper (C function): Iteration<2>.      (line  46)
* gsl_min_fminimizer_goldensection (C variable): Minimization Algorithms.
                                                              (line  15)
* gsl_min_fminimizer_iterate (C function): Iteration<2>.      (line  12)
* gsl_min_fminimizer_name (C function):  Initializing the Minimizer.
                                                              (line  51)
* gsl_min_fminimizer_quad_golden (C variable): Minimization Algorithms.
                                                              (line  50)
* gsl_min_fminimizer_set (C function):   Initializing the Minimizer.
                                                              (line  24)
* gsl_min_fminimizer_set_with_values (C function): Initializing the Minimizer.
                                                              (line  36)
* gsl_min_fminimizer_type (C type):      Minimization Algorithms.
                                                              (line  13)
* gsl_min_fminimizer_x_lower (C function): Iteration<2>.      (line  38)
* gsl_min_fminimizer_x_minimum (C function): Iteration<2>.    (line  32)
* gsl_min_fminimizer_x_upper (C function): Iteration<2>.      (line  38)
* GSL_MIN_INT (C function):              Maximum and Minimum functions.
                                                              (line  36)
* GSL_MIN_LDBL (C function):             Maximum and Minimum functions.
                                                              (line  44)
* gsl_min_test_interval (C function):    Stopping Parameters. (line  18)
* gsl_mode_t (C type):                   Modes.               (line  14)
* gsl_monte_function (C type):           Interface.           (line  27)
* gsl_monte_miser_alloc (C function):    MISER.               (line  48)
* gsl_monte_miser_free (C function):     MISER.               (line  75)
* gsl_monte_miser_init (C function):     MISER.               (line  54)
* gsl_monte_miser_integrate (C function): MISER.              (line  60)
* gsl_monte_miser_params (C type):       MISER.               (line 102)
* gsl_monte_miser_params_get (C function): MISER.             (line  83)
* gsl_monte_miser_params_set (C function): MISER.             (line  89)
* gsl_monte_miser_state (C type):        MISER.               (line  44)
* gsl_monte_plain_alloc (C function):    PLAIN Monte Carlo.   (line  33)
* gsl_monte_plain_free (C function):     PLAIN Monte Carlo.   (line  59)
* gsl_monte_plain_init (C function):     PLAIN Monte Carlo.   (line  38)
* gsl_monte_plain_integrate (C function): PLAIN Monte Carlo.  (line  44)
* gsl_monte_plain_state (C type):        PLAIN Monte Carlo.   (line  29)
* gsl_monte_vegas_alloc (C function):    VEGAS.               (line  54)
* gsl_monte_vegas_chisq (C function):    VEGAS.               (line 123)
* gsl_monte_vegas_free (C function):     VEGAS.               (line  85)
* gsl_monte_vegas_init (C function):     VEGAS.               (line  60)
* gsl_monte_vegas_integrate (C function): VEGAS.              (line  66)
* gsl_monte_vegas_params (C type):       VEGAS.               (line 164)
* gsl_monte_vegas_params_get (C function): VEGAS.             (line 145)
* gsl_monte_vegas_params_set (C function): VEGAS.             (line 151)
* gsl_monte_vegas_runval (C function):   VEGAS.               (line 134)
* gsl_monte_vegas_state (C type):        VEGAS.               (line  50)
* gsl_movstat_accum (C type):            Accumulators.        (line  11)
* gsl_movstat_accum.delete (C member):   Accumulators.        (line  45)
* gsl_movstat_accum.get (C member):      Accumulators.        (line  50)
* gsl_movstat_accum.init (C member):     Accumulators.        (line  32)
* gsl_movstat_accum.insert (C member):   Accumulators.        (line  37)
* gsl_movstat_accum.size (C member):     Accumulators.        (line  27)
* gsl_movstat_accum_max (C variable):    Accumulators.        (line  60)
* gsl_movstat_accum_mean (C variable):   Accumulators.        (line  67)
* gsl_movstat_accum_min (C variable):    Accumulators.        (line  60)
* gsl_movstat_accum_minmax (C variable): Accumulators.        (line  60)
* gsl_movstat_accum_Qn (C variable):     Accumulators.        (line  79)
* gsl_movstat_accum_qqr (C variable):    Accumulators.        (line  89)
* gsl_movstat_accum_sd (C variable):     Accumulators.        (line  67)
* gsl_movstat_accum_Sn (C variable):     Accumulators.        (line  79)
* gsl_movstat_accum_sum (C variable):    Accumulators.        (line  85)
* gsl_movstat_accum_variance (C variable): Accumulators.      (line  67)
* gsl_movstat_alloc (C function):        Allocation for Moving Window Statistics.
                                                              (line  10)
* gsl_movstat_alloc2 (C function):       Allocation for Moving Window Statistics.
                                                              (line  17)
* gsl_movstat_apply (C function):        User-defined Moving Statistics.
                                                              (line  42)
* GSL_MOVSTAT_END_PADVALUE (C macro):    Handling Endpoints.  (line  27)
* GSL_MOVSTAT_END_PADZERO (C macro):     Handling Endpoints.  (line  17)
* gsl_movstat_end_t (C type):            Handling Endpoints.  (line  12)
* GSL_MOVSTAT_END_TRUNCATE (C macro):    Handling Endpoints.  (line  36)
* gsl_movstat_fill (C function):         User-defined Moving Statistics.
                                                              (line  52)
* gsl_movstat_free (C function):         Allocation for Moving Window Statistics.
                                                              (line  25)
* gsl_movstat_function (C type):         User-defined Moving Statistics.
                                                              (line  16)
* gsl_movstat_function.function (C member): User-defined Moving Statistics.
                                                              (line  30)
* gsl_movstat_function.params (C member): User-defined Moving Statistics.
                                                              (line  38)
* gsl_movstat_mad (C function):          Moving MAD.          (line  15)
* gsl_movstat_mad0 (C function):         Moving MAD.          (line  15)
* gsl_movstat_max (C function):          Moving Minimum and Maximum.
                                                              (line  21)
* gsl_movstat_mean (C function):         Moving Mean.         (line  16)
* gsl_movstat_median (C function):       Moving Median.       (line  11)
* gsl_movstat_median (C variable):       Accumulators.        (line  74)
* gsl_movstat_min (C function):          Moving Minimum and Maximum.
                                                              (line  12)
* gsl_movstat_minmax (C function):       Moving Minimum and Maximum.
                                                              (line  30)
* gsl_movstat_Qn (C function):           Moving Q_n.          (line  10)
* gsl_movstat_qqr (C function):          Moving QQR.          (line  19)
* gsl_movstat_sd (C function):           Moving Variance and Standard Deviation.
                                                              (line  23)
* gsl_movstat_Sn (C function):           Moving S_n.          (line  11)
* gsl_movstat_sum (C function):          Moving Sum.          (line  11)
* gsl_movstat_variance (C function):     Moving Variance and Standard Deviation.
                                                              (line  14)
* gsl_movstat_workspace (C type):        Allocation for Moving Window Statistics.
                                                              (line   6)
* gsl_multifit_linear (C function):      Multi-parameter regression.
                                                              (line  75)
* gsl_multifit_linear_alloc (C function): Multi-parameter regression.
                                                              (line  48)
* gsl_multifit_linear_applyW (C function): Regularized regression.
                                                              (line 230)
* gsl_multifit_linear_bsvd (C function): Multi-parameter regression.
                                                              (line  67)
* gsl_multifit_linear_est (C function):  Multi-parameter regression.
                                                              (line 153)
* gsl_multifit_linear_free (C function): Multi-parameter regression.
                                                              (line  56)
* gsl_multifit_linear_gcv (C function):  Regularized regression.
                                                              (line 337)
* gsl_multifit_linear_gcv_calc (C function): Regularized regression.
                                                              (line 330)
* gsl_multifit_linear_gcv_curve (C function): Regularized regression.
                                                              (line 309)
* gsl_multifit_linear_gcv_init (C function): Regularized regression.
                                                              (line 297)
* gsl_multifit_linear_gcv_min (C function): Regularized regression.
                                                              (line 318)
* gsl_multifit_linear_genform1 (C function): Regularized regression.
                                                              (line 200)
* gsl_multifit_linear_genform2 (C function): Regularized regression.
                                                              (line 211)
* gsl_multifit_linear_lcorner (C function): Regularized regression.
                                                              (line 259)
* gsl_multifit_linear_lcorner2 (C function): Regularized regression.
                                                              (line 276)
* gsl_multifit_linear_lcurve (C function): Regularized regression.
                                                              (line 241)
* gsl_multifit_linear_Lk (C function):   Regularized regression.
                                                              (line 349)
* gsl_multifit_linear_Lsobolev (C function): Regularized regression.
                                                              (line 356)
* gsl_multifit_linear_L_decomp (C function): Regularized regression.
                                                              (line 142)
* gsl_multifit_linear_rank (C function): Multi-parameter regression.
                                                              (line 169)
* gsl_multifit_linear_rcond (C function): Regularized regression.
                                                              (line 372)
* gsl_multifit_linear_residuals (C function): Multi-parameter regression.
                                                              (line 162)
* gsl_multifit_linear_solve (C function): Regularized regression.
                                                              (line 180)
* gsl_multifit_linear_stdform1 (C function): Regularized regression.
                                                              (line 117)
* gsl_multifit_linear_stdform2 (C function): Regularized regression.
                                                              (line 155)
* gsl_multifit_linear_svd (C function):  Multi-parameter regression.
                                                              (line  61)
* gsl_multifit_linear_tsvd (C function): Multi-parameter regression.
                                                              (line  97)
* gsl_multifit_linear_wgenform2 (C function): Regularized regression.
                                                              (line 211)
* gsl_multifit_linear_workspace (C type): Multi-parameter regression.
                                                              (line  43)
* gsl_multifit_linear_wstdform1 (C function): Regularized regression.
                                                              (line 117)
* gsl_multifit_linear_wstdform2 (C function): Regularized regression.
                                                              (line 155)
* gsl_multifit_nlinear_alloc (C function): Initializing the Solver<3>.
                                                              (line  17)
* gsl_multifit_nlinear_avratio (C function): Iteration<5>.    (line 108)
* gsl_multifit_nlinear_covar (C function): Covariance matrix of best fit parameters.
                                                              (line   6)
* GSL_MULTIFIT_NLINEAR_CTRDIFF (C macro): Tunable Parameters. (line 213)
* gsl_multifit_nlinear_default_parameters (C function): Initializing the Solver<3>.
                                                              (line  53)
* gsl_multifit_nlinear_driver (C function): High Level Driver.
                                                              (line   9)
* gsl_multifit_nlinear_fdf (C type):     Providing the Function to be Minimized.
                                                              (line  10)
* gsl_multifit_nlinear_fdtype (C type):  Tunable Parameters.  (line 188)
* gsl_multifit_nlinear_free (C function): Initializing the Solver<3>.
                                                              (line  82)
* GSL_MULTIFIT_NLINEAR_FWDIFF (C macro): Tunable Parameters.  (line 195)
* gsl_multifit_nlinear_init (C function): Initializing the Solver<3>.
                                                              (line  63)
* gsl_multifit_nlinear_iterate (C function): Iteration<5>.    (line  10)
* gsl_multifit_nlinear_jac (C function): Iteration<5>.        (line  61)
* gsl_multifit_nlinear_name (C function): Initializing the Solver<3>.
                                                              (line  90)
* gsl_multifit_nlinear_niter (C function): Iteration<5>.      (line  68)
* gsl_multifit_nlinear_parameters (C type): Tunable Parameters.
                                                              (line  11)
* gsl_multifit_nlinear_position (C function): Iteration<5>.   (line  44)
* gsl_multifit_nlinear_rcond (C function): Iteration<5>.      (line  77)
* gsl_multifit_nlinear_residual (C function): Iteration<5>.   (line  52)
* gsl_multifit_nlinear_scale (C type):   Tunable Parameters.  (line  86)
* gsl_multifit_nlinear_scale_levenberg (C variable): Tunable Parameters.
                                                              (line 110)
* gsl_multifit_nlinear_scale_marquardt (C variable): Tunable Parameters.
                                                              (line 121)
* gsl_multifit_nlinear_scale_more (C variable): Tunable Parameters.
                                                              (line  92)
* gsl_multifit_nlinear_solver (C type):  Tunable Parameters.  (line 130)
* gsl_multifit_nlinear_solver_cholesky (C variable): Tunable Parameters.
                                                              (line 149)
* gsl_multifit_nlinear_solver_mcholesky (C variable): Tunable Parameters.
                                                              (line 166)
* gsl_multifit_nlinear_solver_qr (C variable): Tunable Parameters.
                                                              (line 141)
* gsl_multifit_nlinear_solver_svd (C variable): Tunable Parameters.
                                                              (line 181)
* gsl_multifit_nlinear_test (C function): Testing for Convergence.
                                                              (line  19)
* gsl_multifit_nlinear_trs (C type):     Tunable Parameters.  (line  49)
* gsl_multifit_nlinear_trs_ddogleg (C variable): Tunable Parameters.
                                                              (line  71)
* gsl_multifit_nlinear_trs_dogleg (C variable): Tunable Parameters.
                                                              (line  66)
* gsl_multifit_nlinear_trs_lm (C variable): Tunable Parameters.
                                                              (line  55)
* gsl_multifit_nlinear_trs_lmaccel (C variable): Tunable Parameters.
                                                              (line  60)
* gsl_multifit_nlinear_trs_name (C function): Initializing the Solver<3>.
                                                              (line 102)
* gsl_multifit_nlinear_trs_subspace2D (C variable): Tunable Parameters.
                                                              (line  76)
* gsl_multifit_nlinear_trust (C variable): Initializing the Solver<3>.
                                                              (line  12)
* gsl_multifit_nlinear_type (C type):    Initializing the Solver<3>.
                                                              (line   6)
* gsl_multifit_nlinear_winit (C function): Initializing the Solver<3>.
                                                              (line  63)
* gsl_multifit_robust (C function):      Robust linear regression.
                                                              (line 209)
* gsl_multifit_robust_alloc (C function): Robust linear regression.
                                                              (line  81)
* gsl_multifit_robust_bisquare (C variable): Robust linear regression.
                                                              (line  99)
* gsl_multifit_robust_cauchy (C variable): Robust linear regression.
                                                              (line 110)
* gsl_multifit_robust_default (C variable): Robust linear regression.
                                                              (line  93)
* gsl_multifit_robust_est (C function):  Robust linear regression.
                                                              (line 230)
* gsl_multifit_robust_fair (C variable): Robust linear regression.
                                                              (line 123)
* gsl_multifit_robust_free (C function): Robust linear regression.
                                                              (line 168)
* gsl_multifit_robust_huber (C variable): Robust linear regression.
                                                              (line 133)
* gsl_multifit_robust_maxiter (C function): Robust linear regression.
                                                              (line 188)
* gsl_multifit_robust_name (C function): Robust linear regression.
                                                              (line 173)
* gsl_multifit_robust_ols (C variable):  Robust linear regression.
                                                              (line 147)
* gsl_multifit_robust_residuals (C function): Robust linear regression.
                                                              (line 239)
* gsl_multifit_robust_statistics (C function): Robust linear regression.
                                                              (line 250)
* gsl_multifit_robust_stats (C type):    Robust linear regression.
                                                              (line 260)
* gsl_multifit_robust_tune (C function): Robust linear regression.
                                                              (line 179)
* gsl_multifit_robust_type (C type):     Robust linear regression.
                                                              (line  91)
* gsl_multifit_robust_weights (C function): Robust linear regression.
                                                              (line 196)
* gsl_multifit_robust_welsch (C variable): Robust linear regression.
                                                              (line 158)
* gsl_multifit_robust_workspace (C type): Robust linear regression.
                                                              (line  77)
* gsl_multifit_wlinear (C function):     Multi-parameter regression.
                                                              (line 118)
* gsl_multifit_wlinear_tsvd (C function): Multi-parameter regression.
                                                              (line 134)
* gsl_multilarge_linear_accumulate (C function): Large Dense Linear Least Squares Routines.
                                                              (line 112)
* gsl_multilarge_linear_alloc (C function): Large Dense Linear Least Squares Routines.
                                                              (line  11)
* gsl_multilarge_linear_free (C function): Large Dense Linear Least Squares Routines.
                                                              (line  43)
* gsl_multilarge_linear_genform1 (C function): Large Dense Linear Least Squares Routines.
                                                              (line 132)
* gsl_multilarge_linear_genform2 (C function): Large Dense Linear Least Squares Routines.
                                                              (line 143)
* gsl_multilarge_linear_lcurve (C function): Large Dense Linear Least Squares Routines.
                                                              (line 153)
* gsl_multilarge_linear_L_decomp (C function): Large Dense Linear Least Squares Routines.
                                                              (line  82)
* gsl_multilarge_linear_name (C function): Large Dense Linear Least Squares Routines.
                                                              (line  48)
* gsl_multilarge_linear_normal (C variable): Large Dense Linear Least Squares Routines.
                                                              (line  26)
* gsl_multilarge_linear_rcond (C function): Large Dense Linear Least Squares Routines.
                                                              (line 170)
* gsl_multilarge_linear_reset (C function): Large Dense Linear Least Squares Routines.
                                                              (line  54)
* gsl_multilarge_linear_solve (C function): Large Dense Linear Least Squares Routines.
                                                              (line 121)
* gsl_multilarge_linear_stdform1 (C function): Large Dense Linear Least Squares Routines.
                                                              (line  60)
* gsl_multilarge_linear_stdform2 (C function): Large Dense Linear Least Squares Routines.
                                                              (line  92)
* gsl_multilarge_linear_tsqr (C variable): Large Dense Linear Least Squares Routines.
                                                              (line  34)
* gsl_multilarge_linear_type (C type):   Large Dense Linear Least Squares Routines.
                                                              (line  20)
* gsl_multilarge_linear_workspace (C type): Large Dense Linear Least Squares Routines.
                                                              (line   6)
* gsl_multilarge_linear_wstdform1 (C function): Large Dense Linear Least Squares Routines.
                                                              (line  60)
* gsl_multilarge_linear_wstdform2 (C function): Large Dense Linear Least Squares Routines.
                                                              (line  92)
* gsl_multilarge_nlinear_alloc (C function): Initializing the Solver<3>.
                                                              (line  17)
* gsl_multilarge_nlinear_avratio (C function): Iteration<5>.  (line 108)
* gsl_multilarge_nlinear_covar (C function): Covariance matrix of best fit parameters.
                                                              (line   6)
* gsl_multilarge_nlinear_default_parameters (C function): Initializing the Solver<3>.
                                                              (line  53)
* gsl_multilarge_nlinear_driver (C function): High Level Driver.
                                                              (line   9)
* gsl_multilarge_nlinear_fdf (C type):   Providing the Function to be Minimized.
                                                              (line  85)
* gsl_multilarge_nlinear_free (C function): Initializing the Solver<3>.
                                                              (line  82)
* gsl_multilarge_nlinear_init (C function): Initializing the Solver<3>.
                                                              (line  63)
* gsl_multilarge_nlinear_iterate (C function): Iteration<5>.  (line  10)
* gsl_multilarge_nlinear_name (C function): Initializing the Solver<3>.
                                                              (line  90)
* gsl_multilarge_nlinear_niter (C function): Iteration<5>.    (line  68)
* gsl_multilarge_nlinear_parameters (C type): Tunable Parameters.
                                                              (line  30)
* gsl_multilarge_nlinear_position (C function): Iteration<5>. (line  44)
* gsl_multilarge_nlinear_rcond (C function): Iteration<5>.    (line  77)
* gsl_multilarge_nlinear_residual (C function): Iteration<5>. (line  52)
* gsl_multilarge_nlinear_scale (C type): Tunable Parameters.  (line  86)
* gsl_multilarge_nlinear_scale_levenberg (C variable): Tunable Parameters.
                                                              (line 110)
* gsl_multilarge_nlinear_scale_marquardt (C variable): Tunable Parameters.
                                                              (line 121)
* gsl_multilarge_nlinear_scale_more (C variable): Tunable Parameters.
                                                              (line  92)
* gsl_multilarge_nlinear_solver (C type): Tunable Parameters. (line 130)
* gsl_multilarge_nlinear_solver_cholesky (C variable): Tunable Parameters.
                                                              (line 149)
* gsl_multilarge_nlinear_solver_mcholesky (C variable): Tunable Parameters.
                                                              (line 166)
* gsl_multilarge_nlinear_test (C function): Testing for Convergence.
                                                              (line  19)
* gsl_multilarge_nlinear_trs (C type):   Tunable Parameters.  (line  49)
* gsl_multilarge_nlinear_trs_cgst (C variable): Tunable Parameters.
                                                              (line  81)
* gsl_multilarge_nlinear_trs_ddogleg (C variable): Tunable Parameters.
                                                              (line  71)
* gsl_multilarge_nlinear_trs_dogleg (C variable): Tunable Parameters.
                                                              (line  66)
* gsl_multilarge_nlinear_trs_lm (C variable): Tunable Parameters.
                                                              (line  55)
* gsl_multilarge_nlinear_trs_lmaccel (C variable): Tunable Parameters.
                                                              (line  60)
* gsl_multilarge_nlinear_trs_name (C function): Initializing the Solver<3>.
                                                              (line 102)
* gsl_multilarge_nlinear_trs_subspace2D (C variable): Tunable Parameters.
                                                              (line  76)
* gsl_multimin_fdfminimizer (C type):    Initializing the Multidimensional Minimizer.
                                                              (line  10)
* gsl_multimin_fdfminimizer_alloc (C function): Initializing the Multidimensional Minimizer.
                                                              (line  18)
* gsl_multimin_fdfminimizer_conjugate_fr (C variable): Algorithms with Derivatives.
                                                              (line  15)
* gsl_multimin_fdfminimizer_conjugate_pr (C variable): Algorithms with Derivatives.
                                                              (line  34)
* gsl_multimin_fdfminimizer_dx (C function): Iteration<4>.    (line  27)
* gsl_multimin_fdfminimizer_free (C function): Initializing the Multidimensional Minimizer.
                                                              (line  58)
* gsl_multimin_fdfminimizer_gradient (C function): Iteration<4>.
                                                              (line  27)
* gsl_multimin_fdfminimizer_iterate (C function): Iteration<4>.
                                                              (line  11)
* gsl_multimin_fdfminimizer_minimum (C function): Iteration<4>.
                                                              (line  27)
* gsl_multimin_fdfminimizer_name (C function): Initializing the Multidimensional Minimizer.
                                                              (line  66)
* gsl_multimin_fdfminimizer_restart (C function): Iteration<4>.
                                                              (line  47)
* gsl_multimin_fdfminimizer_set (C function): Initializing the Multidimensional Minimizer.
                                                              (line  30)
* gsl_multimin_fdfminimizer_steepest_descent (C variable): Algorithms with Derivatives.
                                                              (line  65)
* gsl_multimin_fdfminimizer_type (C type): Algorithms with Derivatives.
                                                              (line  11)
* gsl_multimin_fdfminimizer_vector_bfgs (C variable): Algorithms with Derivatives.
                                                              (line  43)
* gsl_multimin_fdfminimizer_vector_bfgs2 (C variable): Algorithms with Derivatives.
                                                              (line  43)
* gsl_multimin_fdfminimizer_x (C function): Iteration<4>.     (line  27)
* gsl_multimin_fminimizer (C type):      Initializing the Multidimensional Minimizer.
                                                              (line  14)
* gsl_multimin_fminimizer_alloc (C function): Initializing the Multidimensional Minimizer.
                                                              (line  18)
* gsl_multimin_fminimizer_free (C function): Initializing the Multidimensional Minimizer.
                                                              (line  58)
* gsl_multimin_fminimizer_iterate (C function): Iteration<4>. (line  11)
* gsl_multimin_fminimizer_minimum (C function): Iteration<4>. (line  27)
* gsl_multimin_fminimizer_name (C function): Initializing the Multidimensional Minimizer.
                                                              (line  66)
* gsl_multimin_fminimizer_nmsimplex (C variable): Algorithms without Derivatives<2>.
                                                              (line  14)
* gsl_multimin_fminimizer_nmsimplex2 (C variable): Algorithms without Derivatives<2>.
                                                              (line  14)
* gsl_multimin_fminimizer_nmsimplex2rand (C variable): Algorithms without Derivatives<2>.
                                                              (line  60)
* gsl_multimin_fminimizer_set (C function): Initializing the Multidimensional Minimizer.
                                                              (line  30)
* gsl_multimin_fminimizer_size (C function): Iteration<4>.    (line  27)
* gsl_multimin_fminimizer_type (C type): Algorithms without Derivatives<2>.
                                                              (line   9)
* gsl_multimin_fminimizer_x (C function): Iteration<4>.       (line  27)
* gsl_multimin_function (C type):        Providing a function to minimize.
                                                              (line  53)
* gsl_multimin_function_fdf (C type):    Providing a function to minimize.
                                                              (line  13)
* gsl_multimin_test_gradient (C function): Stopping Criteria. (line  18)
* gsl_multimin_test_size (C function):   Stopping Criteria.   (line  34)
* gsl_multiroot_fdfsolver (C type):      Initializing the Solver<2>.
                                                              (line  16)
* gsl_multiroot_fdfsolver_alloc (C function): Initializing the Solver<2>.
                                                              (line  36)
* gsl_multiroot_fdfsolver_dx (C function): Iteration<3>.      (line  52)
* gsl_multiroot_fdfsolver_f (C function): Iteration<3>.       (line  44)
* gsl_multiroot_fdfsolver_free (C function): Initializing the Solver<2>.
                                                              (line  62)
* gsl_multiroot_fdfsolver_gnewton (C variable): Algorithms using Derivatives.
                                                              (line 102)
* gsl_multiroot_fdfsolver_hybridj (C variable): Algorithms using Derivatives.
                                                              (line  75)
* gsl_multiroot_fdfsolver_hybridsj (C variable): Algorithms using Derivatives.
                                                              (line  18)
* gsl_multiroot_fdfsolver_iterate (C function): Iteration<3>. (line  12)
* gsl_multiroot_fdfsolver_name (C function): Initializing the Solver<2>.
                                                              (line  69)
* gsl_multiroot_fdfsolver_newton (C variable): Algorithms using Derivatives.
                                                              (line  82)
* gsl_multiroot_fdfsolver_root (C function): Iteration<3>.    (line  36)
* gsl_multiroot_fdfsolver_set (C function): Initializing the Solver<2>.
                                                              (line  51)
* gsl_multiroot_fdfsolver_type (C type): Algorithms using Derivatives.
                                                              (line  13)
* gsl_multiroot_fsolver (C type):        Initializing the Solver<2>.
                                                              (line  11)
* gsl_multiroot_fsolver_alloc (C function): Initializing the Solver<2>.
                                                              (line  21)
* gsl_multiroot_fsolver_broyden (C variable): Algorithms without Derivatives.
                                                              (line  54)
* gsl_multiroot_fsolver_dnewton (C variable): Algorithms without Derivatives.
                                                              (line  32)
* gsl_multiroot_fsolver_dx (C function): Iteration<3>.        (line  52)
* gsl_multiroot_fsolver_f (C function):  Iteration<3>.        (line  44)
* gsl_multiroot_fsolver_free (C function): Initializing the Solver<2>.
                                                              (line  62)
* gsl_multiroot_fsolver_hybrid (C variable): Algorithms without Derivatives.
                                                              (line  27)
* gsl_multiroot_fsolver_hybrids (C variable): Algorithms without Derivatives.
                                                              (line  18)
* gsl_multiroot_fsolver_iterate (C function): Iteration<3>.   (line  12)
* gsl_multiroot_fsolver_name (C function): Initializing the Solver<2>.
                                                              (line  69)
* gsl_multiroot_fsolver_root (C function): Iteration<3>.      (line  36)
* gsl_multiroot_fsolver_set (C function): Initializing the Solver<2>.
                                                              (line  51)
* gsl_multiroot_fsolver_type (C type):   Algorithms without Derivatives.
                                                              (line  13)
* gsl_multiroot_function (C type):       Providing the function to solve<2>.
                                                              (line  10)
* gsl_multiroot_function_fdf (C type):   Providing the function to solve<2>.
                                                              (line  62)
* gsl_multiroot_test_delta (C function): Search Stopping Parameters<2>.
                                                              (line  20)
* gsl_multiroot_test_residual (C function): Search Stopping Parameters<2>.
                                                              (line  32)
* gsl_multiset (C type):                 The Multiset struct. (line   6)
* gsl_multiset_alloc (C function):       Multiset allocation. (line   6)
* gsl_multiset_calloc (C function):      Multiset allocation. (line  16)
* gsl_multiset_data (C function):        Multiset properties. (line  15)
* gsl_multiset_fprintf (C function):     Reading and writing multisets.
                                                              (line  28)
* gsl_multiset_fread (C function):       Reading and writing multisets.
                                                              (line  18)
* gsl_multiset_free (C function):        Multiset allocation. (line  33)
* gsl_multiset_fscanf (C function):      Reading and writing multisets.
                                                              (line  38)
* gsl_multiset_fwrite (C function):      Reading and writing multisets.
                                                              (line   9)
* gsl_multiset_get (C function):         Accessing multiset elements.
                                                              (line   8)
* gsl_multiset_init_first (C function):  Multiset allocation. (line  23)
* gsl_multiset_init_last (C function):   Multiset allocation. (line  28)
* gsl_multiset_k (C function):           Multiset properties. (line  10)
* gsl_multiset_memcpy (C function):      Multiset allocation. (line  37)
* gsl_multiset_n (C function):           Multiset properties. (line   6)
* gsl_multiset_next (C function):        Multiset functions.  (line   6)
* gsl_multiset_prev (C function):        Multiset functions.  (line  15)
* gsl_multiset_valid (C function):       Multiset properties. (line  20)
* GSL_NAN (C macro):                     Infinities and Not-a-number.
                                                              (line  16)
* GSL_NEGINF (C macro):                  Infinities and Not-a-number.
                                                              (line  11)
* gsl_ntuple (C type):                   The ntuple struct.   (line   6)
* gsl_ntuple_bookdata (C function):      Writing ntuples.     (line  11)
* gsl_ntuple_close (C function):         Closing an ntuple file.
                                                              (line   6)
* gsl_ntuple_create (C function):        Creating ntuples.    (line   6)
* gsl_ntuple_open (C function):          Opening an existing ntuple file.
                                                              (line   6)
* gsl_ntuple_project (C function):       Histogramming ntuple values.
                                                              (line  41)
* gsl_ntuple_read (C function):          Reading ntuples.     (line   6)
* gsl_ntuple_select_fn (C type):         Histogramming ntuple values.
                                                              (line  13)
* gsl_ntuple_value_fn (C type):          Histogramming ntuple values.
                                                              (line  27)
* gsl_ntuple_write (C function):         Writing ntuples.     (line   6)
* gsl_odeiv2_control (C type):           Adaptive Step-size Control.
                                                              (line  10)
* gsl_odeiv2_control_alloc (C function): Adaptive Step-size Control.
                                                              (line  87)
* gsl_odeiv2_control_errlevel (C function): Adaptive Step-size Control.
                                                              (line 135)
* gsl_odeiv2_control_free (C function):  Adaptive Step-size Control.
                                                              (line 104)
* gsl_odeiv2_control_hadjust (C function): Adaptive Step-size Control.
                                                              (line 109)
* gsl_odeiv2_control_init (C function):  Adaptive Step-size Control.
                                                              (line  95)
* gsl_odeiv2_control_name (C function):  Adaptive Step-size Control.
                                                              (line 125)
* gsl_odeiv2_control_scaled_new (C function): Adaptive Step-size Control.
                                                              (line  72)
* gsl_odeiv2_control_set_driver (C function): Adaptive Step-size Control.
                                                              (line 145)
* gsl_odeiv2_control_standard_new (C function): Adaptive Step-size Control.
                                                              (line  18)
* gsl_odeiv2_control_type (C type):      Adaptive Step-size Control.
                                                              (line  14)
* gsl_odeiv2_control_yp_new (C function): Adaptive Step-size Control.
                                                              (line  63)
* gsl_odeiv2_control_y_new (C function): Adaptive Step-size Control.
                                                              (line  54)
* gsl_odeiv2_driver_alloc_scaled_new (C function): Driver.    (line   9)
* gsl_odeiv2_driver_alloc_standard_new (C function): Driver.  (line   9)
* gsl_odeiv2_driver_alloc_yp_new (C function): Driver.        (line   9)
* gsl_odeiv2_driver_alloc_y_new (C function): Driver.         (line   9)
* gsl_odeiv2_driver_apply (C function):  Driver.              (line  51)
* gsl_odeiv2_driver_apply_fixed_step (C function): Driver.    (line  69)
* gsl_odeiv2_driver_free (C function):   Driver.              (line  92)
* gsl_odeiv2_driver_reset (C function):  Driver.              (line  81)
* gsl_odeiv2_driver_reset_hstart (C function): Driver.        (line  85)
* gsl_odeiv2_driver_set_hmax (C function): Driver.            (line  39)
* gsl_odeiv2_driver_set_hmin (C function): Driver.            (line  33)
* gsl_odeiv2_driver_set_nmax (C function): Driver.            (line  45)
* gsl_odeiv2_evolve (C type):            Evolution.           (line  10)
* gsl_odeiv2_evolve_alloc (C function):  Evolution.           (line  15)
* gsl_odeiv2_evolve_apply (C function):  Evolution.           (line  20)
* gsl_odeiv2_evolve_apply_fixed_step (C function): Evolution. (line  60)
* gsl_odeiv2_evolve_free (C function):   Evolution.           (line  79)
* gsl_odeiv2_evolve_reset (C function):  Evolution.           (line  73)
* gsl_odeiv2_evolve_set_driver (C function): Evolution.       (line  84)
* gsl_odeiv2_step (C type):              Stepping Functions.  (line  10)
* gsl_odeiv2_step_alloc (C function):    Stepping Functions.  (line  14)
* gsl_odeiv2_step_apply (C function):    Stepping Functions.  (line  63)
* gsl_odeiv2_step_bsimp (C variable):    Stepping Functions.  (line 150)
* gsl_odeiv2_step_free (C function):     Stepping Functions.  (line  29)
* gsl_odeiv2_step_msadams (C variable):  Stepping Functions.  (line 156)
* gsl_odeiv2_step_msbdf (C variable):    Stepping Functions.  (line 166)
* gsl_odeiv2_step_name (C function):     Stepping Functions.  (line  34)
* gsl_odeiv2_step_order (C function):    Stepping Functions.  (line  45)
* gsl_odeiv2_step_reset (C function):    Stepping Functions.  (line  23)
* gsl_odeiv2_step_rk1imp (C variable):   Stepping Functions.  (line 127)
* gsl_odeiv2_step_rk2 (C variable):      Stepping Functions.  (line 103)
* gsl_odeiv2_step_rk2imp (C variable):   Stepping Functions.  (line 135)
* gsl_odeiv2_step_rk4 (C variable):      Stepping Functions.  (line 107)
* gsl_odeiv2_step_rk4imp (C variable):   Stepping Functions.  (line 143)
* gsl_odeiv2_step_rk8pd (C variable):    Stepping Functions.  (line 123)
* gsl_odeiv2_step_rkck (C variable):     Stepping Functions.  (line 119)
* gsl_odeiv2_step_rkf45 (C variable):    Stepping Functions.  (line 114)
* gsl_odeiv2_step_set_driver (C function): Stepping Functions.
                                                              (line  53)
* gsl_odeiv2_step_type (C type):         Stepping Functions.  (line 101)
* gsl_odeiv2_system (C type):            Defining the ODE System.
                                                              (line  18)
* gsl_permutation (C type):              The Permutation struct.
                                                              (line   6)
* gsl_permutation_alloc (C function):    Permutation allocation.
                                                              (line   6)
* gsl_permutation_calloc (C function):   Permutation allocation.
                                                              (line  15)
* gsl_permutation_canonical_cycles (C function): Permutations in cyclic form.
                                                              (line  75)
* gsl_permutation_canonical_to_linear (C function): Permutations in cyclic form.
                                                              (line  52)
* gsl_permutation_data (C function):     Permutation properties.
                                                              (line  10)
* gsl_permutation_fprintf (C function):  Reading and writing permutations.
                                                              (line  30)
* gsl_permutation_fread (C function):    Reading and writing permutations.
                                                              (line  18)
* gsl_permutation_free (C function):     Permutation allocation.
                                                              (line  26)
* gsl_permutation_fscanf (C function):   Reading and writing permutations.
                                                              (line  40)
* gsl_permutation_fwrite (C function):   Reading and writing permutations.
                                                              (line   9)
* gsl_permutation_get (C function):      Accessing permutation elements.
                                                              (line   9)
* gsl_permutation_init (C function):     Permutation allocation.
                                                              (line  21)
* gsl_permutation_inverse (C function):  Permutation functions.
                                                              (line  10)
* gsl_permutation_inversions (C function): Permutations in cyclic form.
                                                              (line  59)
* gsl_permutation_linear_cycles (C function): Permutations in cyclic form.
                                                              (line  69)
* gsl_permutation_linear_to_canonical (C function): Permutations in cyclic form.
                                                              (line  45)
* gsl_permutation_memcpy (C function):   Permutation allocation.
                                                              (line  30)
* gsl_permutation_mul (C function):      Applying Permutations.
                                                              (line  52)
* gsl_permutation_next (C function):     Permutation functions.
                                                              (line  16)
* gsl_permutation_prev (C function):     Permutation functions.
                                                              (line  25)
* gsl_permutation_reverse (C function):  Permutation functions.
                                                              (line   6)
* gsl_permutation_size (C function):     Permutation properties.
                                                              (line   6)
* gsl_permutation_swap (C function):     Accessing permutation elements.
                                                              (line  17)
* gsl_permutation_valid (C function):    Permutation properties.
                                                              (line  15)
* gsl_permute (C function):              Applying Permutations.
                                                              (line   9)
* gsl_permute_inverse (C function):      Applying Permutations.
                                                              (line  15)
* gsl_permute_matrix (C function):       Applying Permutations.
                                                              (line  42)
* gsl_permute_vector (C function):       Applying Permutations.
                                                              (line  22)
* gsl_permute_vector_inverse (C function): Applying Permutations.
                                                              (line  31)
* gsl_poly_complex_eval (C function):    Polynomial Evaluation.
                                                              (line  19)
* gsl_poly_complex_solve (C function):   General Polynomial Equations.
                                                              (line  34)
* gsl_poly_complex_solve_cubic (C function): Cubic Equations. (line  25)
* gsl_poly_complex_solve_quadratic (C function): Quadratic Equations.
                                                              (line  30)
* gsl_poly_complex_workspace (C type):   General Polynomial Equations.
                                                              (line  11)
* gsl_poly_complex_workspace_alloc (C function): General Polynomial Equations.
                                                              (line  16)
* gsl_poly_complex_workspace_free (C function): General Polynomial Equations.
                                                              (line  28)
* gsl_poly_dd_eval (C function):         Divided Difference Representation of Polynomials.
                                                              (line  41)
* gsl_poly_dd_hermite_init (C function): Divided Difference Representation of Polynomials.
                                                              (line  61)
* gsl_poly_dd_init (C function):         Divided Difference Representation of Polynomials.
                                                              (line  32)
* gsl_poly_dd_taylor (C function):       Divided Difference Representation of Polynomials.
                                                              (line  49)
* gsl_poly_eval (C function):            Polynomial Evaluation.
                                                              (line  13)
* gsl_poly_eval_derivs (C function):     Polynomial Evaluation.
                                                              (line  32)
* gsl_poly_solve_cubic (C function):     Cubic Equations.     (line   6)
* gsl_poly_solve_quadratic (C function): Quadratic Equations. (line   6)
* GSL_POSINF (C macro):                  Infinities and Not-a-number.
                                                              (line   6)
* gsl_pow_2 (C function):                Small integer powers.
                                                              (line  20)
* gsl_pow_3 (C function):                Small integer powers.
                                                              (line  20)
* gsl_pow_4 (C function):                Small integer powers.
                                                              (line  20)
* gsl_pow_5 (C function):                Small integer powers.
                                                              (line  20)
* gsl_pow_6 (C function):                Small integer powers.
                                                              (line  20)
* gsl_pow_7 (C function):                Small integer powers.
                                                              (line  20)
* gsl_pow_8 (C function):                Small integer powers.
                                                              (line  20)
* gsl_pow_9 (C function):                Small integer powers.
                                                              (line  20)
* gsl_pow_int (C function):              Small integer powers.
                                                              (line  11)
* gsl_pow_uint (C function):             Small integer powers.
                                                              (line  11)
* GSL_PREC_APPROX (C macro):             Modes.               (line  26)
* GSL_PREC_DOUBLE (C macro):             Modes.               (line  16)
* GSL_PREC_SINGLE (C macro):             Modes.               (line  21)
* gsl_qrng (C type):                     Quasi-random number generator initialization.
                                                              (line   6)
* gsl_qrng_alloc (C function):           Quasi-random number generator initialization.
                                                              (line  10)
* gsl_qrng_clone (C function):           Saving and restoring quasi-random number generator state.
                                                              (line  13)
* gsl_qrng_free (C function):            Quasi-random number generator initialization.
                                                              (line  19)
* gsl_qrng_get (C function):             Sampling from a quasi-random number generator.
                                                              (line   6)
* gsl_qrng_halton (C variable):          Quasi-random number generator algorithms.
                                                              (line  22)
* gsl_qrng_init (C function):            Quasi-random number generator initialization.
                                                              (line  24)
* gsl_qrng_memcpy (C function):          Saving and restoring quasi-random number generator state.
                                                              (line   6)
* gsl_qrng_name (C function):            Auxiliary quasi-random number generator functions.
                                                              (line   6)
* gsl_qrng_niederreiter_2 (C variable):  Quasi-random number generator algorithms.
                                                              (line  10)
* gsl_qrng_reversehalton (C variable):   Quasi-random number generator algorithms.
                                                              (line  22)
* gsl_qrng_size (C function):            Auxiliary quasi-random number generator functions.
                                                              (line  10)
* gsl_qrng_sobol (C variable):           Quasi-random number generator algorithms.
                                                              (line  16)
* gsl_qrng_state (C function):           Auxiliary quasi-random number generator functions.
                                                              (line  10)
* gsl_qrng_type (C type):                Quasi-random number generator algorithms.
                                                              (line   8)
* GSL_RANGE_CHECK_OFF (C macro):         Accessing vector elements.
                                                              (line  17)
* gsl_ran_bernoulli (C function):        The Bernoulli Distribution.
                                                              (line   6)
* gsl_ran_bernoulli_pdf (C function):    The Bernoulli Distribution.
                                                              (line  16)
* gsl_ran_beta (C function):             The Beta Distribution.
                                                              (line   6)
* gsl_ran_beta_pdf (C function):         The Beta Distribution.
                                                              (line  16)
* gsl_ran_binomial (C function):         The Binomial Distribution.
                                                              (line   6)
* gsl_ran_binomial_pdf (C function):     The Binomial Distribution.
                                                              (line  18)
* gsl_ran_bivariate_gaussian (C function): The Bivariate Gaussian Distribution.
                                                              (line   6)
* gsl_ran_bivariate_gaussian_pdf (C function): The Bivariate Gaussian Distribution.
                                                              (line  21)
* gsl_ran_cauchy (C function):           The Cauchy Distribution.
                                                              (line   6)
* gsl_ran_cauchy_pdf (C function):       The Cauchy Distribution.
                                                              (line  17)
* gsl_ran_chisq (C function):            The Chi-squared Distribution.
                                                              (line  14)
* gsl_ran_chisq_pdf (C function):        The Chi-squared Distribution.
                                                              (line  24)
* gsl_ran_choose (C function):           Shuffling and Sampling.
                                                              (line  34)
* gsl_ran_dirichlet (C function):        The Dirichlet Distribution.
                                                              (line   6)
* gsl_ran_dirichlet_lnpdf (C function):  The Dirichlet Distribution.
                                                              (line  33)
* gsl_ran_dirichlet_pdf (C function):    The Dirichlet Distribution.
                                                              (line  26)
* gsl_ran_dir_2d (C function):           Spherical Vector Distributions.
                                                              (line  10)
* gsl_ran_dir_2d_trig_method (C function): Spherical Vector Distributions.
                                                              (line  10)
* gsl_ran_dir_3d (C function):           Spherical Vector Distributions.
                                                              (line  32)
* gsl_ran_dir_nd (C function):           Spherical Vector Distributions.
                                                              (line  42)
* gsl_ran_discrete (C function):         General Discrete Distributions.
                                                              (line  65)
* gsl_ran_discrete_free (C function):    General Discrete Distributions.
                                                              (line  80)
* gsl_ran_discrete_pdf (C function):     General Discrete Distributions.
                                                              (line  71)
* gsl_ran_discrete_preproc (C function): General Discrete Distributions.
                                                              (line  54)
* gsl_ran_discrete_t (C type):           General Discrete Distributions.
                                                              (line  49)
* gsl_ran_exponential (C function):      The Exponential Distribution.
                                                              (line   6)
* gsl_ran_exponential_pdf (C function):  The Exponential Distribution.
                                                              (line  15)
* gsl_ran_exppow (C function):           The Exponential Power Distribution.
                                                              (line   6)
* gsl_ran_exppow_pdf (C function):       The Exponential Power Distribution.
                                                              (line  20)
* gsl_ran_fdist (C function):            The F-distribution.  (line  13)
* gsl_ran_fdist_pdf (C function):        The F-distribution.  (line  28)
* gsl_ran_flat (C function):             The Flat Uniform Distribution.
                                                              (line   6)
* gsl_ran_flat_pdf (C function):         The Flat Uniform Distribution.
                                                              (line  16)
* gsl_ran_gamma (C function):            The Gamma Distribution.
                                                              (line   6)
* gsl_ran_gamma_knuth (C function):      The Gamma Distribution.
                                                              (line  23)
* gsl_ran_gamma_pdf (C function):        The Gamma Distribution.
                                                              (line  30)
* gsl_ran_gaussian (C function):         The Gaussian Distribution.
                                                              (line   6)
* gsl_ran_gaussian_pdf (C function):     The Gaussian Distribution.
                                                              (line  20)
* gsl_ran_gaussian_ratio_method (C function): The Gaussian Distribution.
                                                              (line  27)
* gsl_ran_gaussian_tail (C function):    The Gaussian Tail Distribution.
                                                              (line   6)
* gsl_ran_gaussian_tail_pdf (C function): The Gaussian Tail Distribution.
                                                              (line  25)
* gsl_ran_gaussian_ziggurat (C function): The Gaussian Distribution.
                                                              (line  27)
* gsl_ran_geometric (C function):        The Geometric Distribution.
                                                              (line   6)
* gsl_ran_geometric_pdf (C function):    The Geometric Distribution.
                                                              (line  20)
* gsl_ran_gumbel1 (C function):          The Type-1 Gumbel Distribution.
                                                              (line   6)
* gsl_ran_gumbel1_pdf (C function):      The Type-1 Gumbel Distribution.
                                                              (line  17)
* gsl_ran_gumbel2 (C function):          The Type-2 Gumbel Distribution.
                                                              (line   6)
* gsl_ran_gumbel2_pdf (C function):      The Type-2 Gumbel Distribution.
                                                              (line  17)
* gsl_ran_hypergeometric (C function):   The Hypergeometric Distribution.
                                                              (line   6)
* gsl_ran_hypergeometric_pdf (C function): The Hypergeometric Distribution.
                                                              (line  23)
* gsl_ran_landau (C function):           The Landau Distribution.
                                                              (line   6)
* gsl_ran_landau_pdf (C function):       The Landau Distribution.
                                                              (line  19)
* gsl_ran_laplace (C function):          The Laplace Distribution.
                                                              (line   6)
* gsl_ran_laplace_pdf (C function):      The Laplace Distribution.
                                                              (line  15)
* gsl_ran_levy (C function):             The Levy alpha-Stable Distributions.
                                                              (line   6)
* gsl_ran_levy_skew (C function):        The Levy skew alpha-Stable Distribution.
                                                              (line   6)
* gsl_ran_logarithmic (C function):      The Logarithmic Distribution.
                                                              (line   6)
* gsl_ran_logarithmic_pdf (C function):  The Logarithmic Distribution.
                                                              (line  18)
* gsl_ran_logistic (C function):         The Logistic Distribution.
                                                              (line   6)
* gsl_ran_logistic_pdf (C function):     The Logistic Distribution.
                                                              (line  15)
* gsl_ran_lognormal (C function):        The Lognormal Distribution.
                                                              (line   6)
* gsl_ran_lognormal_pdf (C function):    The Lognormal Distribution.
                                                              (line  17)
* gsl_ran_multinomial (C function):      The Multinomial Distribution.
                                                              (line   6)
* gsl_ran_multinomial_lnpdf (C function): The Multinomial Distribution.
                                                              (line  35)
* gsl_ran_multinomial_pdf (C function):  The Multinomial Distribution.
                                                              (line  28)
* gsl_ran_multivariate_gaussian (C function): The Multivariate Gaussian Distribution.
                                                              (line   6)
* gsl_ran_multivariate_gaussian_log_pdf (C function): The Multivariate Gaussian Distribution.
                                                              (line  21)
* gsl_ran_multivariate_gaussian_mean (C function): The Multivariate Gaussian Distribution.
                                                              (line  33)
* gsl_ran_multivariate_gaussian_pdf (C function): The Multivariate Gaussian Distribution.
                                                              (line  21)
* gsl_ran_multivariate_gaussian_vcov (C function): The Multivariate Gaussian Distribution.
                                                              (line  47)
* gsl_ran_negative_binomial (C function): The Negative Binomial Distribution.
                                                              (line   6)
* gsl_ran_negative_binomial_pdf (C function): The Negative Binomial Distribution.
                                                              (line  18)
* gsl_ran_pareto (C function):           The Pareto Distribution.
                                                              (line   6)
* gsl_ran_pareto_pdf (C function):       The Pareto Distribution.
                                                              (line  17)
* gsl_ran_pascal (C function):           The Pascal Distribution.
                                                              (line   6)
* gsl_ran_pascal_pdf (C function):       The Pascal Distribution.
                                                              (line  17)
* gsl_ran_poisson (C function):          The Poisson Distribution.
                                                              (line   6)
* gsl_ran_poisson_pdf (C function):      The Poisson Distribution.
                                                              (line  17)
* gsl_ran_rayleigh (C function):         The Rayleigh Distribution.
                                                              (line   6)
* gsl_ran_rayleigh_pdf (C function):     The Rayleigh Distribution.
                                                              (line  15)
* gsl_ran_rayleigh_tail (C function):    The Rayleigh Tail Distribution.
                                                              (line   6)
* gsl_ran_rayleigh_tail_pdf (C function): The Rayleigh Tail Distribution.
                                                              (line  17)
* gsl_ran_sample (C function):           Shuffling and Sampling.
                                                              (line  64)
* gsl_ran_shuffle (C function):          Shuffling and Sampling.
                                                              (line  13)
* gsl_ran_tdist (C function):            The t-distribution.  (line  14)
* gsl_ran_tdist_pdf (C function):        The t-distribution.  (line  23)
* gsl_ran_ugaussian (C function):        The Gaussian Distribution.
                                                              (line  39)
* gsl_ran_ugaussian_pdf (C function):    The Gaussian Distribution.
                                                              (line  39)
* gsl_ran_ugaussian_ratio_method (C function): The Gaussian Distribution.
                                                              (line  39)
* gsl_ran_ugaussian_tail (C function):   The Gaussian Tail Distribution.
                                                              (line  34)
* gsl_ran_ugaussian_tail_pdf (C function): The Gaussian Tail Distribution.
                                                              (line  34)
* gsl_ran_weibull (C function):          The Weibull Distribution.
                                                              (line   6)
* gsl_ran_weibull_pdf (C function):      The Weibull Distribution.
                                                              (line  17)
* gsl_ran_wishart (C function):          The Wishart Distribution.
                                                              (line   6)
* gsl_ran_wishart_log_pdf (C function):  The Wishart Distribution.
                                                              (line  21)
* gsl_ran_wishart_pdf (C function):      The Wishart Distribution.
                                                              (line  21)
* GSL_REAL (C macro):                    Representation of complex numbers.
                                                              (line  36)
* gsl_rng (C type):                      The Random Number Generator Interface.
                                                              (line  18)
* gsl_rng_alloc (C function):            Random number generator initialization.
                                                              (line   6)
* gsl_rng_borosh13 (C variable):         Other random number generators.
                                                              (line 193)
* gsl_rng_clone (C function):            Copying random number generator state.
                                                              (line  17)
* gsl_rng_cmrg (C variable):             Random number generator algorithms.
                                                              (line 103)
* gsl_rng_coveyou (C variable):          Other random number generators.
                                                              (line 222)
* gsl_rng_default (C variable):          Random number environment variables.
                                                              (line  23)
* gsl_rng_default_seed (C variable):     Random number environment variables.
                                                              (line  31)
* gsl_rng_env_setup (C function):        Random number environment variables.
                                                              (line  40)
* gsl_rng_fishman18 (C variable):        Other random number generators.
                                                              (line 193)
* gsl_rng_fishman20 (C variable):        Other random number generators.
                                                              (line 193)
* gsl_rng_fishman2x (C variable):        Other random number generators.
                                                              (line 211)
* gsl_rng_fread (C function):            Reading and writing random number generator state.
                                                              (line  18)
* gsl_rng_free (C function):             Random number generator initialization.
                                                              (line  47)
* gsl_rng_fwrite (C function):           Reading and writing random number generator state.
                                                              (line   9)
* gsl_rng_get (C function):              Sampling from a random number generator.
                                                              (line  12)
* gsl_rng_gfsr4 (C variable):            Random number generator algorithms.
                                                              (line 181)
* gsl_rng_knuthran (C variable):         Other random number generators.
                                                              (line 183)
* gsl_rng_knuthran2 (C variable):        Other random number generators.
                                                              (line 173)
* gsl_rng_knuthran2002 (C variable):     Other random number generators.
                                                              (line 183)
* gsl_rng_lecuyer21 (C variable):        Other random number generators.
                                                              (line 193)
* gsl_rng_max (C function):              Auxiliary random number generator functions.
                                                              (line  21)
* gsl_rng_memcpy (C function):           Copying random number generator state.
                                                              (line  11)
* gsl_rng_min (C function):              Auxiliary random number generator functions.
                                                              (line  26)
* gsl_rng_minstd (C variable):           Other random number generators.
                                                              (line 123)
* gsl_rng_mrg (C variable):              Random number generator algorithms.
                                                              (line 126)
* gsl_rng_mt19937 (C variable):          Random number generator algorithms.
                                                              (line  19)
* gsl_rng_name (C function):             Auxiliary random number generator functions.
                                                              (line  10)
* gsl_rng_r250 (C variable):             Other random number generators.
                                                              (line  62)
* gsl_rng_rand (C variable):             Unix random number generators.
                                                              (line  17)
* gsl_rng_rand48 (C variable):           Unix random number generators.
                                                              (line  59)
* gsl_rng_random_bsd (C variable):       Unix random number generators.
                                                              (line  27)
* gsl_rng_random_glibc2 (C variable):    Unix random number generators.
                                                              (line  27)
* gsl_rng_random_libc5 (C variable):     Unix random number generators.
                                                              (line  27)
* gsl_rng_randu (C variable):            Other random number generators.
                                                              (line 113)
* gsl_rng_ranf (C variable):             Other random number generators.
                                                              (line  22)
* gsl_rng_ranlux (C variable):           Random number generator algorithms.
                                                              (line  74)
* gsl_rng_ranlux389 (C variable):        Random number generator algorithms.
                                                              (line  74)
* gsl_rng_ranlxd1 (C variable):          Random number generator algorithms.
                                                              (line  67)
* gsl_rng_ranlxd2 (C variable):          Random number generator algorithms.
                                                              (line  67)
* gsl_rng_ranlxs0 (C variable):          Random number generator algorithms.
                                                              (line  47)
* gsl_rng_ranlxs1 (C variable):          Random number generator algorithms.
                                                              (line  47)
* gsl_rng_ranlxs2 (C variable):          Random number generator algorithms.
                                                              (line  47)
* gsl_rng_ranmar (C variable):           Other random number generators.
                                                              (line  55)
* GSL_RNG_SEED (C macro):                Random number environment variables.
                                                              (line  18)
* gsl_rng_set (C function):              Random number generator initialization.
                                                              (line  26)
* gsl_rng_size (C function):             Auxiliary random number generator functions.
                                                              (line  33)
* gsl_rng_slatec (C variable):           Other random number generators.
                                                              (line 153)
* gsl_rng_state (C function):            Auxiliary random number generator functions.
                                                              (line  33)
* gsl_rng_taus (C variable):             Random number generator algorithms.
                                                              (line 144)
* gsl_rng_taus2 (C variable):            Random number generator algorithms.
                                                              (line 144)
* gsl_rng_transputer (C variable):       Other random number generators.
                                                              (line 103)
* gsl_rng_tt800 (C variable):            Other random number generators.
                                                              (line  79)
* GSL_RNG_TYPE (C macro):                Random number environment variables.
                                                              (line  12)
* gsl_rng_type (C type):                 The Random Number Generator Interface.
                                                              (line  18)
* gsl_rng_types_setup (C function):      Auxiliary random number generator functions.
                                                              (line  45)
* gsl_rng_uni (C variable):              Other random number generators.
                                                              (line 145)
* gsl_rng_uni32 (C variable):            Other random number generators.
                                                              (line 145)
* gsl_rng_uniform (C function):          Sampling from a random number generator.
                                                              (line  20)
* gsl_rng_uniform_int (C function):      Sampling from a random number generator.
                                                              (line  40)
* gsl_rng_uniform_pos (C function):      Sampling from a random number generator.
                                                              (line  31)
* gsl_rng_vax (C variable):              Other random number generators.
                                                              (line  93)
* gsl_rng_waterman14 (C variable):       Other random number generators.
                                                              (line 193)
* gsl_rng_zuf (C variable):              Other random number generators.
                                                              (line 158)
* gsl_root_fdfsolver (C type):           Initializing the Solver.
                                                              (line  11)
* gsl_root_fdfsolver_alloc (C function): Initializing the Solver.
                                                              (line  30)
* gsl_root_fdfsolver_free (C function):  Initializing the Solver.
                                                              (line  58)
* gsl_root_fdfsolver_iterate (C function): Iteration.         (line  12)
* gsl_root_fdfsolver_name (C function):  Initializing the Solver.
                                                              (line  63)
* gsl_root_fdfsolver_newton (C variable): Root Finding Algorithms using Derivatives.
                                                              (line  17)
* gsl_root_fdfsolver_root (C function):  Iteration.           (line  35)
* gsl_root_fdfsolver_secant (C variable): Root Finding Algorithms using Derivatives.
                                                              (line  31)
* gsl_root_fdfsolver_set (C function):   Initializing the Solver.
                                                              (line  51)
* gsl_root_fdfsolver_steffenson (C variable): Root Finding Algorithms using Derivatives.
                                                              (line  62)
* gsl_root_fdfsolver_type (C type):      Root Finding Algorithms using Derivatives.
                                                              (line  15)
* gsl_root_fsolver (C type):             Initializing the Solver.
                                                              (line   6)
* gsl_root_fsolver_alloc (C function):   Initializing the Solver.
                                                              (line  16)
* gsl_root_fsolver_bisection (C variable): Root Bracketing Algorithms.
                                                              (line  18)
* gsl_root_fsolver_brent (C variable):   Root Bracketing Algorithms.
                                                              (line  52)
* gsl_root_fsolver_falsepos (C variable): Root Bracketing Algorithms.
                                                              (line  34)
* gsl_root_fsolver_free (C function):    Initializing the Solver.
                                                              (line  58)
* gsl_root_fsolver_iterate (C function): Iteration.           (line  12)
* gsl_root_fsolver_name (C function):    Initializing the Solver.
                                                              (line  63)
* gsl_root_fsolver_root (C function):    Iteration.           (line  35)
* gsl_root_fsolver_set (C function):     Initializing the Solver.
                                                              (line  44)
* gsl_root_fsolver_type (C type):        Root Bracketing Algorithms.
                                                              (line  16)
* gsl_root_fsolver_x_lower (C function): Iteration.           (line  43)
* gsl_root_fsolver_x_upper (C function): Iteration.           (line  43)
* gsl_root_test_delta (C function):      Search Stopping Parameters.
                                                              (line  43)
* gsl_root_test_interval (C function):   Search Stopping Parameters.
                                                              (line  19)
* gsl_root_test_residual (C function):   Search Stopping Parameters.
                                                              (line  54)
* gsl_rstat_add (C function):            Adding Data to the Accumulator.
                                                              (line   6)
* gsl_rstat_alloc (C function):          Initializing the Accumulator.
                                                              (line  12)
* gsl_rstat_free (C function):           Initializing the Accumulator.
                                                              (line  17)
* gsl_rstat_kurtosis (C function):       Current Statistics.  (line  55)
* gsl_rstat_max (C function):            Current Statistics.  (line  10)
* gsl_rstat_mean (C function):           Current Statistics.  (line  14)
* gsl_rstat_median (C function):         Current Statistics.  (line  62)
* gsl_rstat_min (C function):            Current Statistics.  (line   6)
* gsl_rstat_n (C function):              Adding Data to the Accumulator.
                                                              (line  13)
* gsl_rstat_quantile_add (C function):   Quantiles.           (line  36)
* gsl_rstat_quantile_alloc (C function): Quantiles.           (line  18)
* gsl_rstat_quantile_free (C function):  Quantiles.           (line  25)
* gsl_rstat_quantile_get (C function):   Quantiles.           (line  42)
* gsl_rstat_quantile_reset (C function): Quantiles.           (line  30)
* gsl_rstat_quantile_workspace (C type): Quantiles.           (line  13)
* gsl_rstat_reset (C function):          Initializing the Accumulator.
                                                              (line  21)
* gsl_rstat_rms (C function):            Current Statistics.  (line  41)
* gsl_rstat_sd (C function):             Current Statistics.  (line  28)
* gsl_rstat_sd_mean (C function):        Current Statistics.  (line  34)
* gsl_rstat_skew (C function):           Current Statistics.  (line  48)
* gsl_rstat_variance (C function):       Current Statistics.  (line  21)
* gsl_rstat_workspace (C type):          Initializing the Accumulator.
                                                              (line   6)
* GSL_SET_COMPLEX (C macro):             Representation of complex numbers.
                                                              (line  42)
* gsl_set_error_handler (C function):    Error Handlers.      (line  43)
* gsl_set_error_handler_off (C function): Error Handlers.     (line  69)
* GSL_SET_IMAG (C macro):                Representation of complex numbers.
                                                              (line  52)
* GSL_SET_REAL (C macro):                Representation of complex numbers.
                                                              (line  52)
* gsl_sf_airy_Ai (C function):           Airy Functions.      (line   6)
* gsl_sf_airy_Ai_deriv (C function):     Derivatives of Airy Functions.
                                                              (line   6)
* gsl_sf_airy_Ai_deriv_e (C function):   Derivatives of Airy Functions.
                                                              (line   6)
* gsl_sf_airy_Ai_deriv_scaled (C function): Derivatives of Airy Functions.
                                                              (line  20)
* gsl_sf_airy_Ai_deriv_scaled_e (C function): Derivatives of Airy Functions.
                                                              (line  20)
* gsl_sf_airy_Ai_e (C function):         Airy Functions.      (line   6)
* gsl_sf_airy_Ai_scaled (C function):    Airy Functions.      (line  20)
* gsl_sf_airy_Ai_scaled_e (C function):  Airy Functions.      (line  20)
* gsl_sf_airy_Bi (C function):           Airy Functions.      (line  13)
* gsl_sf_airy_Bi_deriv (C function):     Derivatives of Airy Functions.
                                                              (line  13)
* gsl_sf_airy_Bi_deriv_e (C function):   Derivatives of Airy Functions.
                                                              (line  13)
* gsl_sf_airy_Bi_deriv_scaled (C function): Derivatives of Airy Functions.
                                                              (line  29)
* gsl_sf_airy_Bi_deriv_scaled_e (C function): Derivatives of Airy Functions.
                                                              (line  29)
* gsl_sf_airy_Bi_e (C function):         Airy Functions.      (line  13)
* gsl_sf_airy_Bi_scaled (C function):    Airy Functions.      (line  28)
* gsl_sf_airy_Bi_scaled_e (C function):  Airy Functions.      (line  28)
* gsl_sf_airy_zero_Ai (C function):      Zeros of Airy Functions.
                                                              (line   6)
* gsl_sf_airy_zero_Ai_deriv (C function): Zeros of Derivatives of Airy Functions.
                                                              (line   6)
* gsl_sf_airy_zero_Ai_deriv_e (C function): Zeros of Derivatives of Airy Functions.
                                                              (line   6)
* gsl_sf_airy_zero_Ai_e (C function):    Zeros of Airy Functions.
                                                              (line   6)
* gsl_sf_airy_zero_Bi (C function):      Zeros of Airy Functions.
                                                              (line  14)
* gsl_sf_airy_zero_Bi_deriv (C function): Zeros of Derivatives of Airy Functions.
                                                              (line  13)
* gsl_sf_airy_zero_Bi_deriv_e (C function): Zeros of Derivatives of Airy Functions.
                                                              (line  13)
* gsl_sf_airy_zero_Bi_e (C function):    Zeros of Airy Functions.
                                                              (line  14)
* gsl_sf_angle_restrict_pos (C function): Restriction Functions.
                                                              (line  16)
* gsl_sf_angle_restrict_pos_e (C function): Restriction Functions.
                                                              (line  16)
* gsl_sf_angle_restrict_symm (C function): Restriction Functions.
                                                              (line   6)
* gsl_sf_angle_restrict_symm_e (C function): Restriction Functions.
                                                              (line   6)
* gsl_sf_atanint (C function):           Arctangent Integral. (line   6)
* gsl_sf_atanint_e (C function):         Arctangent Integral. (line   6)
* gsl_sf_bessel_I0 (C function):         Regular Modified Cylindrical Bessel Functions.
                                                              (line   6)
* gsl_sf_bessel_I0_e (C function):       Regular Modified Cylindrical Bessel Functions.
                                                              (line   6)
* gsl_sf_bessel_I0_scaled (C function):  Regular Modified Cylindrical Bessel Functions.
                                                              (line  36)
* gsl_sf_bessel_i0_scaled (C function):  Regular Modified Spherical Bessel Functions.
                                                              (line  10)
* gsl_sf_bessel_I0_scaled_e (C function): Regular Modified Cylindrical Bessel Functions.
                                                              (line  36)
* gsl_sf_bessel_i0_scaled_e (C function): Regular Modified Spherical Bessel Functions.
                                                              (line  10)
* gsl_sf_bessel_I1 (C function):         Regular Modified Cylindrical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_I1_e (C function):       Regular Modified Cylindrical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_I1_scaled (C function):  Regular Modified Cylindrical Bessel Functions.
                                                              (line  43)
* gsl_sf_bessel_i1_scaled (C function):  Regular Modified Spherical Bessel Functions.
                                                              (line  17)
* gsl_sf_bessel_I1_scaled_e (C function): Regular Modified Cylindrical Bessel Functions.
                                                              (line  43)
* gsl_sf_bessel_i1_scaled_e (C function): Regular Modified Spherical Bessel Functions.
                                                              (line  17)
* gsl_sf_bessel_i2_scaled (C function):  Regular Modified Spherical Bessel Functions.
                                                              (line  24)
* gsl_sf_bessel_i2_scaled_e (C function): Regular Modified Spherical Bessel Functions.
                                                              (line  24)
* gsl_sf_bessel_il_scaled (C function):  Regular Modified Spherical Bessel Functions.
                                                              (line  31)
* gsl_sf_bessel_il_scaled_array (C function): Regular Modified Spherical Bessel Functions.
                                                              (line  38)
* gsl_sf_bessel_il_scaled_e (C function): Regular Modified Spherical Bessel Functions.
                                                              (line  31)
* gsl_sf_bessel_In (C function):         Regular Modified Cylindrical Bessel Functions.
                                                              (line  18)
* gsl_sf_bessel_Inu (C function):        Regular Modified Bessel Functions—Fractional Order.
                                                              (line   6)
* gsl_sf_bessel_Inu_e (C function):      Regular Modified Bessel Functions—Fractional Order.
                                                              (line   6)
* gsl_sf_bessel_Inu_scaled (C function): Regular Modified Bessel Functions—Fractional Order.
                                                              (line  13)
* gsl_sf_bessel_Inu_scaled_e (C function): Regular Modified Bessel Functions—Fractional Order.
                                                              (line  13)
* gsl_sf_bessel_In_array (C function):   Regular Modified Cylindrical Bessel Functions.
                                                              (line  25)
* gsl_sf_bessel_In_e (C function):       Regular Modified Cylindrical Bessel Functions.
                                                              (line  18)
* gsl_sf_bessel_In_scaled (C function):  Regular Modified Cylindrical Bessel Functions.
                                                              (line  50)
* gsl_sf_bessel_In_scaled_array (C function): Regular Modified Cylindrical Bessel Functions.
                                                              (line  57)
* gsl_sf_bessel_In_scaled_e (C function): Regular Modified Cylindrical Bessel Functions.
                                                              (line  50)
* gsl_sf_bessel_J0 (C function):         Regular Cylindrical Bessel Functions.
                                                              (line   6)
* gsl_sf_bessel_j0 (C function):         Regular Spherical Bessel Functions.
                                                              (line   6)
* gsl_sf_bessel_J0_e (C function):       Regular Cylindrical Bessel Functions.
                                                              (line   6)
* gsl_sf_bessel_j0_e (C function):       Regular Spherical Bessel Functions.
                                                              (line   6)
* gsl_sf_bessel_J1 (C function):         Regular Cylindrical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_j1 (C function):         Regular Spherical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_J1_e (C function):       Regular Cylindrical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_j1_e (C function):       Regular Spherical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_j2 (C function):         Regular Spherical Bessel Functions.
                                                              (line  18)
* gsl_sf_bessel_j2_e (C function):       Regular Spherical Bessel Functions.
                                                              (line  18)
* gsl_sf_bessel_jl (C function):         Regular Spherical Bessel Functions.
                                                              (line  24)
* gsl_sf_bessel_jl_array (C function):   Regular Spherical Bessel Functions.
                                                              (line  31)
* gsl_sf_bessel_jl_e (C function):       Regular Spherical Bessel Functions.
                                                              (line  24)
* gsl_sf_bessel_jl_steed_array (C function): Regular Spherical Bessel Functions.
                                                              (line  40)
* gsl_sf_bessel_Jn (C function):         Regular Cylindrical Bessel Functions.
                                                              (line  18)
* gsl_sf_bessel_Jnu (C function):        Regular Bessel Function—Fractional Order.
                                                              (line   6)
* gsl_sf_bessel_Jnu_e (C function):      Regular Bessel Function—Fractional Order.
                                                              (line   6)
* gsl_sf_bessel_Jn_array (C function):   Regular Cylindrical Bessel Functions.
                                                              (line  25)
* gsl_sf_bessel_Jn_e (C function):       Regular Cylindrical Bessel Functions.
                                                              (line  18)
* gsl_sf_bessel_K0 (C function):         Irregular Modified Cylindrical Bessel Functions.
                                                              (line   6)
* gsl_sf_bessel_K0_e (C function):       Irregular Modified Cylindrical Bessel Functions.
                                                              (line   6)
* gsl_sf_bessel_K0_scaled (C function):  Irregular Modified Cylindrical Bessel Functions.
                                                              (line  37)
* gsl_sf_bessel_k0_scaled (C function):  Irregular Modified Spherical Bessel Functions.
                                                              (line  10)
* gsl_sf_bessel_K0_scaled_e (C function): Irregular Modified Cylindrical Bessel Functions.
                                                              (line  37)
* gsl_sf_bessel_k0_scaled_e (C function): Irregular Modified Spherical Bessel Functions.
                                                              (line  10)
* gsl_sf_bessel_K1 (C function):         Irregular Modified Cylindrical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_K1_e (C function):       Irregular Modified Cylindrical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_K1_scaled (C function):  Irregular Modified Cylindrical Bessel Functions.
                                                              (line  44)
* gsl_sf_bessel_k1_scaled (C function):  Irregular Modified Spherical Bessel Functions.
                                                              (line  17)
* gsl_sf_bessel_K1_scaled_e (C function): Irregular Modified Cylindrical Bessel Functions.
                                                              (line  44)
* gsl_sf_bessel_k1_scaled_e (C function): Irregular Modified Spherical Bessel Functions.
                                                              (line  17)
* gsl_sf_bessel_k2_scaled (C function):  Irregular Modified Spherical Bessel Functions.
                                                              (line  24)
* gsl_sf_bessel_k2_scaled_e (C function): Irregular Modified Spherical Bessel Functions.
                                                              (line  24)
* gsl_sf_bessel_kl_scaled (C function):  Irregular Modified Spherical Bessel Functions.
                                                              (line  31)
* gsl_sf_bessel_kl_scaled_array (C function): Irregular Modified Spherical Bessel Functions.
                                                              (line  38)
* gsl_sf_bessel_kl_scaled_e (C function): Irregular Modified Spherical Bessel Functions.
                                                              (line  31)
* gsl_sf_bessel_Kn (C function):         Irregular Modified Cylindrical Bessel Functions.
                                                              (line  18)
* gsl_sf_bessel_Knu (C function):        Irregular Modified Bessel Functions—Fractional Order.
                                                              (line   6)
* gsl_sf_bessel_Knu_e (C function):      Irregular Modified Bessel Functions—Fractional Order.
                                                              (line   6)
* gsl_sf_bessel_Knu_scaled (C function): Irregular Modified Bessel Functions—Fractional Order.
                                                              (line  21)
* gsl_sf_bessel_Knu_scaled_e (C function): Irregular Modified Bessel Functions—Fractional Order.
                                                              (line  21)
* gsl_sf_bessel_Kn_array (C function):   Irregular Modified Cylindrical Bessel Functions.
                                                              (line  25)
* gsl_sf_bessel_Kn_e (C function):       Irregular Modified Cylindrical Bessel Functions.
                                                              (line  18)
* gsl_sf_bessel_Kn_scaled (C function):  Irregular Modified Cylindrical Bessel Functions.
                                                              (line  51)
* gsl_sf_bessel_Kn_scaled_array (C function): Irregular Modified Cylindrical Bessel Functions.
                                                              (line  58)
* gsl_sf_bessel_Kn_scaled_e (C function): Irregular Modified Cylindrical Bessel Functions.
                                                              (line  51)
* gsl_sf_bessel_lnKnu (C function):      Irregular Modified Bessel Functions—Fractional Order.
                                                              (line  13)
* gsl_sf_bessel_lnKnu_e (C function):    Irregular Modified Bessel Functions—Fractional Order.
                                                              (line  13)
* gsl_sf_bessel_sequence_Jnu_e (C function): Regular Bessel Function—Fractional Order.
                                                              (line  13)
* gsl_sf_bessel_Y0 (C function):         Irregular Cylindrical Bessel Functions.
                                                              (line   6)
* gsl_sf_bessel_y0 (C function):         Irregular Spherical Bessel Functions.
                                                              (line   6)
* gsl_sf_bessel_Y0_e (C function):       Irregular Cylindrical Bessel Functions.
                                                              (line   6)
* gsl_sf_bessel_y0_e (C function):       Irregular Spherical Bessel Functions.
                                                              (line   6)
* gsl_sf_bessel_Y1 (C function):         Irregular Cylindrical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_y1 (C function):         Irregular Spherical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_Y1_e (C function):       Irregular Cylindrical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_y1_e (C function):       Irregular Spherical Bessel Functions.
                                                              (line  12)
* gsl_sf_bessel_y2 (C function):         Irregular Spherical Bessel Functions.
                                                              (line  18)
* gsl_sf_bessel_y2_e (C function):       Irregular Spherical Bessel Functions.
                                                              (line  18)
* gsl_sf_bessel_yl (C function):         Irregular Spherical Bessel Functions.
                                                              (line  24)
* gsl_sf_bessel_yl_array (C function):   Irregular Spherical Bessel Functions.
                                                              (line  31)
* gsl_sf_bessel_yl_e (C function):       Irregular Spherical Bessel Functions.
                                                              (line  24)
* gsl_sf_bessel_Yn (C function):         Irregular Cylindrical Bessel Functions.
                                                              (line  18)
* gsl_sf_bessel_Ynu (C function):        Irregular Bessel Functions—Fractional Order.
                                                              (line   6)
* gsl_sf_bessel_Ynu_e (C function):      Irregular Bessel Functions—Fractional Order.
                                                              (line   6)
* gsl_sf_bessel_Yn_array (C function):   Irregular Cylindrical Bessel Functions.
                                                              (line  25)
* gsl_sf_bessel_Yn_e (C function):       Irregular Cylindrical Bessel Functions.
                                                              (line  18)
* gsl_sf_bessel_zero_J0 (C function):    Zeros of Regular Bessel Functions.
                                                              (line   6)
* gsl_sf_bessel_zero_J0_e (C function):  Zeros of Regular Bessel Functions.
                                                              (line   6)
* gsl_sf_bessel_zero_J1 (C function):    Zeros of Regular Bessel Functions.
                                                              (line  13)
* gsl_sf_bessel_zero_J1_e (C function):  Zeros of Regular Bessel Functions.
                                                              (line  13)
* gsl_sf_bessel_zero_Jnu (C function):   Zeros of Regular Bessel Functions.
                                                              (line  20)
* gsl_sf_bessel_zero_Jnu_e (C function): Zeros of Regular Bessel Functions.
                                                              (line  20)
* gsl_sf_beta (C function):              Beta Functions.      (line   6)
* gsl_sf_beta_e (C function):            Beta Functions.      (line   6)
* gsl_sf_beta_inc (C function):          Incomplete Beta Function.
                                                              (line   6)
* gsl_sf_beta_inc_e (C function):        Incomplete Beta Function.
                                                              (line   6)
* gsl_sf_Chi (C function):               Hyperbolic Integrals.
                                                              (line  13)
* gsl_sf_Chi_e (C function):             Hyperbolic Integrals.
                                                              (line  13)
* gsl_sf_choose (C function):            Factorials.          (line  44)
* gsl_sf_choose_e (C function):          Factorials.          (line  44)
* gsl_sf_Ci (C function):                Trigonometric Integrals.
                                                              (line  13)
* gsl_sf_Ci_e (C function):              Trigonometric Integrals.
                                                              (line  13)
* gsl_sf_clausen (C function):           Clausen Functions.   (line  14)
* gsl_sf_clausen_e (C function):         Clausen Functions.   (line  14)
* gsl_sf_complex_cos_e (C function):     Trigonometric Functions for Complex Arguments.
                                                              (line  12)
* gsl_sf_complex_dilog_e (C function):   Complex Argument.    (line   6)
* gsl_sf_complex_logsin_e (C function):  Trigonometric Functions for Complex Arguments.
                                                              (line  18)
* gsl_sf_complex_log_e (C function):     Logarithm and Related Functions.
                                                              (line  21)
* gsl_sf_complex_sin_e (C function):     Trigonometric Functions for Complex Arguments.
                                                              (line   6)
* gsl_sf_conicalP_0 (C function):        Conical Functions.   (line  24)
* gsl_sf_conicalP_0_e (C function):      Conical Functions.   (line  24)
* gsl_sf_conicalP_1 (C function):        Conical Functions.   (line  31)
* gsl_sf_conicalP_1_e (C function):      Conical Functions.   (line  31)
* gsl_sf_conicalP_cyl_reg (C function):  Conical Functions.   (line  48)
* gsl_sf_conicalP_cyl_reg_e (C function): Conical Functions.  (line  48)
* gsl_sf_conicalP_half (C function):     Conical Functions.   (line  10)
* gsl_sf_conicalP_half_e (C function):   Conical Functions.   (line  10)
* gsl_sf_conicalP_mhalf (C function):    Conical Functions.   (line  17)
* gsl_sf_conicalP_mhalf_e (C function):  Conical Functions.   (line  17)
* gsl_sf_conicalP_sph_reg (C function):  Conical Functions.   (line  38)
* gsl_sf_conicalP_sph_reg_e (C function): Conical Functions.  (line  38)
* gsl_sf_cos (C function):               Circular Trigonometric Functions.
                                                              (line  11)
* gsl_sf_cos_e (C function):             Circular Trigonometric Functions.
                                                              (line  11)
* gsl_sf_cos_err_e (C function):         Trigonometric Functions With Error Estimates.
                                                              (line  15)
* gsl_sf_coulomb_CL_array (C function):  Coulomb Wave Function Normalization Constant.
                                                              (line  15)
* gsl_sf_coulomb_CL_e (C function):      Coulomb Wave Function Normalization Constant.
                                                              (line   9)
* gsl_sf_coulomb_wave_FGp_array (C function): Coulomb Wave Functions.
                                                              (line  55)
* gsl_sf_coulomb_wave_FG_array (C function): Coulomb Wave Functions.
                                                              (line  45)
* gsl_sf_coulomb_wave_FG_e (C function): Coulomb Wave Functions.
                                                              (line  20)
* gsl_sf_coulomb_wave_F_array (C function): Coulomb Wave Functions.
                                                              (line  37)
* gsl_sf_coulomb_wave_sphF_array (C function): Coulomb Wave Functions.
                                                              (line  67)
* gsl_sf_coupling_3j (C function):       3-j Symbols.         (line   6)
* gsl_sf_coupling_3j_e (C function):     3-j Symbols.         (line   6)
* gsl_sf_coupling_6j (C function):       6-j Symbols.         (line   6)
* gsl_sf_coupling_6j_e (C function):     6-j Symbols.         (line   6)
* gsl_sf_coupling_9j (C function):       9-j Symbols.         (line   6)
* gsl_sf_coupling_9j_e (C function):     9-j Symbols.         (line   6)
* gsl_sf_dawson (C function):            Dawson Function.     (line  14)
* gsl_sf_dawson_e (C function):          Dawson Function.     (line  14)
* gsl_sf_debye_1 (C function):           Debye Functions.     (line  13)
* gsl_sf_debye_1_e (C function):         Debye Functions.     (line  13)
* gsl_sf_debye_2 (C function):           Debye Functions.     (line  18)
* gsl_sf_debye_2_e (C function):         Debye Functions.     (line  18)
* gsl_sf_debye_3 (C function):           Debye Functions.     (line  23)
* gsl_sf_debye_3_e (C function):         Debye Functions.     (line  23)
* gsl_sf_debye_4 (C function):           Debye Functions.     (line  28)
* gsl_sf_debye_4_e (C function):         Debye Functions.     (line  28)
* gsl_sf_debye_5 (C function):           Debye Functions.     (line  33)
* gsl_sf_debye_5_e (C function):         Debye Functions.     (line  33)
* gsl_sf_debye_6 (C function):           Debye Functions.     (line  38)
* gsl_sf_debye_6_e (C function):         Debye Functions.     (line  38)
* gsl_sf_dilog (C function):             Real Argument.       (line   6)
* gsl_sf_dilog_e (C function):           Real Argument.       (line   6)
* gsl_sf_doublefact (C function):        Factorials.          (line  19)
* gsl_sf_doublefact_e (C function):      Factorials.          (line  19)
* gsl_sf_ellint_D (C function):          Legendre Form of Incomplete Elliptic Integrals.
                                                              (line  39)
* gsl_sf_ellint_D_e (C function):        Legendre Form of Incomplete Elliptic Integrals.
                                                              (line  39)
* gsl_sf_ellint_E (C function):          Legendre Form of Incomplete Elliptic Integrals.
                                                              (line  16)
* gsl_sf_ellint_Ecomp (C function):      Legendre Form of Complete Elliptic Integrals.
                                                              (line  15)
* gsl_sf_ellint_Ecomp_e (C function):    Legendre Form of Complete Elliptic Integrals.
                                                              (line  15)
* gsl_sf_ellint_E_e (C function):        Legendre Form of Incomplete Elliptic Integrals.
                                                              (line  16)
* gsl_sf_ellint_F (C function):          Legendre Form of Incomplete Elliptic Integrals.
                                                              (line   6)
* gsl_sf_ellint_F_e (C function):        Legendre Form of Incomplete Elliptic Integrals.
                                                              (line   6)
* gsl_sf_ellint_Kcomp (C function):      Legendre Form of Complete Elliptic Integrals.
                                                              (line   6)
* gsl_sf_ellint_Kcomp_e (C function):    Legendre Form of Complete Elliptic Integrals.
                                                              (line   6)
* gsl_sf_ellint_P (C function):          Legendre Form of Incomplete Elliptic Integrals.
                                                              (line  26)
* gsl_sf_ellint_Pcomp (C function):      Legendre Form of Complete Elliptic Integrals.
                                                              (line  24)
* gsl_sf_ellint_Pcomp_e (C function):    Legendre Form of Complete Elliptic Integrals.
                                                              (line  24)
* gsl_sf_ellint_P_e (C function):        Legendre Form of Incomplete Elliptic Integrals.
                                                              (line  26)
* gsl_sf_ellint_RC (C function):         Carlson Forms.       (line   6)
* gsl_sf_ellint_RC_e (C function):       Carlson Forms.       (line   6)
* gsl_sf_ellint_RD (C function):         Carlson Forms.       (line  14)
* gsl_sf_ellint_RD_e (C function):       Carlson Forms.       (line  14)
* gsl_sf_ellint_RF (C function):         Carlson Forms.       (line  24)
* gsl_sf_ellint_RF_e (C function):       Carlson Forms.       (line  24)
* gsl_sf_ellint_RJ (C function):         Carlson Forms.       (line  34)
* gsl_sf_ellint_RJ_e (C function):       Carlson Forms.       (line  34)
* gsl_sf_elljac_e (C function):          Elliptic Functions Jacobi.
                                                              (line  10)
* gsl_sf_erf (C function):               Error Function.      (line   6)
* gsl_sf_erfc (C function):              Complementary Error Function.
                                                              (line   6)
* gsl_sf_erfc_e (C function):            Complementary Error Function.
                                                              (line   6)
* gsl_sf_erf_e (C function):             Error Function.      (line   6)
* gsl_sf_erf_Q (C function):             Probability functions.
                                                              (line  15)
* gsl_sf_erf_Q_e (C function):           Probability functions.
                                                              (line  15)
* gsl_sf_erf_Z (C function):             Probability functions.
                                                              (line   9)
* gsl_sf_erf_Z_e (C function):           Probability functions.
                                                              (line   9)
* gsl_sf_eta (C function):               Eta Function.        (line  15)
* gsl_sf_eta_e (C function):             Eta Function.        (line  15)
* gsl_sf_eta_int (C function):           Eta Function.        (line  10)
* gsl_sf_eta_int_e (C function):         Eta Function.        (line  10)
* gsl_sf_exp (C function):               Exponential Function.
                                                              (line   6)
* gsl_sf_expint_3 (C function):          Ei_3 x.              (line   6)
* gsl_sf_expint_3_e (C function):        Ei_3 x.              (line   6)
* gsl_sf_expint_E1 (C function):         Exponential Integral.
                                                              (line   6)
* gsl_sf_expint_E1_e (C function):       Exponential Integral.
                                                              (line   6)
* gsl_sf_expint_E2 (C function):         Exponential Integral.
                                                              (line  13)
* gsl_sf_expint_E2_e (C function):       Exponential Integral.
                                                              (line  13)
* gsl_sf_expint_Ei (C function):         Ei x.                (line   6)
* gsl_sf_expint_Ei_e (C function):       Ei x.                (line   6)
* gsl_sf_expint_En (C function):         Exponential Integral.
                                                              (line  21)
* gsl_sf_expint_En_e (C function):       Exponential Integral.
                                                              (line  21)
* gsl_sf_expm1 (C function):             Relative Exponential Functions.
                                                              (line   6)
* gsl_sf_expm1_e (C function):           Relative Exponential Functions.
                                                              (line   6)
* gsl_sf_exprel (C function):            Relative Exponential Functions.
                                                              (line  12)
* gsl_sf_exprel_2 (C function):          Relative Exponential Functions.
                                                              (line  20)
* gsl_sf_exprel_2_e (C function):        Relative Exponential Functions.
                                                              (line  20)
* gsl_sf_exprel_e (C function):          Relative Exponential Functions.
                                                              (line  12)
* gsl_sf_exprel_n (C function):          Relative Exponential Functions.
                                                              (line  28)
* gsl_sf_exprel_n_e (C function):        Relative Exponential Functions.
                                                              (line  28)
* gsl_sf_exp_e (C function):             Exponential Function.
                                                              (line   6)
* gsl_sf_exp_e10_e (C function):         Exponential Function.
                                                              (line  12)
* gsl_sf_exp_err_e (C function):         Exponentiation With Error Estimate.
                                                              (line   6)
* gsl_sf_exp_err_e10_e (C function):     Exponentiation With Error Estimate.
                                                              (line  13)
* gsl_sf_exp_mult (C function):          Exponential Function.
                                                              (line  20)
* gsl_sf_exp_mult_e (C function):        Exponential Function.
                                                              (line  20)
* gsl_sf_exp_mult_e10_e (C function):    Exponential Function.
                                                              (line  28)
* gsl_sf_exp_mult_err_e (C function):    Exponentiation With Error Estimate.
                                                              (line  20)
* gsl_sf_exp_mult_err_e10_e (C function): Exponentiation With Error Estimate.
                                                              (line  27)
* gsl_sf_fact (C function):              Factorials.          (line  11)
* gsl_sf_fact_e (C function):            Factorials.          (line  11)
* gsl_sf_fermi_dirac_0 (C function):     Complete Fermi-Dirac Integrals.
                                                              (line  21)
* gsl_sf_fermi_dirac_0_e (C function):   Complete Fermi-Dirac Integrals.
                                                              (line  21)
* gsl_sf_fermi_dirac_1 (C function):     Complete Fermi-Dirac Integrals.
                                                              (line  28)
* gsl_sf_fermi_dirac_1_e (C function):   Complete Fermi-Dirac Integrals.
                                                              (line  28)
* gsl_sf_fermi_dirac_2 (C function):     Complete Fermi-Dirac Integrals.
                                                              (line  35)
* gsl_sf_fermi_dirac_2_e (C function):   Complete Fermi-Dirac Integrals.
                                                              (line  35)
* gsl_sf_fermi_dirac_3half (C function): Complete Fermi-Dirac Integrals.
                                                              (line  64)
* gsl_sf_fermi_dirac_3half_e (C function): Complete Fermi-Dirac Integrals.
                                                              (line  64)
* gsl_sf_fermi_dirac_half (C function):  Complete Fermi-Dirac Integrals.
                                                              (line  57)
* gsl_sf_fermi_dirac_half_e (C function): Complete Fermi-Dirac Integrals.
                                                              (line  57)
* gsl_sf_fermi_dirac_inc_0 (C function): Incomplete Fermi-Dirac Integrals.
                                                              (line  10)
* gsl_sf_fermi_dirac_inc_0_e (C function): Incomplete Fermi-Dirac Integrals.
                                                              (line  10)
* gsl_sf_fermi_dirac_int (C function):   Complete Fermi-Dirac Integrals.
                                                              (line  42)
* gsl_sf_fermi_dirac_int_e (C function): Complete Fermi-Dirac Integrals.
                                                              (line  42)
* gsl_sf_fermi_dirac_m1 (C function):    Complete Fermi-Dirac Integrals.
                                                              (line  13)
* gsl_sf_fermi_dirac_m1_e (C function):  Complete Fermi-Dirac Integrals.
                                                              (line  13)
* gsl_sf_fermi_dirac_mhalf (C function): Complete Fermi-Dirac Integrals.
                                                              (line  50)
* gsl_sf_fermi_dirac_mhalf_e (C function): Complete Fermi-Dirac Integrals.
                                                              (line  50)
* gsl_sf_gamma (C function):             Gamma Functions.     (line  14)
* gsl_sf_gammainv (C function):          Gamma Functions.     (line  54)
* gsl_sf_gammainv_e (C function):        Gamma Functions.     (line  54)
* gsl_sf_gammastar (C function):         Gamma Functions.     (line  43)
* gsl_sf_gammastar_e (C function):       Gamma Functions.     (line  43)
* gsl_sf_gamma_e (C function):           Gamma Functions.     (line  14)
* gsl_sf_gamma_inc (C function):         Incomplete Gamma Functions.
                                                              (line   6)
* gsl_sf_gamma_inc_e (C function):       Incomplete Gamma Functions.
                                                              (line   6)
* gsl_sf_gamma_inc_P (C function):       Incomplete Gamma Functions.
                                                              (line  23)
* gsl_sf_gamma_inc_P_e (C function):     Incomplete Gamma Functions.
                                                              (line  23)
* gsl_sf_gamma_inc_Q (C function):       Incomplete Gamma Functions.
                                                              (line  15)
* gsl_sf_gamma_inc_Q_e (C function):     Incomplete Gamma Functions.
                                                              (line  15)
* gsl_sf_gegenpoly_1 (C function):       Gegenbauer Functions.
                                                              (line  11)
* gsl_sf_gegenpoly_1_e (C function):     Gegenbauer Functions.
                                                              (line  11)
* gsl_sf_gegenpoly_2 (C function):       Gegenbauer Functions.
                                                              (line  11)
* gsl_sf_gegenpoly_2_e (C function):     Gegenbauer Functions.
                                                              (line  11)
* gsl_sf_gegenpoly_3 (C function):       Gegenbauer Functions.
                                                              (line  11)
* gsl_sf_gegenpoly_3_e (C function):     Gegenbauer Functions.
                                                              (line  11)
* gsl_sf_gegenpoly_array (C function):   Gegenbauer Functions.
                                                              (line  32)
* gsl_sf_gegenpoly_n (C function):       Gegenbauer Functions.
                                                              (line  24)
* gsl_sf_gegenpoly_n_e (C function):     Gegenbauer Functions.
                                                              (line  24)
* gsl_sf_hazard (C function):            Probability functions.
                                                              (line  29)
* gsl_sf_hazard_e (C function):          Probability functions.
                                                              (line  29)
* gsl_sf_hermite (C function):           Hermite Polynomials. (line  23)
* gsl_sf_hermite_array (C function):     Hermite Polynomials. (line  31)
* gsl_sf_hermite_array_deriv (C function): Derivatives of Hermite Polynomials.
                                                              (line  14)
* gsl_sf_hermite_deriv (C function):     Derivatives of Hermite Polynomials.
                                                              (line   6)
* gsl_sf_hermite_deriv_array (C function): Derivatives of Hermite Polynomials.
                                                              (line  23)
* gsl_sf_hermite_deriv_e (C function):   Derivatives of Hermite Polynomials.
                                                              (line   6)
* gsl_sf_hermite_e (C function):         Hermite Polynomials. (line  23)
* gsl_sf_hermite_func (C function):      Hermite Functions.   (line  28)
* gsl_sf_hermite_func_array (C function): Hermite Functions.  (line  46)
* gsl_sf_hermite_func_der (C function):  Derivatives of Hermite Functions.
                                                              (line   6)
* gsl_sf_hermite_func_der_e (C function): Derivatives of Hermite Functions.
                                                              (line   6)
* gsl_sf_hermite_func_e (C function):    Hermite Functions.   (line  28)
* gsl_sf_hermite_func_fast (C function): Hermite Functions.   (line  36)
* gsl_sf_hermite_func_fast_e (C function): Hermite Functions. (line  36)
* gsl_sf_hermite_func_series (C function): Hermite Functions. (line  55)
* gsl_sf_hermite_func_series_e (C function): Hermite Functions.
                                                              (line  55)
* gsl_sf_hermite_func_zero (C function): Zeros of Hermite Polynomials and Hermite Functions.
                                                              (line  26)
* gsl_sf_hermite_func_zero_e (C function): Zeros of Hermite Polynomials and Hermite Functions.
                                                              (line  26)
* gsl_sf_hermite_prob (C function):      Hermite Polynomials. (line  47)
* gsl_sf_hermite_prob_array (C function): Hermite Polynomials.
                                                              (line  55)
* gsl_sf_hermite_prob_array_deriv (C function): Derivatives of Hermite Polynomials.
                                                              (line  40)
* gsl_sf_hermite_prob_deriv (C function): Derivatives of Hermite Polynomials.
                                                              (line  32)
* gsl_sf_hermite_prob_deriv_array (C function): Derivatives of Hermite Polynomials.
                                                              (line  50)
* gsl_sf_hermite_prob_deriv_e (C function): Derivatives of Hermite Polynomials.
                                                              (line  32)
* gsl_sf_hermite_prob_e (C function):    Hermite Polynomials. (line  47)
* gsl_sf_hermite_prob_series (C function): Hermite Polynomials.
                                                              (line  63)
* gsl_sf_hermite_prob_series_e (C function): Hermite Polynomials.
                                                              (line  63)
* gsl_sf_hermite_prob_zero (C function): Zeros of Hermite Polynomials and Hermite Functions.
                                                              (line  19)
* gsl_sf_hermite_prob_zero_e (C function): Zeros of Hermite Polynomials and Hermite Functions.
                                                              (line  19)
* gsl_sf_hermite_series (C function):    Hermite Polynomials. (line  38)
* gsl_sf_hermite_series_e (C function):  Hermite Polynomials. (line  38)
* gsl_sf_hermite_zero (C function):      Zeros of Hermite Polynomials and Hermite Functions.
                                                              (line  12)
* gsl_sf_hermite_zero_e (C function):    Zeros of Hermite Polynomials and Hermite Functions.
                                                              (line  12)
* gsl_sf_hydrogenicR (C function):       Normalized Hydrogenic Bound States.
                                                              (line  13)
* gsl_sf_hydrogenicR_1 (C function):     Normalized Hydrogenic Bound States.
                                                              (line   6)
* gsl_sf_hydrogenicR_1_e (C function):   Normalized Hydrogenic Bound States.
                                                              (line   6)
* gsl_sf_hydrogenicR_e (C function):     Normalized Hydrogenic Bound States.
                                                              (line  13)
* gsl_sf_hyperg_0F1 (C function):        Hypergeometric Functions.
                                                              (line  10)
* gsl_sf_hyperg_0F1_e (C function):      Hypergeometric Functions.
                                                              (line  10)
* gsl_sf_hyperg_1F1 (C function):        Hypergeometric Functions.
                                                              (line  28)
* gsl_sf_hyperg_1F1_e (C function):      Hypergeometric Functions.
                                                              (line  28)
* gsl_sf_hyperg_1F1_int (C function):    Hypergeometric Functions.
                                                              (line  18)
* gsl_sf_hyperg_1F1_int_e (C function):  Hypergeometric Functions.
                                                              (line  18)
* gsl_sf_hyperg_2F0 (C function):        Hypergeometric Functions.
                                                              (line 121)
* gsl_sf_hyperg_2F0_e (C function):      Hypergeometric Functions.
                                                              (line 121)
* gsl_sf_hyperg_2F1 (C function):        Hypergeometric Functions.
                                                              (line  66)
* gsl_sf_hyperg_2F1_conj (C function):   Hypergeometric Functions.
                                                              (line  82)
* gsl_sf_hyperg_2F1_conj_e (C function): Hypergeometric Functions.
                                                              (line  82)
* gsl_sf_hyperg_2F1_conj_renorm (C function): Hypergeometric Functions.
                                                              (line 109)
* gsl_sf_hyperg_2F1_conj_renorm_e (C function): Hypergeometric Functions.
                                                              (line 109)
* gsl_sf_hyperg_2F1_e (C function):      Hypergeometric Functions.
                                                              (line  66)
* gsl_sf_hyperg_2F1_renorm (C function): Hypergeometric Functions.
                                                              (line  95)
* gsl_sf_hyperg_2F1_renorm_e (C function): Hypergeometric Functions.
                                                              (line  95)
* gsl_sf_hyperg_U (C function):          Hypergeometric Functions.
                                                              (line  52)
* gsl_sf_hyperg_U_e (C function):        Hypergeometric Functions.
                                                              (line  52)
* gsl_sf_hyperg_U_e10_e (C function):    Hypergeometric Functions.
                                                              (line  59)
* gsl_sf_hyperg_U_int (C function):      Hypergeometric Functions.
                                                              (line  38)
* gsl_sf_hyperg_U_int_e (C function):    Hypergeometric Functions.
                                                              (line  38)
* gsl_sf_hyperg_U_int_e10_e (C function): Hypergeometric Functions.
                                                              (line  45)
* gsl_sf_hypot (C function):             Circular Trigonometric Functions.
                                                              (line  16)
* gsl_sf_hypot_e (C function):           Circular Trigonometric Functions.
                                                              (line  16)
* gsl_sf_hzeta (C function):             Hurwitz Zeta Function.
                                                              (line  10)
* gsl_sf_hzeta_e (C function):           Hurwitz Zeta Function.
                                                              (line  10)
* gsl_sf_laguerre_1 (C function):        Laguerre Functions.  (line  20)
* gsl_sf_laguerre_1_e (C function):      Laguerre Functions.  (line  20)
* gsl_sf_laguerre_2 (C function):        Laguerre Functions.  (line  20)
* gsl_sf_laguerre_2_e (C function):      Laguerre Functions.  (line  20)
* gsl_sf_laguerre_3 (C function):        Laguerre Functions.  (line  20)
* gsl_sf_laguerre_3_e (C function):      Laguerre Functions.  (line  20)
* gsl_sf_laguerre_n (C function):        Laguerre Functions.  (line  33)
* gsl_sf_laguerre_n_e (C function):      Laguerre Functions.  (line  33)
* gsl_sf_lambert_W0 (C function):        Lambert W Functions. (line  13)
* gsl_sf_lambert_W0_e (C function):      Lambert W Functions. (line  13)
* gsl_sf_lambert_Wm1 (C function):       Lambert W Functions. (line  19)
* gsl_sf_lambert_Wm1_e (C function):     Lambert W Functions. (line  19)
* gsl_sf_legendre_array (C function):    Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line  84)
* gsl_sf_legendre_array_e (C function):  Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line  84)
* gsl_sf_legendre_array_index (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 195)
* gsl_sf_legendre_array_n (C function):  Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 187)
* gsl_sf_legendre_array_size (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 242)
* gsl_sf_legendre_deriv2_alt_array (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 163)
* gsl_sf_legendre_deriv2_alt_array_e (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 163)
* gsl_sf_legendre_deriv2_array (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 141)
* gsl_sf_legendre_deriv2_array_e (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 141)
* gsl_sf_legendre_deriv_alt_array (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 121)
* gsl_sf_legendre_deriv_alt_array_e (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 121)
* gsl_sf_legendre_deriv_array (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 102)
* gsl_sf_legendre_deriv_array_e (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 102)
* gsl_sf_legendre_H3d (C function):      Radial Functions for Hyperbolic Space.
                                                              (line  35)
* gsl_sf_legendre_H3d_0 (C function):    Radial Functions for Hyperbolic Space.
                                                              (line  11)
* gsl_sf_legendre_H3d_0_e (C function):  Radial Functions for Hyperbolic Space.
                                                              (line  11)
* gsl_sf_legendre_H3d_1 (C function):    Radial Functions for Hyperbolic Space.
                                                              (line  23)
* gsl_sf_legendre_H3d_1_e (C function):  Radial Functions for Hyperbolic Space.
                                                              (line  23)
* gsl_sf_legendre_H3d_array (C function): Radial Functions for Hyperbolic Space.
                                                              (line  45)
* gsl_sf_legendre_H3d_e (C function):    Radial Functions for Hyperbolic Space.
                                                              (line  35)
* gsl_sf_legendre_P1 (C function):       Legendre Polynomials.
                                                              (line   6)
* gsl_sf_legendre_P1_e (C function):     Legendre Polynomials.
                                                              (line   6)
* gsl_sf_legendre_P2 (C function):       Legendre Polynomials.
                                                              (line   6)
* gsl_sf_legendre_P2_e (C function):     Legendre Polynomials.
                                                              (line   6)
* gsl_sf_legendre_P3 (C function):       Legendre Polynomials.
                                                              (line   6)
* gsl_sf_legendre_P3_e (C function):     Legendre Polynomials.
                                                              (line   6)
* gsl_sf_legendre_Pl (C function):       Legendre Polynomials.
                                                              (line  19)
* gsl_sf_legendre_Plm (C function):      Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 204)
* gsl_sf_legendre_Plm_array (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 221)
* gsl_sf_legendre_Plm_deriv_array (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 221)
* gsl_sf_legendre_Plm_e (C function):    Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 204)
* gsl_sf_legendre_Pl_array (C function): Legendre Polynomials.
                                                              (line  27)
* gsl_sf_legendre_Pl_deriv_array (C function): Legendre Polynomials.
                                                              (line  27)
* gsl_sf_legendre_Pl_e (C function):     Legendre Polynomials.
                                                              (line  19)
* gsl_sf_legendre_Q0 (C function):       Legendre Polynomials.
                                                              (line  36)
* gsl_sf_legendre_Q0_e (C function):     Legendre Polynomials.
                                                              (line  36)
* gsl_sf_legendre_Q1 (C function):       Legendre Polynomials.
                                                              (line  43)
* gsl_sf_legendre_Q1_e (C function):     Legendre Polynomials.
                                                              (line  43)
* gsl_sf_legendre_Ql (C function):       Legendre Polynomials.
                                                              (line  50)
* gsl_sf_legendre_Ql_e (C function):     Legendre Polynomials.
                                                              (line  50)
* gsl_sf_legendre_sphPlm (C function):   Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 211)
* gsl_sf_legendre_sphPlm_array (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 232)
* gsl_sf_legendre_sphPlm_deriv_array (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 232)
* gsl_sf_legendre_sphPlm_e (C function): Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line 211)
* gsl_sf_legendre_t (C type):            Associated Legendre Polynomials and Spherical Harmonics.
                                                              (line  66)
* gsl_sf_lnbeta (C function):            Beta Functions.      (line  14)
* gsl_sf_lnbeta_e (C function):          Beta Functions.      (line  14)
* gsl_sf_lnchoose (C function):          Factorials.          (line  51)
* gsl_sf_lnchoose_e (C function):        Factorials.          (line  51)
* gsl_sf_lncosh (C function):            Hyperbolic Trigonometric Functions.
                                                              (line  11)
* gsl_sf_lncosh_e (C function):          Hyperbolic Trigonometric Functions.
                                                              (line  11)
* gsl_sf_lndoublefact (C function):      Factorials.          (line  36)
* gsl_sf_lndoublefact_e (C function):    Factorials.          (line  36)
* gsl_sf_lnfact (C function):            Factorials.          (line  27)
* gsl_sf_lnfact_e (C function):          Factorials.          (line  27)
* gsl_sf_lngamma (C function):           Gamma Functions.     (line  23)
* gsl_sf_lngamma_complex_e (C function): Gamma Functions.     (line  60)
* gsl_sf_lngamma_e (C function):         Gamma Functions.     (line  23)
* gsl_sf_lngamma_sgn_e (C function):     Gamma Functions.     (line  32)
* gsl_sf_lnpoch (C function):            Pochhammer Symbol.   (line  16)
* gsl_sf_lnpoch_e (C function):          Pochhammer Symbol.   (line  16)
* gsl_sf_lnpoch_sgn_e (C function):      Pochhammer Symbol.   (line  23)
* gsl_sf_lnsinh (C function):            Hyperbolic Trigonometric Functions.
                                                              (line   6)
* gsl_sf_lnsinh_e (C function):          Hyperbolic Trigonometric Functions.
                                                              (line   6)
* gsl_sf_log (C function):               Logarithm and Related Functions.
                                                              (line  10)
* gsl_sf_log_1plusx (C function):        Logarithm and Related Functions.
                                                              (line  28)
* gsl_sf_log_1plusx_e (C function):      Logarithm and Related Functions.
                                                              (line  28)
* gsl_sf_log_1plusx_mx (C function):     Logarithm and Related Functions.
                                                              (line  34)
* gsl_sf_log_1plusx_mx_e (C function):   Logarithm and Related Functions.
                                                              (line  34)
* gsl_sf_log_abs (C function):           Logarithm and Related Functions.
                                                              (line  15)
* gsl_sf_log_abs_e (C function):         Logarithm and Related Functions.
                                                              (line  15)
* gsl_sf_log_e (C function):             Logarithm and Related Functions.
                                                              (line  10)
* gsl_sf_log_erfc (C function):          Log Complementary Error Function.
                                                              (line   6)
* gsl_sf_log_erfc_e (C function):        Log Complementary Error Function.
                                                              (line   6)
* gsl_sf_mathieu_a (C function):         Mathieu Function Characteristic Values.
                                                              (line   6)
* gsl_sf_mathieu_alloc (C function):     Mathieu Function Workspace.
                                                              (line  14)
* gsl_sf_mathieu_a_array (C function):   Mathieu Function Characteristic Values.
                                                              (line  16)
* gsl_sf_mathieu_a_e (C function):       Mathieu Function Characteristic Values.
                                                              (line   6)
* gsl_sf_mathieu_b (C function):         Mathieu Function Characteristic Values.
                                                              (line   6)
* gsl_sf_mathieu_b_array (C function):   Mathieu Function Characteristic Values.
                                                              (line  16)
* gsl_sf_mathieu_b_e (C function):       Mathieu Function Characteristic Values.
                                                              (line   6)
* gsl_sf_mathieu_ce (C function):        Angular Mathieu Functions.
                                                              (line   6)
* gsl_sf_mathieu_ce_array (C function):  Angular Mathieu Functions.
                                                              (line  16)
* gsl_sf_mathieu_ce_e (C function):      Angular Mathieu Functions.
                                                              (line   6)
* gsl_sf_mathieu_free (C function):      Mathieu Function Workspace.
                                                              (line  22)
* gsl_sf_mathieu_Mc (C function):        Radial Mathieu Functions.
                                                              (line   6)
* gsl_sf_mathieu_Mc_array (C function):  Radial Mathieu Functions.
                                                              (line  21)
* gsl_sf_mathieu_Mc_e (C function):      Radial Mathieu Functions.
                                                              (line   6)
* gsl_sf_mathieu_Ms (C function):        Radial Mathieu Functions.
                                                              (line   6)
* gsl_sf_mathieu_Ms_array (C function):  Radial Mathieu Functions.
                                                              (line  21)
* gsl_sf_mathieu_Ms_e (C function):      Radial Mathieu Functions.
                                                              (line   6)
* gsl_sf_mathieu_se (C function):        Angular Mathieu Functions.
                                                              (line   6)
* gsl_sf_mathieu_se_array (C function):  Angular Mathieu Functions.
                                                              (line  16)
* gsl_sf_mathieu_se_e (C function):      Angular Mathieu Functions.
                                                              (line   6)
* gsl_sf_mathieu_workspace (C type):     Mathieu Function Workspace.
                                                              (line  10)
* gsl_sf_multiply (C function):          Elementary Operations.
                                                              (line  10)
* gsl_sf_multiply_e (C function):        Elementary Operations.
                                                              (line  10)
* gsl_sf_multiply_err_e (C function):    Elementary Operations.
                                                              (line  18)
* gsl_sf_poch (C function):              Pochhammer Symbol.   (line   6)
* gsl_sf_pochrel (C function):           Pochhammer Symbol.   (line  31)
* gsl_sf_pochrel_e (C function):         Pochhammer Symbol.   (line  31)
* gsl_sf_poch_e (C function):            Pochhammer Symbol.   (line   6)
* gsl_sf_polar_to_rect (C function):     Conversion Functions.
                                                              (line   6)
* gsl_sf_pow_int (C function):           Power Function.      (line  10)
* gsl_sf_pow_int_e (C function):         Power Function.      (line  10)
* gsl_sf_psi (C function):               Digamma Function.    (line  12)
* gsl_sf_psi_1 (C function):             Trigamma Function.   (line  12)
* gsl_sf_psi_1piy (C function):          Digamma Function.    (line  18)
* gsl_sf_psi_1piy_e (C function):        Digamma Function.    (line  18)
* gsl_sf_psi_1_e (C function):           Trigamma Function.   (line  12)
* gsl_sf_psi_1_int (C function):         Trigamma Function.   (line   6)
* gsl_sf_psi_1_int_e (C function):       Trigamma Function.   (line   6)
* gsl_sf_psi_e (C function):             Digamma Function.    (line  12)
* gsl_sf_psi_int (C function):           Digamma Function.    (line   6)
* gsl_sf_psi_int_e (C function):         Digamma Function.    (line   6)
* gsl_sf_psi_n (C function):             Polygamma Function.  (line   6)
* gsl_sf_psi_n_e (C function):           Polygamma Function.  (line   6)
* gsl_sf_rect_to_polar (C function):     Conversion Functions.
                                                              (line  13)
* gsl_sf_result (C type):                The gsl_sf_result struct.
                                                              (line  13)
* gsl_sf_result_e10 (C type):            The gsl_sf_result struct.
                                                              (line  31)
* gsl_sf_Shi (C function):               Hyperbolic Integrals.
                                                              (line   6)
* gsl_sf_Shi_e (C function):             Hyperbolic Integrals.
                                                              (line   6)
* gsl_sf_Si (C function):                Trigonometric Integrals.
                                                              (line   6)
* gsl_sf_sin (C function):               Circular Trigonometric Functions.
                                                              (line   6)
* gsl_sf_sinc (C function):              Circular Trigonometric Functions.
                                                              (line  23)
* gsl_sf_sinc_e (C function):            Circular Trigonometric Functions.
                                                              (line  23)
* gsl_sf_sin_e (C function):             Circular Trigonometric Functions.
                                                              (line   6)
* gsl_sf_sin_err_e (C function):         Trigonometric Functions With Error Estimates.
                                                              (line   6)
* gsl_sf_Si_e (C function):              Trigonometric Integrals.
                                                              (line   6)
* gsl_sf_synchrotron_1 (C function):     Synchrotron Functions.
                                                              (line   9)
* gsl_sf_synchrotron_1_e (C function):   Synchrotron Functions.
                                                              (line   9)
* gsl_sf_synchrotron_2 (C function):     Synchrotron Functions.
                                                              (line  16)
* gsl_sf_synchrotron_2_e (C function):   Synchrotron Functions.
                                                              (line  16)
* gsl_sf_taylorcoeff (C function):       Factorials.          (line  58)
* gsl_sf_taylorcoeff_e (C function):     Factorials.          (line  58)
* gsl_sf_transport_2 (C function):       Transport Functions. (line  13)
* gsl_sf_transport_2_e (C function):     Transport Functions. (line  13)
* gsl_sf_transport_3 (C function):       Transport Functions. (line  19)
* gsl_sf_transport_3_e (C function):     Transport Functions. (line  19)
* gsl_sf_transport_4 (C function):       Transport Functions. (line  25)
* gsl_sf_transport_4_e (C function):     Transport Functions. (line  25)
* gsl_sf_transport_5 (C function):       Transport Functions. (line  31)
* gsl_sf_transport_5_e (C function):     Transport Functions. (line  31)
* gsl_sf_zeta (C function):              Riemann Zeta Function.
                                                              (line  16)
* gsl_sf_zetam1 (C function):            Riemann Zeta Function Minus One.
                                                              (line  15)
* gsl_sf_zetam1_e (C function):          Riemann Zeta Function Minus One.
                                                              (line  15)
* gsl_sf_zetam1_int (C function):        Riemann Zeta Function Minus One.
                                                              (line  10)
* gsl_sf_zetam1_int_e (C function):      Riemann Zeta Function Minus One.
                                                              (line  10)
* gsl_sf_zeta_e (C function):            Riemann Zeta Function.
                                                              (line  16)
* gsl_sf_zeta_int (C function):          Riemann Zeta Function.
                                                              (line  10)
* gsl_sf_zeta_int_e (C function):        Riemann Zeta Function.
                                                              (line  10)
* GSL_SIGN (C macro):                    Testing the Sign of Numbers.
                                                              (line   6)
* gsl_siman_copy_construct_t (C type):   Simulated Annealing functions.
                                                              (line  93)
* gsl_siman_copy_t (C type):             Simulated Annealing functions.
                                                              (line  86)
* gsl_siman_destroy_t (C type):          Simulated Annealing functions.
                                                              (line 100)
* gsl_siman_Efunc_t (C type):            Simulated Annealing functions.
                                                              (line  56)
* gsl_siman_metric_t (C type):           Simulated Annealing functions.
                                                              (line  72)
* gsl_siman_params_t (C type):           Simulated Annealing functions.
                                                              (line 107)
* gsl_siman_print_t (C type):            Simulated Annealing functions.
                                                              (line  79)
* gsl_siman_solve (C function):          Simulated Annealing functions.
                                                              (line   6)
* gsl_siman_step_t (C type):             Simulated Annealing functions.
                                                              (line  63)
* gsl_sort (C function):                 Sorting vectors.     (line  22)
* gsl_sort2 (C function):                Sorting vectors.     (line  28)
* gsl_sort_index (C function):           Sorting vectors.     (line  47)
* gsl_sort_largest (C function):         Selecting the k smallest or largest elements.
                                                              (line  24)
* gsl_sort_largest_index (C function):   Selecting the k smallest or largest elements.
                                                              (line  53)
* gsl_sort_smallest (C function):        Selecting the k smallest or largest elements.
                                                              (line  16)
* gsl_sort_smallest_index (C function):  Selecting the k smallest or largest elements.
                                                              (line  44)
* gsl_sort_vector (C function):          Sorting vectors.     (line  36)
* gsl_sort_vector2 (C function):         Sorting vectors.     (line  41)
* gsl_sort_vector_index (C function):    Sorting vectors.     (line  59)
* gsl_sort_vector_largest (C function):  Selecting the k smallest or largest elements.
                                                              (line  32)
* gsl_sort_vector_largest_index (C function): Selecting the k smallest or largest elements.
                                                              (line  62)
* gsl_sort_vector_smallest (C function): Selecting the k smallest or largest elements.
                                                              (line  32)
* gsl_sort_vector_smallest_index (C function): Selecting the k smallest or largest elements.
                                                              (line  62)
* gsl_spblas_dgemm (C function):         Sparse BLAS operations.
                                                              (line  17)
* gsl_spblas_dgemv (C function):         Sparse BLAS operations.
                                                              (line   6)
* gsl_splinalg_itersolve_alloc (C function): Iterating the Sparse Linear System.
                                                              (line   9)
* gsl_splinalg_itersolve_free (C function): Iterating the Sparse Linear System.
                                                              (line  20)
* gsl_splinalg_itersolve_gmres (C variable): Types of Sparse Iterative Solvers.
                                                              (line  11)
* gsl_splinalg_itersolve_iterate (C function): Iterating the Sparse Linear System.
                                                              (line  31)
* gsl_splinalg_itersolve_name (C function): Iterating the Sparse Linear System.
                                                              (line  26)
* gsl_splinalg_itersolve_normr (C function): Iterating the Sparse Linear System.
                                                              (line  52)
* gsl_splinalg_itersolve_type (C type):  Types of Sparse Iterative Solvers.
                                                              (line   9)
* gsl_spline (C type):                   1D Higher-level Interface.
                                                              (line  14)
* gsl_spline2d (C type):                 2D Higher-level Interface.
                                                              (line  14)
* gsl_spline2d_alloc (C function):       2D Higher-level Interface.
                                                              (line  19)
* gsl_spline2d_eval (C function):        2D Higher-level Interface.
                                                              (line  35)
* gsl_spline2d_eval_deriv_x (C function): 2D Higher-level Interface.
                                                              (line  42)
* gsl_spline2d_eval_deriv_xx (C function): 2D Higher-level Interface.
                                                              (line  56)
* gsl_spline2d_eval_deriv_xx_e (C function): 2D Higher-level Interface.
                                                              (line  56)
* gsl_spline2d_eval_deriv_xy (C function): 2D Higher-level Interface.
                                                              (line  70)
* gsl_spline2d_eval_deriv_xy_e (C function): 2D Higher-level Interface.
                                                              (line  70)
* gsl_spline2d_eval_deriv_x_e (C function): 2D Higher-level Interface.
                                                              (line  42)
* gsl_spline2d_eval_deriv_y (C function): 2D Higher-level Interface.
                                                              (line  49)
* gsl_spline2d_eval_deriv_yy (C function): 2D Higher-level Interface.
                                                              (line  63)
* gsl_spline2d_eval_deriv_yy_e (C function): 2D Higher-level Interface.
                                                              (line  63)
* gsl_spline2d_eval_deriv_y_e (C function): 2D Higher-level Interface.
                                                              (line  49)
* gsl_spline2d_eval_e (C function):      2D Higher-level Interface.
                                                              (line  35)
* gsl_spline2d_free (C function):        2D Higher-level Interface.
                                                              (line  27)
* gsl_spline2d_get (C function):         2D Higher-level Interface.
                                                              (line  80)
* gsl_spline2d_init (C function):        2D Higher-level Interface.
                                                              (line  22)
* gsl_spline2d_min_size (C function):    2D Higher-level Interface.
                                                              (line  32)
* gsl_spline2d_name (C function):        2D Higher-level Interface.
                                                              (line  29)
* gsl_spline2d_set (C function):         2D Higher-level Interface.
                                                              (line  77)
* gsl_spline_alloc (C function):         1D Higher-level Interface.
                                                              (line  19)
* gsl_spline_eval (C function):          1D Higher-level Interface.
                                                              (line  33)
* gsl_spline_eval_deriv (C function):    1D Higher-level Interface.
                                                              (line  38)
* gsl_spline_eval_deriv2 (C function):   1D Higher-level Interface.
                                                              (line  45)
* gsl_spline_eval_deriv2_e (C function): 1D Higher-level Interface.
                                                              (line  45)
* gsl_spline_eval_deriv_e (C function):  1D Higher-level Interface.
                                                              (line  38)
* gsl_spline_eval_e (C function):        1D Higher-level Interface.
                                                              (line  33)
* gsl_spline_eval_integ (C function):    1D Higher-level Interface.
                                                              (line  52)
* gsl_spline_eval_integ_e (C function):  1D Higher-level Interface.
                                                              (line  52)
* gsl_spline_free (C function):          1D Higher-level Interface.
                                                              (line  26)
* gsl_spline_init (C function):          1D Higher-level Interface.
                                                              (line  23)
* gsl_spline_min_size (C function):      1D Higher-level Interface.
                                                              (line  30)
* gsl_spline_name (C function):          1D Higher-level Interface.
                                                              (line  28)
* gsl_spmatrix (C type):                 Overview<8>.         (line  10)
* gsl_spmatrix_add (C function):         Matrix Operations.   (line  43)
* gsl_spmatrix_add_to_dense (C function): Matrix Operations.  (line  51)
* gsl_spmatrix_alloc (C function):       Allocation.          (line  12)
* gsl_spmatrix_alloc_nzmax (C function): Allocation.          (line  28)
* gsl_spmatrix_compress (C function):    Compressed Format.   (line  27)
* GSL_SPMATRIX_COO (C macro):            Allocation.          (line  44)
* gsl_spmatrix_csc (C function):         Compressed Format.   (line   9)
* GSL_SPMATRIX_CSC (C macro):            Allocation.          (line  48)
* gsl_spmatrix_csr (C function):         Compressed Format.   (line  18)
* GSL_SPMATRIX_CSR (C macro):            Allocation.          (line  52)
* gsl_spmatrix_d2sp (C function):        Conversion Between Sparse and Dense Matrices.
                                                              (line  10)
* gsl_spmatrix_equal (C function):       Matrix Properties.   (line  27)
* gsl_spmatrix_fprintf (C function):     Reading and Writing Matrices.
                                                              (line  32)
* gsl_spmatrix_fread (C function):       Reading and Writing Matrices.
                                                              (line  18)
* gsl_spmatrix_free (C function):        Allocation.          (line  71)
* gsl_spmatrix_fscanf (C function):      Reading and Writing Matrices.
                                                              (line  46)
* gsl_spmatrix_fwrite (C function):      Reading and Writing Matrices.
                                                              (line   6)
* gsl_spmatrix_get (C function):         Accessing Matrix Elements.
                                                              (line   6)
* gsl_spmatrix_memcpy (C function):      Copying Matrices.    (line   6)
* gsl_spmatrix_minmax (C function):      Finding Maximum and Minimum Elements.
                                                              (line   6)
* gsl_spmatrix_min_index (C function):   Finding Maximum and Minimum Elements.
                                                              (line  16)
* gsl_spmatrix_nnz (C function):         Matrix Properties.   (line  20)
* gsl_spmatrix_ptr (C function):         Accessing Matrix Elements.
                                                              (line  23)
* gsl_spmatrix_realloc (C function):     Allocation.          (line  59)
* gsl_spmatrix_scale (C function):       Matrix Operations.   (line   6)
* gsl_spmatrix_scale_columns (C function): Matrix Operations. (line  15)
* gsl_spmatrix_scale_rows (C function):  Matrix Operations.   (line  29)
* gsl_spmatrix_set (C function):         Accessing Matrix Elements.
                                                              (line  15)
* gsl_spmatrix_set_zero (C function):    Initializing Matrix Elements.
                                                              (line  11)
* gsl_spmatrix_sp2d (C function):        Conversion Between Sparse and Dense Matrices.
                                                              (line  19)
* gsl_spmatrix_transpose (C function):   Exchanging Rows and Columns.
                                                              (line  17)
* gsl_spmatrix_transpose_memcpy (C function): Exchanging Rows and Columns.
                                                              (line   6)
* gsl_spmatrix_type (C function):        Matrix Properties.   (line   6)
* gsl_stats_absdev (C function):         Absolute deviation.  (line   6)
* gsl_stats_absdev_m (C function):       Absolute deviation.  (line  20)
* gsl_stats_correlation (C function):    Correlation.         (line   6)
* gsl_stats_covariance (C function):     Covariance.          (line   6)
* gsl_stats_covariance_m (C function):   Covariance.          (line  16)
* gsl_stats_gastwirth_from_sorted_data (C function): Gastwirth Estimator.
                                                              (line  15)
* gsl_stats_kurtosis (C function):       Higher moments skewness and kurtosis.
                                                              (line  33)
* gsl_stats_kurtosis_m_sd (C function):  Higher moments skewness and kurtosis.
                                                              (line  45)
* gsl_stats_lag1_autocorrelation (C function): Autocorrelation.
                                                              (line   6)
* gsl_stats_lag1_autocorrelation_m (C function): Autocorrelation.
                                                              (line  16)
* gsl_stats_mad (C function):            Median Absolute Deviation MAD.
                                                              (line  20)
* gsl_stats_mad0 (C function):           Median Absolute Deviation MAD.
                                                              (line  17)
* gsl_stats_max (C function):            Maximum and Minimum values.
                                                              (line  12)
* gsl_stats_max_index (C function):      Maximum and Minimum values.
                                                              (line  43)
* gsl_stats_mean (C function):           Mean Standard Deviation and Variance.
                                                              (line   6)
* gsl_stats_median (C function):         Median and Percentiles.
                                                              (line  29)
* gsl_stats_median_from_sorted_data (C function): Median and Percentiles.
                                                              (line  13)
* gsl_stats_min (C function):            Maximum and Minimum values.
                                                              (line  24)
* gsl_stats_minmax (C function):         Maximum and Minimum values.
                                                              (line  36)
* gsl_stats_minmax_index (C function):   Maximum and Minimum values.
                                                              (line  61)
* gsl_stats_min_index (C function):      Maximum and Minimum values.
                                                              (line  52)
* gsl_stats_Qn0_from_sorted_data (C function): Q_n Statistic. (line  19)
* gsl_stats_Qn_from_sorted_data (C function): Q_n Statistic.  (line  19)
* gsl_stats_quantile_from_sorted_data (C function): Median and Percentiles.
                                                              (line  38)
* gsl_stats_sd (C function):             Mean Standard Deviation and Variance.
                                                              (line  47)
* gsl_stats_sd_m (C function):           Mean Standard Deviation and Variance.
                                                              (line  47)
* gsl_stats_sd_with_fixed_mean (C function): Mean Standard Deviation and Variance.
                                                              (line  82)
* gsl_stats_select (C function):         Order Statistics.    (line  14)
* gsl_stats_skew (C function):           Higher moments skewness and kurtosis.
                                                              (line   6)
* gsl_stats_skew_m_sd (C function):      Higher moments skewness and kurtosis.
                                                              (line  21)
* gsl_stats_Sn0_from_sorted_data (C function): S_n Statistic. (line  17)
* gsl_stats_Sn_from_sorted_data (C function): S_n Statistic.  (line  22)
* gsl_stats_spearman (C function):       Correlation.         (line  21)
* gsl_stats_trmean_from_sorted_data (C function): Trimmed Mean.
                                                              (line  14)
* gsl_stats_tss (C function):            Mean Standard Deviation and Variance.
                                                              (line  57)
* gsl_stats_tss_m (C function):          Mean Standard Deviation and Variance.
                                                              (line  57)
* gsl_stats_variance (C function):       Mean Standard Deviation and Variance.
                                                              (line  19)
* gsl_stats_variance_m (C function):     Mean Standard Deviation and Variance.
                                                              (line  38)
* gsl_stats_variance_with_fixed_mean (C function): Mean Standard Deviation and Variance.
                                                              (line  71)
* gsl_stats_wabsdev (C function):        Weighted Samples.    (line  96)
* gsl_stats_wabsdev_m (C function):      Weighted Samples.    (line 105)
* gsl_stats_wkurtosis (C function):      Weighted Samples.    (line 129)
* gsl_stats_wkurtosis_m_sd (C function): Weighted Samples.    (line 136)
* gsl_stats_wmean (C function):          Weighted Samples.    (line  14)
* gsl_stats_wsd (C function):            Weighted Samples.    (line  46)
* gsl_stats_wsd_m (C function):          Weighted Samples.    (line  54)
* gsl_stats_wsd_with_fixed_mean (C function): Weighted Samples.
                                                              (line  72)
* gsl_stats_wskew (C function):          Weighted Samples.    (line 113)
* gsl_stats_wskew_m_sd (C function):     Weighted Samples.    (line 120)
* gsl_stats_wtss (C function):           Weighted Samples.    (line  81)
* gsl_stats_wtss_m (C function):         Weighted Samples.    (line  81)
* gsl_stats_wvariance (C function):      Weighted Samples.    (line  23)
* gsl_stats_wvariance_m (C function):    Weighted Samples.    (line  38)
* gsl_stats_wvariance_with_fixed_mean (C function): Weighted Samples.
                                                              (line  60)
* gsl_strerror (C function):             Error Codes.         (line  41)
* gsl_sum_levin_utrunc_accel (C function): Acceleration functions without error estimation.
                                                              (line  38)
* gsl_sum_levin_utrunc_alloc (C function): Acceleration functions without error estimation.
                                                              (line  26)
* gsl_sum_levin_utrunc_free (C function): Acceleration functions without error estimation.
                                                              (line  33)
* gsl_sum_levin_utrunc_workspace (C type): Acceleration functions without error estimation.
                                                              (line  22)
* gsl_sum_levin_u_accel (C function):    Acceleration functions.
                                                              (line  37)
* gsl_sum_levin_u_alloc (C function):    Acceleration functions.
                                                              (line  27)
* gsl_sum_levin_u_free (C function):     Acceleration functions.
                                                              (line  33)
* gsl_sum_levin_u_workspace (C type):    Acceleration functions.
                                                              (line  23)
* gsl_vector (C type):                   Vectors.             (line  17)
* gsl_vector_add (C function):           Vector operations.   (line   6)
* gsl_vector_add_constant (C function):  Vector operations.   (line  39)
* gsl_vector_alloc (C function):         Vector allocation.   (line  14)
* gsl_vector_axpby (C function):         Vector operations.   (line  45)
* gsl_vector_calloc (C function):        Vector allocation.   (line  23)
* gsl_vector_complex_const_imag (C function): Vector views.   (line 114)
* gsl_vector_complex_const_real (C function): Vector views.   (line 102)
* gsl_vector_complex_imag (C function):  Vector views.        (line 114)
* gsl_vector_complex_real (C function):  Vector views.        (line 102)
* gsl_vector_const_ptr (C function):     Accessing vector elements.
                                                              (line  65)
* gsl_vector_const_subvector (C function): Vector views.      (line  31)
* gsl_vector_const_subvector_with_stride (C function): Vector views.
                                                              (line  63)
* gsl_vector_const_view (C type):        Vector views.        (line  12)
* gsl_vector_const_view_array (C function): Vector views.     (line 126)
* gsl_vector_const_view_array_with_stride (C function): Vector views.
                                                              (line 152)
* gsl_vector_div (C function):           Vector operations.   (line  27)
* gsl_vector_equal (C function):         Vector properties.   (line  19)
* gsl_vector_fprintf (C function):       Reading and writing vectors.
                                                              (line  28)
* gsl_vector_fread (C function):         Reading and writing vectors.
                                                              (line  18)
* gsl_vector_free (C function):          Vector allocation.   (line  28)
* gsl_vector_fscanf (C function):        Reading and writing vectors.
                                                              (line  37)
* gsl_vector_fwrite (C function):        Reading and writing vectors.
                                                              (line   9)
* gsl_vector_get (C function):           Accessing vector elements.
                                                              (line  48)
* gsl_vector_isneg (C function):         Vector properties.   (line  10)
* gsl_vector_isnonneg (C function):      Vector properties.   (line  10)
* gsl_vector_isnull (C function):        Vector properties.   (line  10)
* gsl_vector_ispos (C function):         Vector properties.   (line  10)
* gsl_vector_max (C function):           Finding maximum and minimum elements of vectors.
                                                              (line   8)
* gsl_vector_max_index (C function):     Finding maximum and minimum elements of vectors.
                                                              (line  22)
* gsl_vector_memcpy (C function):        Copying vectors.     (line  12)
* gsl_vector_min (C function):           Finding maximum and minimum elements of vectors.
                                                              (line  12)
* gsl_vector_minmax (C function):        Finding maximum and minimum elements of vectors.
                                                              (line  16)
* gsl_vector_minmax_index (C function):  Finding maximum and minimum elements of vectors.
                                                              (line  34)
* gsl_vector_min_index (C function):     Finding maximum and minimum elements of vectors.
                                                              (line  28)
* gsl_vector_mul (C function):           Vector operations.   (line  20)
* gsl_vector_ptr (C function):           Accessing vector elements.
                                                              (line  65)
* gsl_vector_reverse (C function):       Exchanging elements. (line  16)
* gsl_vector_scale (C function):         Vector operations.   (line  34)
* gsl_vector_set (C function):           Accessing vector elements.
                                                              (line  56)
* gsl_vector_set_all (C function):       Initializing vector elements.
                                                              (line   6)
* gsl_vector_set_basis (C function):     Initializing vector elements.
                                                              (line  15)
* gsl_vector_set_zero (C function):      Initializing vector elements.
                                                              (line  11)
* gsl_vector_sub (C function):           Vector operations.   (line  13)
* gsl_vector_subvector (C function):     Vector views.        (line  31)
* gsl_vector_subvector_with_stride (C function): Vector views.
                                                              (line  63)
* gsl_vector_swap (C function):          Copying vectors.     (line  18)
* gsl_vector_swap_elements (C function): Exchanging elements. (line   9)
* gsl_vector_view (C type):              Vector views.        (line  12)
* gsl_vector_view_array (C function):    Vector views.        (line 126)
* gsl_vector_view_array_with_stride (C function): Vector views.
                                                              (line 152)
* gsl_wavelet (C type):                  Initialization.      (line   6)
* gsl_wavelet2d_nstransform (C function): Wavelet transforms in two dimension.
                                                              (line  67)
* gsl_wavelet2d_nstransform_forward (C function): Wavelet transforms in two dimension.
                                                              (line  67)
* gsl_wavelet2d_nstransform_inverse (C function): Wavelet transforms in two dimension.
                                                              (line  67)
* gsl_wavelet2d_nstransform_matrix (C function): Wavelet transforms in two dimension.
                                                              (line  83)
* gsl_wavelet2d_nstransform_matrix_forward (C function): Wavelet transforms in two dimension.
                                                              (line  83)
* gsl_wavelet2d_nstransform_matrix_inverse (C function): Wavelet transforms in two dimension.
                                                              (line  83)
* gsl_wavelet2d_transform (C function):  Wavelet transforms in two dimension.
                                                              (line  29)
* gsl_wavelet2d_transform_forward (C function): Wavelet transforms in two dimension.
                                                              (line  29)
* gsl_wavelet2d_transform_inverse (C function): Wavelet transforms in two dimension.
                                                              (line  29)
* gsl_wavelet2d_transform_matrix (C function): Wavelet transforms in two dimension.
                                                              (line  56)
* gsl_wavelet2d_transform_matrix_forward (C function): Wavelet transforms in two dimension.
                                                              (line  56)
* gsl_wavelet2d_transform_matrix_inverse (C function): Wavelet transforms in two dimension.
                                                              (line  56)
* gsl_wavelet_alloc (C function):        Initialization.      (line  11)
* gsl_wavelet_bspline (C variable):      Initialization.      (line  36)
* gsl_wavelet_bspline_centered (C variable): Initialization.  (line  36)
* gsl_wavelet_daubechies (C variable):   Initialization.      (line  23)
* gsl_wavelet_daubechies_centered (C variable): Initialization.
                                                              (line  23)
* gsl_wavelet_free (C function):         Initialization.      (line  53)
* gsl_wavelet_haar (C variable):         Initialization.      (line  30)
* gsl_wavelet_haar_centered (C variable): Initialization.     (line  30)
* gsl_wavelet_name (C function):         Initialization.      (line  48)
* gsl_wavelet_transform (C function):    Wavelet transforms in one dimension.
                                                              (line   6)
* gsl_wavelet_transform_forward (C function): Wavelet transforms in one dimension.
                                                              (line   6)
* gsl_wavelet_transform_inverse (C function): Wavelet transforms in one dimension.
                                                              (line   6)
* gsl_wavelet_type (C type):             Initialization.      (line  21)
* gsl_wavelet_workspace (C type):        Initialization.      (line  57)
* gsl_wavelet_workspace_alloc (C function): Initialization.   (line  62)
* gsl_wavelet_workspace_free (C function): Initialization.    (line  74)
* Gumbel distribution (Type 1):          The Type-1 Gumbel Distribution.
                                                              (line   6)
* Gumbel distribution (Type 2):          The Type-2 Gumbel Distribution.
                                                              (line   6)
* Haar wavelets:                         Initialization.      (line  30)
* Hankel transforms, discrete:           References and Further Reading<27>.
                                                              (line  54)
* HAVE_INLINE:                           ANSI C Compliance.   (line  25)
* hazard function, normal distribution:  Probability functions.
                                                              (line  21)
* HBOOK:                                 References and Further Reading<19>.
                                                              (line   6)
* header files, including:               An Example Program.  (line  28)
* heapsort:                              Examples<6>.         (line 111)
* HEMM, Level-3 BLAS:                    Level 3.             (line  63)
* HEMV, Level-2 BLAS:                    Level 2.             (line  88)
* HER, Level-2 BLAS:                     Level 2.             (line 145)
* HER2, Level-2 BLAS:                    Level 2.             (line 171)
* HER2K, Level-3 BLAS:                   Level 3.             (line 217)
* HERK, Level-3 BLAS:                    Level 3.             (line 170)
* Hermite functions:                     Derivatives of Hermite Polynomials.
                                                              (line  58)
* Hermite functions, derivatives:        Derivatives of Hermite Functions.
                                                              (line   6)
* Hermite functions, zeros:              Zeros of Hermite Polynomials and Hermite Functions.
                                                              (line   6)
* Hermite polynomials:                   Hermite Polynomials and Functions.
                                                              (line  11)
* Hermite polynomials, derivatives:      Derivatives of Hermite Polynomials.
                                                              (line   6)
* Hermite polynomials, zeros:            Zeros of Hermite Polynomials and Hermite Functions.
                                                              (line   6)
* hermitian matrix, complex, eigensystem: Complex Hermitian Matrices.
                                                              (line   9)
* Hessenberg decomposition:              Tridiagonal Decomposition of Hermitian Matrices.
                                                              (line  44)
* Hessenberg triangular decomposition:   Hessenberg Decomposition of Real Matrices.
                                                              (line  53)
* histogram statistics:                  Searching histogram ranges.
                                                              (line  22)
* histogram, from ntuple:                Histogramming ntuple values.
                                                              (line  41)
* histograms:                            References and Further Reading<18>.
                                                              (line  19)
* histograms, random sampling from:      Resampling from histograms.
                                                              (line  17)
* Householder linear solver:             Householder Transformations.
                                                              (line  56)
* Householder matrix:                    Givens Rotations.    (line  35)
* Householder transformation:            Givens Rotations.    (line  35)
* how to report:                         No Warranty.         (line  12)
* Hurwitz Zeta Function:                 Hurwitz Zeta Function.
                                                              (line   6)
* HYBRID algorithm, unscaled without derivatives: Algorithms without Derivatives.
                                                              (line  27)
* HYBRID algorithms for nonlinear systems: Algorithms using Derivatives.
                                                              (line  18)
* HYBRIDJ algorithm:                     Algorithms using Derivatives.
                                                              (line  75)
* HYBRIDS algorithm, scaled without derivatives: Algorithms without Derivatives.
                                                              (line  18)
* HYBRIDSJ algorithm:                    Algorithms using Derivatives.
                                                              (line  18)
* hydrogen atom:                         Coulomb Functions.   (line   6)
* hyperbolic cosine, inverse:            Elementary Functions.
                                                              (line  37)
* hyperbolic functions, complex numbers: Inverse Complex Trigonometric Functions.
                                                              (line  65)
* hyperbolic integrals:                  Hyperbolic Integrals.
                                                              (line   6)
* hyperbolic sine, inverse:              Elementary Functions.
                                                              (line  42)
* hyperbolic space:                      Legendre Functions and Spherical Harmonics.
                                                              (line   6)
* hyperbolic tangent, inverse:           Elementary Functions.
                                                              (line  47)
* hypergeometric functions:              Hypergeometric Functions.
                                                              (line   6)
* hypergeometric random variates:        The Hypergeometric Distribution.
                                                              (line   6)
* hypot:                                 Elementary Functions.
                                                              (line  24)
* hypot function, special functions:     Circular Trigonometric Functions.
                                                              (line  16)
* I(x), Bessel Functions:                Regular Modified Cylindrical Bessel Functions.
                                                              (line   6)
* i(x), Bessel Functions:                Regular Modified Spherical Bessel Functions.
                                                              (line   6)
* identity matrix:                       Accessing matrix elements.
                                                              (line  49)
* identity permutation:                  Permutation allocation.
                                                              (line  21)
* IEEE exceptions:                       Representation of floating point numbers.
                                                              (line 136)
* IEEE floating point:                   References and Further Reading<39>.
                                                              (line  18)
* IEEE format for floating point numbers: IEEE floating-point arithmetic.
                                                              (line  11)
* IEEE infinity, defined as a macro:     Mathematical Constants.
                                                              (line  59)
* IEEE NaN, defined as a macro:          Infinities and Not-a-number.
                                                              (line  16)
* illumination, units of:                Viscosity.           (line  13)
* imperial units:                        Measurement of Time. (line  22)
* Implicit Euler method:                 Stepping Functions.  (line 127)
* Implicit Runge-Kutta method:           Stepping Functions.  (line 135)
* importance sampling, VEGAS:            MISER.               (line 158)
* including GSL header files:            An Example Program.  (line  29)
* incomplete Beta function, normalized:  Incomplete Beta Function.
                                                              (line   6)
* incomplete Fermi-Dirac integral:       Incomplete Fermi-Dirac Integrals.
                                                              (line   6)
* incomplete Gamma function:             Incomplete Gamma Functions.
                                                              (line  15)
* indirect sorting:                      Sorting objects.     (line  57)
* indirect sorting, of vector elements:  Sorting vectors.     (line  47)
* infinity, defined as a macro:          Mathematical Constants.
                                                              (line  60)
* infinity, IEEE format:                 Representation of floating point numbers.
                                                              (line  30)
* info-gsl mailing list:                 GSL is Free Software.
                                                              (line  45)
* initial value problems, differential equations: References and Further Reading<21>.
                                                              (line  11)
* initializing matrices:                 Accessing matrix elements.
                                                              (line  49)
* initializing vectors:                  Accessing vector elements.
                                                              (line  74)
* inline functions:                      ANSI C Compliance.   (line  25)
* integer powers:                        Power Function.      (line   6)
* integrals, exponential:                Exponential Integrals.
                                                              (line   6)
* integration, numerical (quadrature):   References and Further Reading<11>.
                                                              (line  67)
* interpolating quadrature:              Fixed point quadratures.
                                                              (line   6)
* interpolation:                         References and Further Reading<22>.
                                                              (line  47)
* interpolation, using Chebyshev polynomials: References and Further Reading<24>.
                                                              (line  14)
* inverse complex trigonometric functions: Complex Trigonometric Functions.
                                                              (line  35)
* inverse cumulative distribution functions: References.      (line  14)
* inverse hyperbolic cosine:             Elementary Functions.
                                                              (line  37)
* inverse hyperbolic functions, complex numbers: Complex Hyperbolic Functions.
                                                              (line  35)
* inverse hyperbolic sine:               Elementary Functions.
                                                              (line  42)
* inverse hyperbolic tangent:            Elementary Functions.
                                                              (line  47)
* inverse of a matrix, by LU decomposition: LU Decomposition. (line  79)
* inverting a permutation:               Permutation functions.
                                                              (line  10)
* Irregular Cylindrical Bessel Functions: Irregular Cylindrical Bessel Functions.
                                                              (line   6)
* Irregular Modified Bessel Functions, Fractional Order: Irregular Modified Bessel Functions—Fractional Order.
                                                              (line   6)
* Irregular Modified Cylindrical Bessel Functions: Irregular Modified Cylindrical Bessel Functions.
                                                              (line   6)
* Irregular Modified Spherical Bessel Functions: Irregular Modified Spherical Bessel Functions.
                                                              (line   6)
* Irregular Spherical Bessel Functions:  Irregular Spherical Bessel Functions.
                                                              (line   6)
* iterating through combinations:        Combination functions.
                                                              (line   6)
* iterating through multisets:           Multiset functions.  (line   6)
* iterating through permutations:        Permutation functions.
                                                              (line  16)
* iterations (C variable):               VEGAS.               (line 173)
* iterative refinement of solutions in linear systems: LU Decomposition.
                                                              (line  66)
* J(x), Bessel Functions:                Regular Cylindrical Bessel Functions.
                                                              (line   6)
* j(x), Bessel Functions:                Regular Spherical Bessel Functions.
                                                              (line   6)
* Jacobi elliptic functions:             Elliptic Functions Jacobi.
                                                              (line   6)
* Jacobi orthogonalization:              Singular Value Decomposition.
                                                              (line  61)
* Jacobian matrix, ODEs:                 Defining the ODE System.
                                                              (line  42)
* Jacobian matrix, root finding:         Overview<3>.         (line  41)
* K(x), Bessel Functions:                Irregular Modified Cylindrical Bessel Functions.
                                                              (line   6)
* k(x), Bessel Functions:                Irregular Modified Spherical Bessel Functions.
                                                              (line   6)
* knots, basis splines:                  Initializing the B-splines solver.
                                                              (line  23)
* kurtosis:                              Absolute deviation.  (line  32)
* Laguerre functions:                    Laguerre Functions.  (line   6)
* Lambert function:                      Lambert W Functions. (line   6)
* Landau distribution:                   The Landau Distribution.
                                                              (line   6)
* LAPACK:                                References and Further Reading<10>.
                                                              (line  18)
* Laplace distribution:                  The Laplace Distribution.
                                                              (line   6)
* large dense linear least squares:      Robust linear regression.
                                                              (line 323)
* large linear least squares, normal equations: Large dense linear systems.
                                                              (line  46)
* large linear least squares, routines:  Large Dense Linear Systems Solution Steps.
                                                              (line  30)
* large linear least squares, steps:     Tall Skinny QR TSQR Approach.
                                                              (line  38)
* large linear least squares, TSQR:      Normal Equations Approach.
                                                              (line  30)
* ldexp:                                 Elementary Functions.
                                                              (line  52)
* LDL decomposition:                     Modified Cholesky Decomposition.
                                                              (line  69)
* LDLT decomposition:                    Modified Cholesky Decomposition.
                                                              (line  69)
* LDLT decomposition, banded:            Banded Cholesky Decomposition.
                                                              (line  74)
* LD_LIBRARY_PATH:                       Linking with an alternative BLAS library.
                                                              (line  25)
* leading dimension, matrices:           Example programs for vectors.
                                                              (line 100)
* least squares fit:                     References and Further Reading<32>.
                                                              (line  24)
* least squares troubleshooting:         Large Dense Linear Least Squares Routines.
                                                              (line 180)
* least squares, covariance of best-fit parameters: High Level Driver.
                                                              (line  39)
* least squares, nonlinear:              References and Further Reading<33>.
                                                              (line  55)
* least squares, regularized:            Multi-parameter regression.
                                                              (line 177)
* least squares, robust:                 Regularized regression.
                                                              (line 380)
* Legendre forms of elliptic integrals:  Definition of Legendre Forms.
                                                              (line   6)
* Legendre functions:                    Legendre Functions and Spherical Harmonics.
                                                              (line   6)
* Legendre polynomials:                  Legendre Functions and Spherical Harmonics.
                                                              (line   6)
* length, computed accurately using hypot: Elementary Functions.
                                                              (line  24)
* length, computed accurately using hypot3: Elementary Functions.
                                                              (line  30)
* Levenberg-Marquardt algorithm:         Solving the Trust Region Subproblem TRS.
                                                              (line  29)
* Levenberg-Marquardt algorithm, geodesic acceleration: Levenberg-Marquardt.
                                                              (line  34)
* Levin u-transform:                     References and Further Reading<25>.
                                                              (line  12)
* Levy distribution:                     The Levy alpha-Stable Distributions.
                                                              (line   6)
* Levy distribution, skew:               The Levy skew alpha-Stable Distribution.
                                                              (line   6)
* libraries, linking with:               Compiling and Linking.
                                                              (line  24)
* libraries, shared:                     Linking with an alternative BLAS library.
                                                              (line  26)
* license of GSL:                        Top.                 (line  12)
* light, units of:                       Viscosity.           (line  14)
* linear algebra:                        References and Further Reading<8>.
                                                              (line  33)
* linear algebra, BLAS:                  References and Further Reading<7>.
                                                              (line  15)
* linear algebra, sparse:                References and Further Reading<37>.
                                                              (line  13)
* linear interpolation:                  1D Interpolation Types.
                                                              (line  10)
* linear least squares, large:           Robust linear regression.
                                                              (line 322)
* linear regression:                     Overview<5>.         (line  60)
* linear systems:                        LU Decomposition.    (line  44)
* linear systems, refinement of solutions: LU Decomposition.  (line  66)
* linking with GSL libraries:            Compiling and Linking.
                                                              (line  24)
* location estimation:                   Order Statistics.    (line  22)
* log1p:                                 Elementary Functions.
                                                              (line  12)
* logarithm and related functions:       Logarithm and Related Functions.
                                                              (line   6)
* logarithm of Beta function:            Beta Functions.      (line  14)
* logarithm of combinatorial factor C(m: Factorials.          (line  51)
* logarithm of complex number:           Elementary Complex Functions.
                                                              (line  33)
* logarithm of cosh function, special functions: Hyperbolic Trigonometric Functions.
                                                              (line  11)
* logarithm of double factorial:         Factorials.          (line  36)
* logarithm of factorial:                Factorials.          (line  27)
* logarithm of Gamma function:           Gamma Functions.     (line  23)
* logarithm of Pochhammer symbol:        Pochhammer Symbol.   (line  16)
* logarithm of sinh function:            Hyperbolic Trigonometric Functions.
                                                              (line   6)
* logarithm of the determinant of a matrix: LU Decomposition. (line 121)
* logarithm, computed accurately near 1: Elementary Functions.
                                                              (line  12)
* Logarithmic random variates:           The Hypergeometric Distribution.
                                                              (line  41)
* Logistic distribution:                 The Logistic Distribution.
                                                              (line   6)
* Lognormal distribution:                The Lognormal Distribution.
                                                              (line   6)
* low discrepancy sequences:             Acknowledgements.    (line  12)
* Low-level CBLAS:                       Autoconf Macros.     (line 107)
* LQ decomposition:                      QR Decomposition with Column Pivoting.
                                                              (line 165)
* LU decomposition:                      Linear Algebra.      (line  15)
* macros for mathematical constants:     Mathematical Functions.
                                                              (line  14)
* magnitude of complex number:           Properties of complex numbers.
                                                              (line  11)
* mailing list:                          No Warranty.         (line  12)
* mailing list archives:                 Reporting Bugs.      (line  31)
* mailing list for GSL announcements:    GSL is Free Software.
                                                              (line  46)
* mantissa, IEEE format:                 IEEE floating-point arithmetic.
                                                              (line  11)
* mass, units of:                        Volume Area and Length.
                                                              (line  42)
* mathematical constants, defined as macros: Mathematical Functions.
                                                              (line  14)
* mathematical functions, elementary:    Examples.            (line  38)
* Mathieu Function Characteristic Values: Mathieu Function Characteristic Values.
                                                              (line   6)
* Mathieu functions:                     Mathieu Functions.   (line   6)
* matrices:                              Example programs for vectors.
                                                              (line 100)
* matrices, banded:                      Triangular Systems.  (line  44)
* matrices, initializing:                Accessing matrix elements.
                                                              (line  49)
* matrices, range-checking:              Matrix allocation.   (line  38)
* matrices, sparse:                      References and Further Reading<35>.
                                                              (line  22)
* matrix determinant:                    LU Decomposition.    (line 111)
* matrix diagonal:                       Creating row and column views.
                                                              (line  67)
* matrix factorization:                  References and Further Reading<8>.
                                                              (line  33)
* matrix inverse:                        LU Decomposition.    (line  79)
* matrix square root, Cholesky decomposition: Singular Value Decomposition.
                                                              (line  94)
* matrix subdiagonal:                    Creating row and column views.
                                                              (line  80)
* matrix superdiagonal:                  Creating row and column views.
                                                              (line  94)
* matrix, constant:                      Accessing matrix elements.
                                                              (line  48)
* matrix, identity:                      Accessing matrix elements.
                                                              (line  49)
* matrix, operations:                    References and Further Reading<7>.
                                                              (line  15)
* matrix, zero:                          Accessing matrix elements.
                                                              (line  49)
* max:                                   References and Further Reading<14>.
                                                              (line  51)
* maximal phase, Daubechies wavelets:    Initialization.      (line  23)
* maximization, see minimization:        References and Further Reading<29>.
                                                              (line  17)
* maximum of two numbers:                Maximum and Minimum functions.
                                                              (line  10)
* maximum value, from histogram:         Searching histogram ranges.
                                                              (line  22)
* mean:                                  References and Further Reading<14>.
                                                              (line  52)
* mean value, from histogram:            Histogram Statistics.
                                                              (line  28)
* mean, trimmed:                         Robust Location Estimates.
                                                              (line  16)
* mean, truncated:                       Robust Location Estimates.
                                                              (line  16)
* median absolute deviation:             Robust Scale Estimates.
                                                              (line  12)
* Mills’ ratio, inverse:                 Probability functions.
                                                              (line  21)
* min:                                   References and Further Reading<14>.
                                                              (line  52)
* minimization, BFGS algorithm:          Algorithms with Derivatives.
                                                              (line  43)
* minimization, caveats:                 Overview<2>.         (line  45)
* minimization, conjugate gradient algorithm: Algorithms with Derivatives.
                                                              (line  15)
* minimization, multidimensional:        References and Further Reading<31>.
                                                              (line  29)
* minimization, one-dimensional:         References and Further Reading<29>.
                                                              (line  17)
* minimization, overview:                One Dimensional Minimization.
                                                              (line  20)
* minimization, Polak-Ribiere algorithm: Algorithms with Derivatives.
                                                              (line  34)
* minimization, providing a function to minimize: Initializing the Minimizer.
                                                              (line  60)
* minimization, simplex algorithm:       Algorithms without Derivatives<2>.
                                                              (line  14)
* minimization, steepest descent algorithm: Algorithms with Derivatives.
                                                              (line  65)
* minimization, stopping parameters:     Iteration<2>.        (line  56)
* minimum finding, Brent’s method:       Minimization Algorithms.
                                                              (line  33)
* minimum finding, golden section algorithm: Minimization Algorithms.
                                                              (line  15)
* minimum of two numbers:                Maximum and Minimum functions.
                                                              (line  15)
* minimum value, from histogram:         Searching histogram ranges.
                                                              (line  21)
* MINPACK, minimization algorithms:      Algorithms using Derivatives.
                                                              (line  18)
* min_calls (C variable):                MISER.               (line 111)
* min_calls_per_bisection (C variable):  MISER.               (line 121)
* MISCFUN:                               References and Further Reading<3>.
                                                              (line  15)
* MISER monte carlo integration:         PLAIN Monte Carlo.   (line  64)
* Mixed-radix FFT, complex data:         Radix-2 FFT routines for complex data.
                                                              (line 128)
* Mixed-radix FFT, real data:            Radix-2 FFT routines for real data.
                                                              (line 101)
* mode (C variable):                     VEGAS.               (line 194)
* Modified Bessel Functions, Fractional Order: Regular Modified Bessel Functions—Fractional Order.
                                                              (line   6)
* Modified Cholesky Decomposition:       Pivoted Cholesky Decomposition.
                                                              (line 106)
* Modified Clenshaw-Curtis quadrature:   Integrands with weight functions.
                                                              (line   6)
* Modified Cylindrical Bessel Functions: Regular Modified Cylindrical Bessel Functions.
                                                              (line   6)
* Modified Givens Rotation, BLAS:        Level 1.             (line 153)
* Modified Newton’s method for nonlinear systems: Algorithms using Derivatives.
                                                              (line 102)
* Modified Spherical Bessel Functions:   Regular Modified Spherical Bessel Functions.
                                                              (line   6)
* Monte Carlo integration:               References and Further Reading<19>.
                                                              (line   9)
* moving maximum:                        Moving Variance and Standard Deviation.
                                                              (line  32)
* moving mean:                           Allocation for Moving Window Statistics.
                                                              (line  29)
* moving median:                         Moving Sum.          (line  20)
* moving median absolute deviation:      Robust Scale Estimation.
                                                              (line  17)
* moving minimum:                        Moving Variance and Standard Deviation.
                                                              (line  32)
* moving quantile range:                 Moving MAD.          (line  30)
* moving standard deviation:             Moving Mean.         (line  25)
* moving sum:                            Moving Minimum and Maximum.
                                                              (line  39)
* moving variance:                       Moving Mean.         (line  25)
* moving window accumulators:            Moving Q_n.          (line  20)
* moving window statistics:              References and Further Reading<16>.
                                                              (line  14)
* moving window, allocation:             Handling Endpoints.  (line  40)
* MRG, multiple recursive random number generator: Random number generator algorithms.
                                                              (line 126)
* MT19937 random number generator:       Random number generator algorithms.
                                                              (line  19)
* multi-parameter regression:            Linear regression without a constant term.
                                                              (line  48)
* multidimensional integration:          References and Further Reading<19>.
                                                              (line   8)
* multidimensional root finding, Broyden algorithm: Algorithms without Derivatives.
                                                              (line  54)
* multidimensional root finding, overview: Multidimensional Root-Finding.
                                                              (line  20)
* multidimensional root finding, providing a function to solve: Initializing the Solver<2>.
                                                              (line  80)
* Multimin, caveats:                     Overview<4>.         (line  47)
* Multinomial distribution:              The Multinomial Distribution.
                                                              (line   6)
* multiplication:                        Elementary Operations.
                                                              (line   6)
* multisets:                             References and Further Reading<6>.
                                                              (line  11)
* multistep methods, ODEs:               Stepping Functions.  (line 156)
* n):                                    Factorials.          (line  44)
* n) <1>:                                Factorials.          (line  51)
* N-dimensional random direction vector: Spherical Vector Distributions.
                                                              (line  42)
* NaN, defined as a macro:               Infinities and Not-a-number.
                                                              (line  16)
* nautical units:                        Imperial Units.      (line  25)
* Negative Binomial distribution, random variates: The Negative Binomial Distribution.
                                                              (line   6)
* Nelder-Mead simplex algorithm for minimization: Algorithms without Derivatives<2>.
                                                              (line  14)
* Newton algorithm, discrete:            Algorithms without Derivatives.
                                                              (line  32)
* Newton algorithm, globally convergent: Algorithms using Derivatives.
                                                              (line 102)
* Newton’s method for finding roots:     Root Finding Algorithms using Derivatives.
                                                              (line  17)
* Newton’s method for systems of nonlinear equations: Algorithms using Derivatives.
                                                              (line  82)
* Niederreiter sequence:                 Acknowledgements.    (line  11)
* NIST Statistical Reference Datasets:   References and Further Reading<33>.
                                                              (line  16)
* non-normalized incomplete Gamma function: Incomplete Gamma Functions.
                                                              (line   6)
* nonlinear equation, solutions of:      References and Further Reading<28>.
                                                              (line  14)
* nonlinear fitting, stopping parameters, convergence: Iteration<5>.
                                                              (line 119)
* nonlinear functions, minimization:     References and Further Reading<29>.
                                                              (line  16)
* nonlinear least squares:               References and Further Reading<33>.
                                                              (line  56)
* nonlinear least squares, dogleg:       Levenberg-Marquardt with Geodesic Acceleration.
                                                              (line  29)
* nonlinear least squares, double dogleg: Dogleg.             (line  26)
* nonlinear least squares, levenberg-marquardt: Solving the Trust Region Subproblem TRS.
                                                              (line  29)
* nonlinear least squares, levenberg-marquardt, geodesic acceleration: Levenberg-Marquardt.
                                                              (line  33)
* nonlinear least squares, overview:     Nonlinear Least-Squares Fitting.
                                                              (line  27)
* nonlinear systems of equations, solution of: References and Further Reading<30>.
                                                              (line  13)
* nonsymmetric matrix, real, eigensystem: Real Nonsymmetric Matrices.
                                                              (line   6)
* Nordsieck form:                        Stepping Functions.  (line 156)
* normalized form, IEEE format:          Representation of floating point numbers.
                                                              (line  14)
* normalized incomplete Beta function:   Incomplete Beta Function.
                                                              (line   6)
* Not-a-number, defined as a macro:      Infinities and Not-a-number.
                                                              (line  16)
* NRM2, Level-1 BLAS:                    Level 1.             (line  40)
* ntuples:                               Example programs for 2D histograms.
                                                              (line  72)
* nuclear physics, constants:            Astronomy and Astrophysics.
                                                              (line  29)
* numerical constants, defined as macros: Mathematical Functions.
                                                              (line  14)
* numerical derivatives:                 References and Further Reading<23>.
                                                              (line  18)
* numerical integration (quadrature):    References and Further Reading<11>.
                                                              (line  67)
* obtaining GSL:                         GSL is Free Software.
                                                              (line  46)
* ODEs, initial value problems:          References and Further Reading<21>.
                                                              (line  10)
* online statistics:                     References and Further Reading<15>.
                                                              (line  39)
* online statistics <1>:                 References and Further Reading<16>.
                                                              (line  13)
* optimization, combinatorial:           References and Further Reading<20>.
                                                              (line  21)
* optimization, see minimization:        References and Further Reading<29>.
                                                              (line  17)
* optimized functions, alternatives:     Portability functions.
                                                              (line  33)
* ordering, matrix elements:             Example programs for vectors.
                                                              (line  99)
* ordinary differential equations, initial value problem: References and Further Reading<21>.
                                                              (line  11)
* oscillatory functions, numerical integration of: QAWO adaptive integration for oscillatory functions.
                                                              (line   6)
* ostream (C variable):                  VEGAS.               (line 204)
* overflow, IEEE exceptions:             Representation of floating point numbers.
                                                              (line 135)
* Pareto distribution:                   The Pareto Distribution.
                                                              (line   6)
* PAW:                                   References and Further Reading<19>.
                                                              (line   6)
* permutations:                          References and Further Reading<4>.
                                                              (line  12)
* physical constants:                    References and Further Reading<38>.
                                                              (line  15)
* physical dimension, matrices:          Example programs for vectors.
                                                              (line 100)
* pi, defined as a macro:                Mathematical Constants.
                                                              (line   9)
* Pivoted Cholesky Decomposition:        Cholesky Decomposition.
                                                              (line 149)
* plain Monte Carlo:                     Interface.           (line  80)
* Pochhammer symbol:                     Pochhammer Symbol.   (line   6)
* Poisson random numbers:                The Poisson Distribution.
                                                              (line   6)
* Polak-Ribiere algorithm, minimization: Algorithms with Derivatives.
                                                              (line  34)
* polar form of complex numbers:         Complex Numbers.     (line  21)
* polar to rectangular conversion:       Conversion Functions.
                                                              (line   6)
* polygamma functions:                   Psi Digamma Function.
                                                              (line   6)
* polynomial evaluation:                 Polynomials.         (line  13)
* polynomial interpolation:              1D Interpolation Types.
                                                              (line  15)
* polynomials, roots of:                 References and Further Reading.
                                                              (line  32)
* power function:                        Power Function.      (line   6)
* power of complex number:               Elementary Complex Functions.
                                                              (line  17)
* power, units of:                       Mass and Weight.     (line  50)
* precision, IEEE arithmetic:            Representation of floating point numbers.
                                                              (line 136)
* predictor-corrector method, ODEs:      Stepping Functions.  (line 156)
* prefixes:                              Force and Energy.    (line  22)
* pressure:                              Thermal Energy and Power.
                                                              (line  22)
* Prince-Dormand, Runge-Kutta method:    Stepping Functions.  (line 123)
* printers units:                        Speed and Nautical Units.
                                                              (line  25)
* probability distribution, from histogram: Resampling from histograms.
                                                              (line  18)
* probability distributions, from histograms: Reading and writing histograms.
                                                              (line  64)
* projection of ntuples:                 Histogramming ntuple values.
                                                              (line  41)
* psi function:                          Psi Digamma Function.
                                                              (line   6)
* QAG quadrature algorithm:              QAG adaptive integration.
                                                              (line   6)
* QAGI quadrature algorithm:             QAGI adaptive integration on infinite intervals.
                                                              (line   6)
* QAGP quadrature algorithm:             QAGP adaptive integration with known singular points.
                                                              (line   6)
* QAGS quadrature algorithm:             QAGS adaptive integration with singularities.
                                                              (line   6)
* QAWC quadrature algorithm:             QAWC adaptive integration for Cauchy principal values.
                                                              (line   6)
* QAWF quadrature algorithm:             QAWF adaptive integration for Fourier integrals.
                                                              (line   6)
* QAWO quadrature algorithm:             QAWO adaptive integration for oscillatory functions.
                                                              (line   6)
* QAWS quadrature algorithm:             QAWS adaptive integration for singular functions.
                                                              (line   6)
* Qn statistic:                          S_n Statistic.       (line  36)
* QNG quadrature algorithm:              QNG non-adaptive Gauss-Kronrod integration.
                                                              (line   6)
* QR decomposition:                      LU Decomposition.    (line 136)
* QR decomposition with column pivoting: Triangle on Top of Rectangle.
                                                              (line  33)
* QUADPACK:                              References and Further Reading<11>.
                                                              (line  66)
* quadratic equation, solving:           Divided Difference Representation of Polynomials.
                                                              (line  79)
* quadrature:                            References and Further Reading<11>.
                                                              (line  67)
* quadrature, fixed point:               Fixed point quadratures.
                                                              (line   6)
* quadrature, interpolating:             Fixed point quadratures.
                                                              (line   6)
* quantile functions:                    References.          (line  13)
* quasi-random sequences:                Acknowledgements.    (line  12)
* R250 shift-register random number generator: Other random number generators.
                                                              (line  62)
* Racah coefficients:                    Coupling Coefficients.
                                                              (line   6)
* Radial Mathieu Functions:              Radial Mathieu Functions.
                                                              (line   6)
* radioactivity:                         Light and Illumination.
                                                              (line  34)
* Radix-2 FFT for real data:             Overview of real data FFTs.
                                                              (line  49)
* Radix-2 FFT, complex data:             Overview of complex data FFTs.
                                                              (line  72)
* rand, BSD random number generator:     Unix random number generators.
                                                              (line  17)
* rand48 random number generator:        Unix random number generators.
                                                              (line  59)
* random number distributions:           References.          (line  14)
* random number generators:              References and Further Reading<12>.
                                                              (line  33)
* random sampling from histograms:       Resampling from histograms.
                                                              (line  18)
* RANDU random number generator:         Other random number generators.
                                                              (line 113)
* RANF random number generator:          Other random number generators.
                                                              (line  22)
* range:                                 References and Further Reading<14>.
                                                              (line  52)
* range-checking for matrices:           Matrix allocation.   (line  37)
* range-checking for vectors:            Vector allocation.   (line  36)
* RANLUX random number generator:        Random number generator algorithms.
                                                              (line  74)
* RANLXD random number generator:        Random number generator algorithms.
                                                              (line  67)
* RANLXS random number generator:        Random number generator algorithms.
                                                              (line  47)
* RANMAR random number generator:        Other random number generators.
                                                              (line  55)
* RANMAR random number generator <1>:    Other random number generators.
                                                              (line 123)
* Rayleigh distribution:                 The Rayleigh Distribution.
                                                              (line   6)
* Rayleigh Tail distribution:            The Rayleigh Tail Distribution.
                                                              (line   6)
* real nonsymmetric matrix, eigensystem: Real Nonsymmetric Matrices.
                                                              (line   6)
* real symmetric matrix, eigensystem:    Real Symmetric Matrices.
                                                              (line   6)
* Reciprocal Gamma function:             Gamma Functions.     (line  54)
* rectangular to polar conversion:       Conversion Functions.
                                                              (line   6)
* recursive stratified sampling, MISER:  PLAIN Monte Carlo.   (line  63)
* reduction of angular variables:        Restriction Functions.
                                                              (line   6)
* refinement of solutions in linear systems: LU Decomposition.
                                                              (line  66)
* regression, least squares:             References and Further Reading<32>.
                                                              (line  24)
* regression, ridge:                     Multi-parameter regression.
                                                              (line 178)
* regression, robust:                    Regularized regression.
                                                              (line 381)
* regression, Tikhonov:                  Multi-parameter regression.
                                                              (line 178)
* Regular Bessel Functions, Fractional Order: Regular Bessel Function—Fractional Order.
                                                              (line   6)
* Regular Bessel Functions, Zeros of:    Zeros of Regular Bessel Functions.
                                                              (line   6)
* Regular Cylindrical Bessel Functions:  Regular Cylindrical Bessel Functions.
                                                              (line   6)
* Regular Modified Bessel Functions, Fractional Order: Regular Modified Bessel Functions—Fractional Order.
                                                              (line   6)
* Regular Modified Cylindrical Bessel Functions: Regular Modified Cylindrical Bessel Functions.
                                                              (line   6)
* Regular Modified Spherical Bessel Functions: Regular Modified Spherical Bessel Functions.
                                                              (line   6)
* Regular Spherical Bessel Functions:    Regular Spherical Bessel Functions.
                                                              (line   6)
* Regulated Gamma function:              Gamma Functions.     (line  43)
* relative Pochhammer symbol:            Pochhammer Symbol.   (line  31)
* reporting bugs in GSL:                 No Warranty.         (line  12)
* representations of complex numbers:    Complex Numbers.     (line  21)
* resampling from histograms:            Reading and writing histograms.
                                                              (line  65)
* residual, in nonlinear systems of equations: Search Stopping Parameters<2>.
                                                              (line  32)
* reversing a permutation:               Permutation functions.
                                                              (line   6)
* ridge regression:                      Multi-parameter regression.
                                                              (line 178)
* Riemann Zeta Function:                 Riemann Zeta Function.
                                                              (line   6)
* RK2, Runge-Kutta method:               Stepping Functions.  (line 103)
* RK4, Runge-Kutta method:               Stepping Functions.  (line 107)
* RKF45, Runge-Kutta-Fehlberg method:    Stepping Functions.  (line 114)
* robust location estimators:            Order Statistics.    (line  22)
* robust regression:                     Regularized regression.
                                                              (line 381)
* robust scale estimators:               Gastwirth Estimator. (line  24)
* rolling maximum:                       Moving Variance and Standard Deviation.
                                                              (line  31)
* rolling mean:                          Allocation for Moving Window Statistics.
                                                              (line  28)
* rolling median:                        Moving Sum.          (line  19)
* rolling median absolute deviation:     Robust Scale Estimation.
                                                              (line  17)
* rolling minimum:                       Moving Variance and Standard Deviation.
                                                              (line  32)
* rolling quantile range:                Moving MAD.          (line  29)
* rolling standard deviation:            Moving Mean.         (line  24)
* rolling sum:                           Moving Minimum and Maximum.
                                                              (line  38)
* rolling variance:                      Moving Mean.         (line  25)
* rolling window accumulators:           Moving Q_n.          (line  19)
* root finding:                          References and Further Reading<28>.
                                                              (line  15)
* root finding, bisection algorithm:     Root Bracketing Algorithms.
                                                              (line  18)
* root finding, Brent’s method:          Root Bracketing Algorithms.
                                                              (line  52)
* root finding, caveats:                 Overview.            (line  39)
* root finding, false position algorithm: Root Bracketing Algorithms.
                                                              (line  34)
* root finding, initial guess:           Providing the function to solve.
                                                              (line 121)
* root finding, Newton’s method:         Root Finding Algorithms using Derivatives.
                                                              (line  17)
* root finding, overview:                One Dimensional Root-Finding.
                                                              (line  19)
* root finding, providing a function to solve: Initializing the Solver.
                                                              (line  74)
* root finding, search bounds:           Providing the function to solve.
                                                              (line 122)
* root finding, secant method:           Root Finding Algorithms using Derivatives.
                                                              (line  31)
* root finding, Steffenson’s method:     Root Finding Algorithms using Derivatives.
                                                              (line  62)
* root finding, stopping parameters:     Iteration.           (line  50)
* root finding, stopping parameters <1>: Iteration<3>.        (line  59)
* roots:                                 References and Further Reading<28>.
                                                              (line  15)
* ROTG, Level-1 BLAS:                    Level 1.             (line 128)
* rounding mode:                         Representation of floating point numbers.
                                                              (line 136)
* Runge-Kutta Cash-Karp method:          Stepping Functions.  (line 119)
* Runge-Kutta methods, ordinary differential equations: Stepping Functions.
                                                              (line 103)
* Runge-Kutta Prince-Dormand method:     Stepping Functions.  (line 123)
* running statistics:                    References and Further Reading<15>.
                                                              (line  40)
* safe comparison of floating point numbers: Approximate Comparison of Floating Point Numbers.
                                                              (line  12)
* safeguarded step-length algorithm:     Minimization Algorithms.
                                                              (line  50)
* sampling from histograms:              Reading and writing histograms.
                                                              (line  65)
* sampling from histograms <1>:          Resampling from histograms.
                                                              (line  18)
* SAXPY, Level-1 BLAS:                   Level 1.             (line 102)
* SCAL, Level-1 BLAS:                    Level 1.             (line 114)
* scale estimation:                      Gastwirth Estimator. (line  24)
* schedule, cooling:                     Simulated Annealing algorithm.
                                                              (line  20)
* se(q,x), Mathieu function:             Angular Mathieu Functions.
                                                              (line   6)
* secant method for finding roots:       Root Finding Algorithms using Derivatives.
                                                              (line  31)
* selection function, ntuples:           Histogramming ntuple values.
                                                              (line  13)
* series, acceleration:                  References and Further Reading<25>.
                                                              (line  12)
* shared libraries:                      Linking with an alternative BLAS library.
                                                              (line  26)
* shell prompt:                          Further Information. (line  27)
* Shi(x):                                Hyperbolic Integrals.
                                                              (line   6)
* shift-register random number generator: Other random number generators.
                                                              (line  62)
* Si(x):                                 Trigonometric Integrals.
                                                              (line   6)
* sign bit, IEEE format:                 IEEE floating-point arithmetic.
                                                              (line  11)
* sign of the determinant of a matrix:   LU Decomposition.    (line 130)
* simplex algorithm, minimization:       Algorithms without Derivatives<2>.
                                                              (line  14)
* simulated annealing:                   References and Further Reading<20>.
                                                              (line  21)
* sin, of complex number:                Complex Trigonometric Functions.
                                                              (line   6)
* sine function, special functions:      Circular Trigonometric Functions.
                                                              (line   6)
* single precision, IEEE format:         Representation of floating point numbers.
                                                              (line  34)
* singular functions, numerical integration of: QAWS adaptive integration for singular functions.
                                                              (line   6)
* singular points, specifying positions in quadrature: QAGP adaptive integration with known singular points.
                                                              (line   6)
* singular value decomposition:          Complete Orthogonal Decomposition.
                                                              (line 116)
* Skew Levy distribution:                The Levy skew alpha-Stable Distribution.
                                                              (line   6)
* skewness:                              Absolute deviation.  (line  33)
* slope, see numerical derivative:       References and Further Reading<23>.
                                                              (line  17)
* Sn statistic:                          Median Absolute Deviation MAD.
                                                              (line  29)
* Sobol sequence:                        Acknowledgements.    (line  12)
* solution of:                           LU Decomposition.    (line  44)
* solution of linear system by Householder transformations: Householder Transformations.
                                                              (line  57)
* solution of linear systems, Ax=b:      References and Further Reading<8>.
                                                              (line  33)
* solving a nonlinear equation:          References and Further Reading<28>.
                                                              (line  15)
* solving nonlinear systems of equations: References and Further Reading<30>.
                                                              (line  13)
* sorting:                               Examples<6>.         (line 112)
* sorting eigenvalues and eigenvectors:  Sorting Eigenvalues and Eigenvectors.
                                                              (line   6)
* sorting vector elements:               Sorting vectors.     (line  22)
* source code, reuse in applications:    Deprecated Functions.
                                                              (line  13)
* sparse BLAS:                           References and Further Reading<36>.
                                                              (line  14)
* sparse BLAS, references:               Sparse BLAS operations.
                                                              (line  22)
* sparse linear algebra:                 References and Further Reading<37>.
                                                              (line  14)
* sparse linear algebra, examples:       Iterating the Sparse Linear System.
                                                              (line  58)
* sparse linear algebra, iterative solvers: Overview<9>.      (line  20)
* sparse linear algebra, overview:       Sparse Linear Algebra.
                                                              (line  14)
* sparse linear algebra, references:     Examples<35>.        (line 136)
* sparse matrices:                       References and Further Reading<35>.
                                                              (line  23)
* sparse matrices, accessing elements:   Allocation.          (line  78)
* sparse matrices, allocation:           Overview<8>.         (line  59)
* sparse matrices, BLAS operations:      Sparse BLAS Support. (line  14)
* sparse matrices, compressed column storage: Coordinate Storage COO.
                                                              (line  39)
* sparse matrices, compressed row storage: Compressed Sparse Column CSC.
                                                              (line  27)
* sparse matrices, compressed sparse column: Coordinate Storage COO.
                                                              (line  40)
* sparse matrices, compressed sparse row: Compressed Sparse Column CSC.
                                                              (line  28)
* sparse matrices, compression:          Finding Maximum and Minimum Elements.
                                                              (line  26)
* sparse matrices, conversion:           Compressed Format.   (line  37)
* sparse matrices, coordinate format:    Sparse Matrix Storage Formats.
                                                              (line  18)
* sparse matrices, copying:              Reading and Writing Matrices.
                                                              (line  54)
* sparse matrices, data types:           Sparse Matrices.     (line  18)
* sparse matrices, examples:             Conversion Between Sparse and Dense Matrices.
                                                              (line  28)
* sparse matrices, exchanging rows and columns: Copying Matrices.
                                                              (line  15)
* sparse matrices, initializing elements: Accessing Matrix Elements.
                                                              (line  32)
* sparse matrices, iterative solvers:    Overview<9>.         (line  20)
* sparse matrices, min/max elements:     Matrix Properties.   (line  36)
* sparse matrices, operations:           Exchanging Rows and Columns.
                                                              (line  30)
* sparse matrices, overview:             Compressed Sparse Row CSR.
                                                              (line  26)
* sparse matrices, properties:           Matrix Operations.   (line  61)
* sparse matrices, reading:              Initializing Matrix Elements.
                                                              (line  21)
* sparse matrices, references:           Examples<34>.        (line 133)
* sparse matrices, storage formats:      Data types<2>.       (line  56)
* sparse matrices, triplet format:       Sparse Matrix Storage Formats.
                                                              (line  18)
* sparse matrices, writing:              Initializing Matrix Elements.
                                                              (line  20)
* sparse, iterative solvers:             Overview<9>.         (line  19)
* special functions:                     References and Further Reading<2>.
                                                              (line  28)
* special functions <1>:                 Hyperbolic Trigonometric Functions.
                                                              (line   6)
* Spherical Bessel Functions:            Regular Spherical Bessel Functions.
                                                              (line   6)
* spherical harmonics:                   Legendre Functions and Spherical Harmonics.
                                                              (line   6)
* spherical random variates, 2D:         Spherical Vector Distributions.
                                                              (line  10)
* spherical random variates, 3D:         Spherical Vector Distributions.
                                                              (line  32)
* spherical random variates, N-dimensional: Spherical Vector Distributions.
                                                              (line  42)
* spline:                                References and Further Reading<22>.
                                                              (line  46)
* splines, basis:                        References and Further Reading<34>.
                                                              (line  37)
* square root of a matrix, Cholesky decomposition: Singular Value Decomposition.
                                                              (line  95)
* square root of complex number:         Elementary Complex Functions.
                                                              (line   6)
* stage (C variable):                    VEGAS.               (line 178)
* standard deviation:                    References and Further Reading<14>.
                                                              (line  52)
* standard deviation, from histogram:    Histogram Statistics.
                                                              (line  35)
* standards conformance, ANSI C:         Conventions used in this manual.
                                                              (line  20)
* Statistical Reference Datasets (StRD): References and Further Reading<33>.
                                                              (line  16)
* statistics:                            References and Further Reading<14>.
                                                              (line  52)
* statistics, from histogram:            Searching histogram ranges.
                                                              (line  22)
* statistics, moving window:             References and Further Reading<16>.
                                                              (line  14)
* steepest descent algorithm, minimization: Algorithms with Derivatives.
                                                              (line  65)
* Steffenson’s method for finding roots: Root Finding Algorithms using Derivatives.
                                                              (line  62)
* stratified sampling in Monte Carlo integration: References and Further Reading<19>.
                                                              (line   9)
* stride, of vector index:               Example programs for blocks.
                                                              (line  27)
* Student t-distribution:                The t-distribution.  (line  14)
* subdiagonal, of a matrix:              Creating row and column views.
                                                              (line  80)
* summation, acceleration:               References and Further Reading<25>.
                                                              (line  12)
* superdiagonal, matrix:                 Creating row and column views.
                                                              (line  94)
* SVD:                                   Complete Orthogonal Decomposition.
                                                              (line 117)
* SWAP, Level-1 BLAS:                    Level 1.             (line  81)
* swapping permutation elements:         Accessing permutation elements.
                                                              (line  17)
* SYMM, Level-3 BLAS:                    Level 3.             (line  36)
* symmetric matrices, banded:            General Banded Format.
                                                              (line  26)
* symmetric matrix, real, eigensystem:   Real Symmetric Matrices.
                                                              (line   6)
* SYMV, Level-2 BLAS:                    Level 2.             (line  73)
* synchrotron functions:                 Synchrotron Functions.
                                                              (line   6)
* SYR, Level-2 BLAS:                     Level 2.             (line 133)
* SYR2, Level-2 BLAS:                    Level 2.             (line 158)
* SYR2K, Level-3 BLAS:                   Level 3.             (line 187)
* SYRK, Level-3 BLAS:                    Level 3.             (line 145)
* systems of equations, nonlinear:       References and Further Reading<30>.
                                                              (line  12)
* t-distribution:                        The t-distribution.  (line  14)
* t-test:                                References and Further Reading<14>.
                                                              (line  52)
* tangent of complex number:             Complex Trigonometric Functions.
                                                              (line  16)
* Tausworthe random number generator:    Random number generator algorithms.
                                                              (line 144)
* Taylor coefficients, computation of:   Factorials.          (line  58)
* testing combination for validity:      Combination properties.
                                                              (line  20)
* testing multiset for validity:         Multiset properties. (line  20)
* testing permutation for validity:      Permutation properties.
                                                              (line  15)
* thermal energy, units of:              Mass and Weight.     (line  49)
* Tikhonov regression:                   Multi-parameter regression.
                                                              (line 178)
* time units:                            Atomic and Nuclear Physics.
                                                              (line  79)
* trailing dimension, matrices:          Example programs for vectors.
                                                              (line 100)
* transformation, Householder:           Givens Rotations.    (line  34)
* transforms, Hankel:                    References and Further Reading<27>.
                                                              (line  53)
* transforms, wavelet:                   References and Further Reading<26>.
                                                              (line  24)
* transport functions:                   Transport Functions. (line   6)
* traveling salesman problem:            Trivial example.     (line 114)
* triangular systems:                    Tridiagonal Systems. (line  71)
* tridiagonal decomposition:             LDLT Decomposition.  (line  60)
* tridiagonal decomposition <1>:         Tridiagonal Decomposition of Real Symmetric Matrices.
                                                              (line  43)
* tridiagonal systems:                   Householder solver for linear systems.
                                                              (line  21)
* trigonometric functions:               Trigonometric Functions.
                                                              (line   6)
* trigonometric functions of complex numbers: Elementary Complex Functions.
                                                              (line  50)
* trigonometric integrals:               Trigonometric Integrals.
                                                              (line   6)
* trimmed mean:                          Robust Location Estimates.
                                                              (line  16)
* TRMM, Level-3 BLAS:                    Level 3.             (line  82)
* TRMV, Level-2 BLAS:                    Level 2.             (line  28)
* TRSM, Level-3 BLAS:                    Level 3.             (line 113)
* TRSV, Level-2 BLAS:                    Level 2.             (line  51)
* truncated mean:                        Robust Location Estimates.
                                                              (line  16)
* TSP:                                   Trivial example.     (line 115)
* TT800 random number generator:         Other random number generators.
                                                              (line  79)
* two dimensional Gaussian distribution: The Bivariate Gaussian Distribution.
                                                              (line   6)
* two dimensional Gaussian distribution <1>: The Multivariate Gaussian Distribution.
                                                              (line   6)
* two dimensional histograms:            Example programs for histograms.
                                                              (line  65)
* two-sided exponential distribution:    The Laplace Distribution.
                                                              (line   6)
* Type 1 Gumbel distribution, random variates: The Type-1 Gumbel Distribution.
                                                              (line   6)
* Type 2 Gumbel distribution:            The Type-2 Gumbel Distribution.
                                                              (line   6)
* u-transform for series:                References and Further Reading<25>.
                                                              (line  12)
* underflow, IEEE exceptions:            Representation of floating point numbers.
                                                              (line 136)
* uniform distribution:                  The Flat Uniform Distribution.
                                                              (line   6)
* units of:                              Thermal Energy and Power.
                                                              (line  21)
* units of <1>:                          Pressure.            (line  33)
* units of <2>:                          Light and Illumination.
                                                              (line  33)
* units of <3>:                          Radioactivity.       (line  17)
* units, conversion of:                  References and Further Reading<38>.
                                                              (line  14)
* units, imperial:                       Measurement of Time. (line  21)
* Unix random number generators, rand:   Unix random number generators.
                                                              (line  17)
* Unix random number generators, rand48: Unix random number generators.
                                                              (line  17)
* unnormalized incomplete Gamma function: Incomplete Gamma Functions.
                                                              (line   6)
* unweighted linear fits:                References and Further Reading<32>.
                                                              (line  23)
* value function, ntuples:               Histogramming ntuple values.
                                                              (line  27)
* Van der Pol oscillator, example:       Examples<22>.        (line   6)
* variance:                              References and Further Reading<14>.
                                                              (line  52)
* variance, from histogram:              Histogram Statistics.
                                                              (line  35)
* variance-covariance matrix, linear fits: Overview<5>.       (line  48)
* VAX random number generator:           Other random number generators.
                                                              (line  93)
* vector, operations:                    References and Further Reading<7>.
                                                              (line  15)
* vector, sorting elements of:           Sorting vectors.     (line  22)
* vectors:                               Example programs for blocks.
                                                              (line  28)
* vectors, initializing:                 Accessing vector elements.
                                                              (line  75)
* vectors, range-checking:               Vector allocation.   (line  36)
* VEGAS Monte Carlo integration:         MISER.               (line 159)
* verbose (C variable):                  VEGAS.               (line 204)
* viscosity:                             Pressure.            (line  34)
* volume units:                          Printers Units.      (line  13)
* W function:                            Lambert W Functions. (line   6)
* warning options:                       Handling floating point exceptions.
                                                              (line  31)
* warranty (none):                       Obtaining GSL.       (line  25)
* wavelet transforms:                    References and Further Reading<26>.
                                                              (line  25)
* website, developer information:        Reporting Bugs.      (line  31)
* Weibull distribution:                  The Weibull Distribution.
                                                              (line   6)
* weight, units of:                      Volume Area and Length.
                                                              (line  41)
* weighted linear fits:                  References and Further Reading<32>.
                                                              (line  24)
* Wigner coefficients:                   Coupling Coefficients.
                                                              (line   6)
* Wishart random variates:               The Logarithmic Distribution.
                                                              (line  24)
* Y(x), Bessel Functions:                Irregular Cylindrical Bessel Functions.
                                                              (line   6)
* y(x), Bessel Functions:                Irregular Spherical Bessel Functions.
                                                              (line   6)
* zero finding:                          References and Further Reading<28>.
                                                              (line  15)
* zero matrix:                           Accessing matrix elements.
                                                              (line  49)
* zero, IEEE format:                     Representation of floating point numbers.
                                                              (line  30)
* Zeros of Regular Bessel Functions:     Zeros of Regular Bessel Functions.
                                                              (line   6)
* Zeta functions:                        Zeta Functions.      (line   6)
* Ziggurat method:                       The Gaussian Distribution.
                                                              (line  27)



Tag Table:
Node: Top324
Ref: index doc527
Ref: 0527
Node: Introduction26167
Ref: intro doc26258
Ref: 126258
Ref: intro gnu-scientific-library26258
Ref: 226258
Ref: intro introduction26258
Ref: 326258
Node: Routines available in GSL26819
Ref: intro routines-available-in-gsl26923
Ref: 426923
Node: GSL is Free Software29960
Ref: intro gsl-is-free-software30086
Ref: 530086
Ref: GSL is Free Software-Footnote-132299
Ref: GSL is Free Software-Footnote-232417
Node: Obtaining GSL32466
Ref: intro obtaining-gsl32578
Ref: 632578
Node: No Warranty33490
Ref: intro no-warranty33596
Ref: 733596
Ref: No Warranty-Footnote-134002
Node: Reporting Bugs34120
Ref: intro reporting-bugs34232
Ref: 834232
Ref: Reporting Bugs-Footnote-135076
Node: Further Information35130
Ref: intro further-information35262
Ref: 935262
Node: Conventions used in this manual36147
Ref: intro conventions-used-in-this-manual36256
Ref: a36256
Node: Using the Library36910
Ref: usage doc37012
Ref: b37012
Ref: usage using-the-library37012
Ref: c37012
Node: An Example Program37496
Ref: usage an-example-program37599
Ref: d37599
Ref: An Example Program-Footnote-138240
Node: Compiling and Linking38347
Ref: usage compiling-and-linking38475
Ref: e38475
Node: Linking programs with the library39456
Ref: usage linking-programs-with-the-library39597
Ref: f39597
Ref: Linking programs with the library-Footnote-140890
Ref: Linking programs with the library-Footnote-240926
Node: Linking with an alternative BLAS library40983
Ref: usage linking-with-an-alternative-blas-library41124
Ref: 1041124
Node: Shared Libraries42077
Ref: usage shared-libraries42204
Ref: 1242204
Ref: Shared Libraries-Footnote-143569
Node: ANSI C Compliance43620
Ref: usage ansi-c-compliance43742
Ref: 1343742
Node: Inline functions44712
Ref: usage inline-functions44829
Ref: 1444829
Ref: usage sec-inline-functions44829
Ref: 1544829
Node: Long double46291
Ref: usage long-double46412
Ref: 1746412
Node: Portability functions47919
Ref: usage id548055
Ref: 1848055
Ref: usage portability-functions48055
Ref: 1948055
Node: Alternative optimized functions49354
Ref: usage alternative-optimized-functions49514
Ref: 1b49514
Node: Support for different numeric types50816
Ref: usage support-for-different-numeric-types50977
Ref: 1c50977
Node: Compatibility with C++53921
Ref: usage compatibility-with-c54069
Ref: 1e54069
Node: Aliasing of arrays54502
Ref: usage aliasing-of-arrays54628
Ref: 1f54628
Ref: usage id654628
Ref: 2054628
Node: Thread-safety55257
Ref: usage thread-safety55381
Ref: 2155381
Node: Deprecated Functions56346
Ref: usage deprecated-functions56462
Ref: 2256462
Node: Code Reuse56975
Ref: usage code-reuse57069
Ref: 2357069
Node: Error Handling57628
Ref: err doc57740
Ref: 2457740
Ref: err error-handling57740
Ref: 2557740
Node: Error Reporting58363
Ref: err error-reporting58450
Ref: 2658450
Node: Error Codes60263
Ref: err error-codes60373
Ref: 2760373
Ref: err c GSL_EDOM60821
Ref: 2860821
Ref: err c GSL_ERANGE61033
Ref: 2961033
Ref: err c GSL_ENOMEM61236
Ref: 2a61236
Ref: err c GSL_EINVAL61538
Ref: 2b61538
Ref: err c gsl_strerror61847
Ref: 2c61847
Node: Error Handlers62201
Ref: err error-handlers62343
Ref: 2d62343
Ref: err c gsl_error_handler_t63282
Ref: 2e63282
Ref: err c gsl_set_error_handler64201
Ref: 2f64201
Ref: err c gsl_set_error_handler_off65280
Ref: 3065280
Node: Using GSL error reporting in your own functions65896
Ref: err using-gsl-error-reporting-in-your-own-functions66035
Ref: 3166035
Ref: err c GSL_ERROR66610
Ref: 3266610
Ref: err c GSL_ERROR_VAL67361
Ref: 3367361
Node: Examples67860
Ref: err examples67976
Ref: 3467976
Node: Mathematical Functions69069
Ref: math doc69179
Ref: 3669179
Ref: math mathematical-functions69179
Ref: 3769179
Node: Mathematical Constants69831
Ref: math mathematical-constants69949
Ref: 3869949
Node: Infinities and Not-a-number72103
Ref: math infinities-and-not-a-number72250
Ref: 3972250
Ref: math c GSL_POSINF72315
Ref: 3a72315
Ref: math c GSL_NEGINF72476
Ref: 3b72476
Ref: math c GSL_NAN72637
Ref: 3c72637
Ref: math c gsl_isnan72797
Ref: 3d72797
Ref: math c gsl_isinf72901
Ref: 3e72901
Ref: math c gsl_finite73073
Ref: 3f73073
Ref: Infinities and Not-a-number-Footnote-173261
Node: Elementary Functions73697
Ref: math elementary-functions73842
Ref: 4073842
Ref: math c gsl_log1p74222
Ref: 4174222
Ref: math c gsl_expm174447
Ref: 4274447
Ref: math c gsl_hypot74672
Ref: 1a74672
Ref: math c gsl_hypot374908
Ref: 4374908
Ref: math c gsl_acosh75101
Ref: 4475101
Ref: math c gsl_asinh75286
Ref: 4575286
Ref: math c gsl_atanh75471
Ref: 4675471
Ref: math c gsl_ldexp75656
Ref: 4775656
Ref: math c gsl_frexp75838
Ref: 4875838
Node: Small integer powers76230
Ref: math small-integer-powers76375
Ref: 4976375
Ref: math c gsl_pow_int76690
Ref: 4a76690
Ref: math c gsl_pow_uint76741
Ref: 4b76741
Ref: math c gsl_pow_277132
Ref: 4d77132
Ref: math c gsl_pow_377180
Ref: 4e77180
Ref: math c gsl_pow_477228
Ref: 4f77228
Ref: math c gsl_pow_577276
Ref: 5077276
Ref: math c gsl_pow_677324
Ref: 5177324
Ref: math c gsl_pow_777372
Ref: 5277372
Ref: math c gsl_pow_877420
Ref: 5377420
Ref: math c gsl_pow_977468
Ref: 5477468
Node: Testing the Sign of Numbers77912
Ref: math testing-the-sign-of-numbers78069
Ref: 5578069
Ref: math c GSL_SIGN78134
Ref: 5678134
Node: Testing for Odd and Even Numbers78356
Ref: math testing-for-odd-and-even-numbers78522
Ref: 5778522
Ref: math c GSL_IS_ODD78597
Ref: 5878597
Ref: math c GSL_IS_EVEN78752
Ref: 5978752
Node: Maximum and Minimum functions78958
Ref: math maximum-and-minimum-functions79145
Ref: 5a79145
Ref: math c GSL_MAX79407
Ref: 5b79407
Ref: math c GSL_MIN79547
Ref: 5c79547
Ref: math c GSL_MAX_DBL79687
Ref: 5d79687
Ref: math c GSL_MIN_DBL80106
Ref: 5e80106
Ref: math c GSL_MAX_INT80525
Ref: 5f80525
Ref: math c GSL_MIN_INT80584
Ref: 6080584
Ref: math c GSL_MAX_LDBL80915
Ref: 6180915
Ref: math c GSL_MIN_LDBL81009
Ref: 6281009
Node: Approximate Comparison of Floating Point Numbers81379
Ref: math approximate-comparison-of-floating-point-numbers81525
Ref: 6381525
Ref: math c gsl_fcmp81941
Ref: 6481941
Node: Complex Numbers82772
Ref: complex doc82879
Ref: 6582879
Ref: complex complex-numbers82879
Ref: 6682879
Ref: Complex Numbers-Footnote-184047
Node: Representation of complex numbers84111
Ref: complex representation-of-complex-numbers84235
Ref: 6784235
Ref: complex c gsl_complex_rect85008
Ref: 6885008
Ref: complex c gsl_complex_polar85272
Ref: 6985272
Ref: complex c GSL_REAL85510
Ref: 6a85510
Ref: complex c GSL_IMAG85534
Ref: 6b85534
Ref: complex c GSL_SET_COMPLEX85650
Ref: 6c85650
Ref: complex c GSL_SET_REAL85925
Ref: 6d85925
Ref: complex c GSL_SET_IMAG85957
Ref: 6e85957
Node: Properties of complex numbers86119
Ref: complex properties-of-complex-numbers86280
Ref: 6f86280
Ref: complex c gsl_complex_arg86349
Ref: 7086349
Ref: complex c gsl_complex_abs86518
Ref: 7186518
Ref: complex c gsl_complex_abs286650
Ref: 7286650
Ref: complex c gsl_complex_logabs86798
Ref: 7386798
Node: Complex arithmetic operators87157
Ref: complex complex-arithmetic-operators87313
Ref: 7487313
Ref: complex c gsl_complex_add87380
Ref: 7587380
Ref: complex c gsl_complex_sub87546
Ref: 7687546
Ref: complex c gsl_complex_mul87719
Ref: 7787719
Ref: complex c gsl_complex_div87888
Ref: 7887888
Ref: complex c gsl_complex_add_real88059
Ref: 7988059
Ref: complex c gsl_complex_sub_real88240
Ref: 7a88240
Ref: complex c gsl_complex_mul_real88428
Ref: 7b88428
Ref: complex c gsl_complex_div_real88612
Ref: 7c88612
Ref: complex c gsl_complex_add_imag88798
Ref: 7d88798
Ref: complex c gsl_complex_sub_imag88980
Ref: 7e88980
Ref: complex c gsl_complex_mul_imag89169
Ref: 7f89169
Ref: complex c gsl_complex_div_imag89357
Ref: 8089357
Ref: complex c gsl_complex_conjugate89546
Ref: 8189546
Ref: complex c gsl_complex_inverse89712
Ref: 8289712
Ref: complex c gsl_complex_negative89896
Ref: 8389896
Node: Elementary Complex Functions90056
Ref: complex elementary-complex-functions90214
Ref: 8490214
Ref: complex c gsl_complex_sqrt90281
Ref: 8590281
Ref: complex c gsl_complex_sqrt_real90542
Ref: 8690542
Ref: complex c gsl_complex_pow90718
Ref: 8790718
Ref: complex c gsl_complex_pow_real90986
Ref: 8890986
Ref: complex c gsl_complex_exp91159
Ref: 8991159
Ref: complex c gsl_complex_log91315
Ref: 8a91315
Ref: complex c gsl_complex_log1091534
Ref: 8b91534
Ref: complex c gsl_complex_log_b91704
Ref: 8c91704
Node: Complex Trigonometric Functions91962
Ref: complex complex-trigonometric-functions92131
Ref: 8d92131
Ref: complex c gsl_complex_sin92204
Ref: 8e92204
Ref: complex c gsl_complex_cos92383
Ref: 8f92383
Ref: complex c gsl_complex_tan92561
Ref: 9092561
Ref: complex c gsl_complex_sec92731
Ref: 9192731
Ref: complex c gsl_complex_csc92894
Ref: 9292894
Ref: complex c gsl_complex_cot93059
Ref: 9393059
Node: Inverse Complex Trigonometric Functions93225
Ref: complex inverse-complex-trigonometric-functions93394
Ref: 9493394
Ref: complex c gsl_complex_arcsin93483
Ref: 9593483
Ref: complex c gsl_complex_arcsin_real93718
Ref: 9693718
Ref: complex c gsl_complex_arccos94158
Ref: 9794158
Ref: complex c gsl_complex_arccos_real94395
Ref: 9894395
Ref: complex c gsl_complex_arctan94800
Ref: 9994800
Ref: complex c gsl_complex_arcsec95032
Ref: 9a95032
Ref: complex c gsl_complex_arcsec_real95207
Ref: 9b95207
Ref: complex c gsl_complex_arccsc95379
Ref: 9c95379
Ref: complex c gsl_complex_arccsc_real95556
Ref: 9d95556
Ref: complex c gsl_complex_arccot95730
Ref: 9e95730
Node: Complex Hyperbolic Functions95908
Ref: complex complex-hyperbolic-functions96082
Ref: 9f96082
Ref: complex c gsl_complex_sinh96149
Ref: a096149
Ref: complex c gsl_complex_cosh96336
Ref: a196336
Ref: complex c gsl_complex_tanh96525
Ref: a296525
Ref: complex c gsl_complex_sech96710
Ref: a396710
Ref: complex c gsl_complex_csch96887
Ref: a496887
Ref: complex c gsl_complex_coth97066
Ref: a597066
Node: Inverse Complex Hyperbolic Functions97246
Ref: complex inverse-complex-hyperbolic-functions97411
Ref: a697411
Ref: complex c gsl_complex_arcsinh97494
Ref: a797494
Ref: complex c gsl_complex_arccosh97736
Ref: a897736
Ref: complex c gsl_complex_arccosh_real98112
Ref: a998112
Ref: complex c gsl_complex_arctanh98282
Ref: aa98282
Ref: complex c gsl_complex_arctanh_real98533
Ref: ab98533
Ref: complex c gsl_complex_arcsech98704
Ref: ac98704
Ref: complex c gsl_complex_arccsch98893
Ref: ad98893
Ref: complex c gsl_complex_arccoth99084
Ref: ae99084
Node: References and Further Reading99276
Ref: complex references-and-further-reading99404
Ref: af99404
Node: Polynomials100686
Ref: poly doc100788
Ref: b0100788
Ref: poly polynomials100788
Ref: b1100788
Node: Polynomial Evaluation101461
Ref: poly polynomial-evaluation101588
Ref: b2101588
Ref: poly c gsl_poly_eval101875
Ref: b3101875
Ref: poly c gsl_poly_complex_eval102073
Ref: b4102073
Ref: poly c gsl_complex_poly_complex_eval102292
Ref: b5102292
Ref: poly c gsl_poly_eval_derivs102527
Ref: b6102527
Node: Divided Difference Representation of Polynomials102931
Ref: poly divided-difference-representation-of-polynomials103086
Ref: b7103086
Ref: poly c gsl_poly_dd_init104413
Ref: b8104413
Ref: poly c gsl_poly_dd_eval104902
Ref: b9104902
Ref: poly c gsl_poly_dd_taylor105276
Ref: ba105276
Ref: poly c gsl_poly_dd_hermite_init105875
Ref: bb105875
Node: Quadratic Equations106993
Ref: poly quadratic-equations107142
Ref: bc107142
Ref: poly c gsl_poly_solve_quadratic107191
Ref: bd107191
Ref: poly c gsl_poly_complex_solve_quadratic108410
Ref: be108410
Node: Cubic Equations108995
Ref: poly cubic-equations109124
Ref: bf109124
Ref: poly c gsl_poly_solve_cubic109165
Ref: c0109165
Ref: poly c gsl_poly_complex_solve_cubic110110
Ref: c1110110
Node: General Polynomial Equations110643
Ref: poly general-polynomial-equations110764
Ref: c2110764
Ref: poly c gsl_poly_complex_workspace111104
Ref: c3111104
Ref: poly c gsl_poly_complex_workspace_alloc111234
Ref: c4111234
Ref: poly c gsl_poly_complex_workspace_free111741
Ref: c6111741
Ref: poly c gsl_poly_complex_solve111921
Ref: c5111921
Node: Examples<2>113195
Ref: poly examples113334
Ref: c7113334
Node: References and Further Reading<2>114523
Ref: poly references-and-further-reading114625
Ref: c8114625
Node: Special Functions115543
Ref: specfunc doc115650
Ref: c9115650
Ref: specfunc special-functions115650
Ref: ca115650
Node: Usage117642
Ref: specfunc usage117735
Ref: cb117735
Node: The gsl_sf_result struct118814
Ref: specfunc the-gsl-sf-result-struct118921
Ref: cc118921
Ref: specfunc c gsl_sf_result119309
Ref: cd119309
Ref: specfunc c gsl_sf_result_e10119991
Ref: ce119991
Node: Modes120161
Ref: specfunc modes120293
Ref: cf120293
Ref: specfunc c gsl_mode_t120769
Ref: d0120769
Ref: specfunc c GSL_PREC_DOUBLE120791
Ref: d1120791
Ref: specfunc c GSL_PREC_SINGLE120914
Ref: d2120914
Ref: specfunc c GSL_PREC_APPROX121032
Ref: d3121032
Node: Airy Functions and Derivatives121234
Ref: specfunc airy-functions-and-derivatives121358
Ref: d4121358
Node: Airy Functions121917
Ref: specfunc airy-functions122037
Ref: d5122037
Ref: specfunc c gsl_sf_airy_Ai122080
Ref: d6122080
Ref: specfunc c gsl_sf_airy_Ai_e122144
Ref: d7122144
Ref: specfunc c gsl_sf_airy_Bi122346
Ref: d9122346
Ref: specfunc c gsl_sf_airy_Bi_e122410
Ref: da122410
Ref: specfunc c gsl_sf_airy_Ai_scaled122612
Ref: db122612
Ref: specfunc c gsl_sf_airy_Ai_scaled_e122683
Ref: dc122683
Ref: specfunc c gsl_sf_airy_Bi_scaled122961
Ref: dd122961
Ref: specfunc c gsl_sf_airy_Bi_scaled_e123032
Ref: de123032
Node: Derivatives of Airy Functions123309
Ref: specfunc derivatives-of-airy-functions123461
Ref: df123461
Ref: specfunc c gsl_sf_airy_Ai_deriv123534
Ref: e0123534
Ref: specfunc c gsl_sf_airy_Ai_deriv_e123604
Ref: e1123604
Ref: specfunc c gsl_sf_airy_Bi_deriv123824
Ref: e2123824
Ref: specfunc c gsl_sf_airy_Bi_deriv_e123894
Ref: e3123894
Ref: specfunc c gsl_sf_airy_Ai_deriv_scaled124114
Ref: e4124114
Ref: specfunc c gsl_sf_airy_Ai_deriv_scaled_e124201
Ref: e5124201
Ref: specfunc c gsl_sf_airy_Bi_deriv_scaled124484
Ref: e6124484
Ref: specfunc c gsl_sf_airy_Bi_deriv_scaled_e124571
Ref: e7124571
Node: Zeros of Airy Functions124853
Ref: specfunc zeros-of-airy-functions125029
Ref: e8125029
Ref: specfunc c gsl_sf_airy_zero_Ai125090
Ref: e9125090
Ref: specfunc c gsl_sf_airy_zero_Ai_e125148
Ref: ea125148
Ref: specfunc c gsl_sf_airy_zero_Bi125338
Ref: eb125338
Ref: specfunc c gsl_sf_airy_zero_Bi_e125396
Ref: ec125396
Node: Zeros of Derivatives of Airy Functions125586
Ref: specfunc zeros-of-derivatives-of-airy-functions125724
Ref: ed125724
Ref: specfunc c gsl_sf_airy_zero_Ai_deriv125815
Ref: ee125815
Ref: specfunc c gsl_sf_airy_zero_Ai_deriv_e125879
Ref: ef125879
Ref: specfunc c gsl_sf_airy_zero_Bi_deriv126087
Ref: f0126087
Ref: specfunc c gsl_sf_airy_zero_Bi_deriv_e126151
Ref: f1126151
Node: Bessel Functions126359
Ref: specfunc bessel-functions126495
Ref: f2126495
Node: Regular Cylindrical Bessel Functions127555
Ref: specfunc regular-cylindrical-bessel-functions127692
Ref: f3127692
Ref: specfunc c gsl_sf_bessel_J0127779
Ref: f4127779
Ref: specfunc c gsl_sf_bessel_J0_e127828
Ref: f5127828
Ref: specfunc c gsl_sf_bessel_J1128000
Ref: f6128000
Ref: specfunc c gsl_sf_bessel_J1_e128049
Ref: f7128049
Ref: specfunc c gsl_sf_bessel_Jn128220
Ref: f8128220
Ref: specfunc c gsl_sf_bessel_Jn_e128276
Ref: f9128276
Ref: specfunc c gsl_sf_bessel_Jn_array128466
Ref: fa128466
Node: Irregular Cylindrical Bessel Functions128907
Ref: specfunc irregular-cylindrical-bessel-functions129098
Ref: fb129098
Ref: specfunc c gsl_sf_bessel_Y0129189
Ref: fc129189
Ref: specfunc c gsl_sf_bessel_Y0_e129238
Ref: fd129238
Ref: specfunc c gsl_sf_bessel_Y1129421
Ref: fe129421
Ref: specfunc c gsl_sf_bessel_Y1_e129470
Ref: ff129470
Ref: specfunc c gsl_sf_bessel_Yn129652
Ref: 100129652
Ref: specfunc c gsl_sf_bessel_Yn_e129708
Ref: 101129708
Ref: specfunc c gsl_sf_bessel_Yn_array129909
Ref: 102129909
Node: Regular Modified Cylindrical Bessel Functions130393
Ref: specfunc regular-modified-cylindrical-bessel-functions130595
Ref: 103130595
Ref: specfunc c gsl_sf_bessel_I0130700
Ref: 104130700
Ref: specfunc c gsl_sf_bessel_I0_e130749
Ref: 105130749
Ref: specfunc c gsl_sf_bessel_I1130930
Ref: 106130930
Ref: specfunc c gsl_sf_bessel_I1_e130979
Ref: 107130979
Ref: specfunc c gsl_sf_bessel_In131159
Ref: 108131159
Ref: specfunc c gsl_sf_bessel_In_e131215
Ref: 109131215
Ref: specfunc c gsl_sf_bessel_In_array131414
Ref: 10a131414
Ref: specfunc c gsl_sf_bessel_I0_scaled131930
Ref: 10b131930
Ref: specfunc c gsl_sf_bessel_I0_scaled_e131986
Ref: 10c131986
Ref: specfunc c gsl_sf_bessel_I1_scaled132201
Ref: 10d132201
Ref: specfunc c gsl_sf_bessel_I1_scaled_e132257
Ref: 10e132257
Ref: specfunc c gsl_sf_bessel_In_scaled132471
Ref: 10f132471
Ref: specfunc c gsl_sf_bessel_In_scaled_e132534
Ref: 110132534
Ref: specfunc c gsl_sf_bessel_In_scaled_array132757
Ref: 111132757
Node: Irregular Modified Cylindrical Bessel Functions133289
Ref: specfunc irregular-modified-cylindrical-bessel-functions133487
Ref: 112133487
Ref: specfunc c gsl_sf_bessel_K0133596
Ref: 113133596
Ref: specfunc c gsl_sf_bessel_K0_e133645
Ref: 114133645
Ref: specfunc c gsl_sf_bessel_K1133839
Ref: 115133839
Ref: specfunc c gsl_sf_bessel_K1_e133888
Ref: 116133888
Ref: specfunc c gsl_sf_bessel_Kn134081
Ref: 117134081
Ref: specfunc c gsl_sf_bessel_Kn_e134137
Ref: 118134137
Ref: specfunc c gsl_sf_bessel_Kn_array134349
Ref: 119134349
Ref: specfunc c gsl_sf_bessel_K0_scaled134908
Ref: 11a134908
Ref: specfunc c gsl_sf_bessel_K0_scaled_e134964
Ref: 11b134964
Ref: specfunc c gsl_sf_bessel_K1_scaled135186
Ref: 11c135186
Ref: specfunc c gsl_sf_bessel_K1_scaled_e135242
Ref: 11d135242
Ref: specfunc c gsl_sf_bessel_Kn_scaled135463
Ref: 11e135463
Ref: specfunc c gsl_sf_bessel_Kn_scaled_e135526
Ref: 11f135526
Ref: specfunc c gsl_sf_bessel_Kn_scaled_array135758
Ref: 120135758
Node: Regular Spherical Bessel Functions136329
Ref: specfunc regular-spherical-bessel-functions136518
Ref: 121136518
Ref: specfunc c gsl_sf_bessel_j0136601
Ref: 122136601
Ref: specfunc c gsl_sf_bessel_j0_e136650
Ref: 123136650
Ref: specfunc c gsl_sf_bessel_j1136832
Ref: 124136832
Ref: specfunc c gsl_sf_bessel_j1_e136881
Ref: 125136881
Ref: specfunc c gsl_sf_bessel_j2137076
Ref: 126137076
Ref: specfunc c gsl_sf_bessel_j2_e137125
Ref: 127137125
Ref: specfunc c gsl_sf_bessel_jl137333
Ref: 128137333
Ref: specfunc c gsl_sf_bessel_jl_e137389
Ref: 129137389
Ref: specfunc c gsl_sf_bessel_jl_array137604
Ref: 12a137604
Ref: specfunc c gsl_sf_bessel_jl_steed_array138053
Ref: 12b138053
Node: Irregular Spherical Bessel Functions138595
Ref: specfunc irregular-spherical-bessel-functions138780
Ref: 12c138780
Ref: specfunc c gsl_sf_bessel_y0138867
Ref: 12d138867
Ref: specfunc c gsl_sf_bessel_y0_e138916
Ref: 12e138916
Ref: specfunc c gsl_sf_bessel_y1139101
Ref: 12f139101
Ref: specfunc c gsl_sf_bessel_y1_e139150
Ref: 130139150
Ref: specfunc c gsl_sf_bessel_y2139348
Ref: 131139348
Ref: specfunc c gsl_sf_bessel_y2_e139397
Ref: 132139397
Ref: specfunc c gsl_sf_bessel_yl139610
Ref: 133139610
Ref: specfunc c gsl_sf_bessel_yl_e139666
Ref: 134139666
Ref: specfunc c gsl_sf_bessel_yl_array139870
Ref: 135139870
Node: Regular Modified Spherical Bessel Functions140308
Ref: specfunc regular-modified-spherical-bessel-functions140504
Ref: 136140504
Ref: specfunc c gsl_sf_bessel_i0_scaled140765
Ref: 137140765
Ref: specfunc c gsl_sf_bessel_i0_scaled_e140821
Ref: 138140821
Ref: specfunc c gsl_sf_bessel_i1_scaled141035
Ref: 139141035
Ref: specfunc c gsl_sf_bessel_i1_scaled_e141091
Ref: 13a141091
Ref: specfunc c gsl_sf_bessel_i2_scaled141304
Ref: 13b141304
Ref: specfunc c gsl_sf_bessel_i2_scaled_e141360
Ref: 13c141360
Ref: specfunc c gsl_sf_bessel_il_scaled141573
Ref: 13d141573
Ref: specfunc c gsl_sf_bessel_il_scaled_e141636
Ref: 13e141636
Ref: specfunc c gsl_sf_bessel_il_scaled_array141857
Ref: 13f141857
Node: Irregular Modified Spherical Bessel Functions142332
Ref: specfunc irregular-modified-spherical-bessel-functions142534
Ref: 140142534
Ref: specfunc c gsl_sf_bessel_k0_scaled142812
Ref: 141142812
Ref: specfunc c gsl_sf_bessel_k0_scaled_e142868
Ref: 142142868
Ref: specfunc c gsl_sf_bessel_k1_scaled143090
Ref: 143143090
Ref: specfunc c gsl_sf_bessel_k1_scaled_e143146
Ref: 144143146
Ref: specfunc c gsl_sf_bessel_k2_scaled143367
Ref: 145143367
Ref: specfunc c gsl_sf_bessel_k2_scaled_e143423
Ref: 146143423
Ref: specfunc c gsl_sf_bessel_kl_scaled143645
Ref: 147143645
Ref: specfunc c gsl_sf_bessel_kl_scaled_e143708
Ref: 148143708
Ref: specfunc c gsl_sf_bessel_kl_scaled_array143938
Ref: 149143938
Node: Regular Bessel Function—Fractional Order144420
Ref: specfunc regular-bessel-function-fractional-order144624
Ref: 14a144624
Ref: specfunc c gsl_sf_bessel_Jnu144721
Ref: 14b144721
Ref: specfunc c gsl_sf_bessel_Jnu_e144782
Ref: 14c144782
Ref: specfunc c gsl_sf_bessel_sequence_Jnu_e144986
Ref: 14d144986
Node: Irregular Bessel Functions—Fractional Order145432
Ref: specfunc irregular-bessel-functions-fractional-order145643
Ref: 14e145643
Ref: specfunc c gsl_sf_bessel_Ynu145748
Ref: 14f145748
Ref: specfunc c gsl_sf_bessel_Ynu_e145809
Ref: 150145809
Node: Regular Modified Bessel Functions—Fractional Order146015
Ref: specfunc regular-modified-bessel-functions-fractional-order146238
Ref: 151146238
Ref: specfunc c gsl_sf_bessel_Inu146357
Ref: 152146357
Ref: specfunc c gsl_sf_bessel_Inu_e146418
Ref: 153146418
Ref: specfunc c gsl_sf_bessel_Inu_scaled146634
Ref: 154146634
Ref: specfunc c gsl_sf_bessel_Inu_scaled_e146702
Ref: 155146702
Node: Irregular Modified Bessel Functions—Fractional Order146942
Ref: specfunc irregular-modified-bessel-functions-fractional-order147153
Ref: 156147153
Ref: specfunc c gsl_sf_bessel_Knu147276
Ref: 157147276
Ref: specfunc c gsl_sf_bessel_Knu_e147337
Ref: 158147337
Ref: specfunc c gsl_sf_bessel_lnKnu147555
Ref: 159147555
Ref: specfunc c gsl_sf_bessel_lnKnu_e147618
Ref: 15a147618
Ref: specfunc c gsl_sf_bessel_Knu_scaled147865
Ref: 15b147865
Ref: specfunc c gsl_sf_bessel_Knu_scaled_e147933
Ref: 15c147933
Node: Zeros of Regular Bessel Functions148181
Ref: specfunc zeros-of-regular-bessel-functions148331
Ref: 15d148331
Ref: specfunc c gsl_sf_bessel_zero_J0148414
Ref: 15e148414
Ref: specfunc c gsl_sf_bessel_zero_J0_e148474
Ref: 15f148474
Ref: specfunc c gsl_sf_bessel_zero_J1148678
Ref: 160148678
Ref: specfunc c gsl_sf_bessel_zero_J1_e148738
Ref: 161148738
Ref: specfunc c gsl_sf_bessel_zero_Jnu148942
Ref: 162148942
Ref: specfunc c gsl_sf_bessel_zero_Jnu_e149014
Ref: 163149014
Node: Clausen Functions149311
Ref: specfunc clausen-functions149434
Ref: 164149434
Ref: specfunc c gsl_sf_clausen149749
Ref: 166149749
Ref: specfunc c gsl_sf_clausen_e149796
Ref: 167149796
Node: Coulomb Functions149926
Ref: specfunc coulomb-functions150054
Ref: 168150054
Node: Normalized Hydrogenic Bound States150379
Ref: specfunc normalized-hydrogenic-bound-states150499
Ref: 169150499
Ref: specfunc c gsl_sf_hydrogenicR_1150582
Ref: 16a150582
Ref: specfunc c gsl_sf_hydrogenicR_1_e150645
Ref: 16b150645
Ref: specfunc c gsl_sf_hydrogenicR150878
Ref: 16c150878
Ref: specfunc c gsl_sf_hydrogenicR_e150963
Ref: 16d150963
Node: Coulomb Wave Functions151476
Ref: specfunc coulomb-wave-functions151649
Ref: 16f151649
Ref: specfunc c gsl_sf_coulomb_wave_FG_e152309
Ref: 170152309
Ref: specfunc c gsl_sf_coulomb_wave_F_array153124
Ref: 171153124
Ref: specfunc c gsl_sf_coulomb_wave_FG_array153485
Ref: 172153485
Ref: specfunc c gsl_sf_coulomb_wave_FGp_array153944
Ref: 173153944
Ref: specfunc c gsl_sf_coulomb_wave_sphF_array154542
Ref: 174154542
Node: Coulomb Wave Function Normalization Constant155031
Ref: specfunc coulomb-wave-function-normalization-constant155161
Ref: 175155161
Ref: specfunc c gsl_sf_coulomb_CL_e155347
Ref: 176155347
Ref: specfunc c gsl_sf_coulomb_CL_array155548
Ref: 177155548
Node: Coupling Coefficients155786
Ref: specfunc coupling-coefficients155912
Ref: 178155912
Node: 3-j Symbols156513
Ref: specfunc j-symbols156603
Ref: 179156603
Ref: specfunc c gsl_sf_coupling_3j156640
Ref: 17a156640
Ref: specfunc c gsl_sf_coupling_3j_e156763
Ref: 17b156763
Node: 6-j Symbols157137
Ref: specfunc id1157247
Ref: 17c157247
Ref: specfunc c gsl_sf_coupling_6j157284
Ref: 17d157284
Ref: specfunc c gsl_sf_coupling_6j_e157407
Ref: 17e157407
Node: 9-j Symbols157781
Ref: specfunc id2157871
Ref: 17f157871
Ref: specfunc c gsl_sf_coupling_9j157908
Ref: 180157908
Ref: specfunc c gsl_sf_coupling_9j_e158077
Ref: 181158077
Node: Dawson Function158511
Ref: specfunc dawson-function158635
Ref: 182158635
Ref: specfunc c gsl_sf_dawson158907
Ref: 183158907
Ref: specfunc c gsl_sf_dawson_e158953
Ref: 184158953
Node: Debye Functions159098
Ref: specfunc debye-functions159212
Ref: 185159212
Ref: specfunc c gsl_sf_debye_1159510
Ref: 186159510
Ref: specfunc c gsl_sf_debye_1_e159557
Ref: 187159557
Ref: specfunc c gsl_sf_debye_2159696
Ref: 188159696
Ref: specfunc c gsl_sf_debye_2_e159743
Ref: 189159743
Ref: specfunc c gsl_sf_debye_3159883
Ref: 18a159883
Ref: specfunc c gsl_sf_debye_3_e159930
Ref: 18b159930
Ref: specfunc c gsl_sf_debye_4160069
Ref: 18c160069
Ref: specfunc c gsl_sf_debye_4_e160116
Ref: 18d160116
Ref: specfunc c gsl_sf_debye_5160256
Ref: 18e160256
Ref: specfunc c gsl_sf_debye_5_e160303
Ref: 18f160303
Ref: specfunc c gsl_sf_debye_6160442
Ref: 190160442
Ref: specfunc c gsl_sf_debye_6_e160489
Ref: 191160489
Node: Dilogarithm160628
Ref: specfunc dilog-function160748
Ref: 165160748
Ref: specfunc dilogarithm160748
Ref: 192160748
Node: Real Argument161001
Ref: specfunc real-argument161088
Ref: 193161088
Ref: specfunc c gsl_sf_dilog161131
Ref: 194161131
Ref: specfunc c gsl_sf_dilog_e161176
Ref: 195161176
Node: Complex Argument161708
Ref: specfunc complex-argument161795
Ref: 196161795
Ref: specfunc c gsl_sf_complex_dilog_e161844
Ref: 197161844
Node: Elementary Operations162191
Ref: specfunc elementary-operations162314
Ref: 198162314
Ref: specfunc c gsl_sf_multiply162549
Ref: 199162549
Ref: specfunc c gsl_sf_multiply_e162607
Ref: 19a162607
Ref: specfunc c gsl_sf_multiply_err_e162817
Ref: 19b162817
Node: Elliptic Integrals163127
Ref: specfunc elliptic-integrals163264
Ref: 19c163264
Node: Definition of Legendre Forms163701
Ref: specfunc definition-of-legendre-forms163821
Ref: 19d163821
Node: Definition of Carlson Forms164516
Ref: specfunc definition-of-carlson-forms164689
Ref: 19e164689
Node: Legendre Form of Complete Elliptic Integrals165202
Ref: specfunc legendre-form-of-complete-elliptic-integrals165393
Ref: 19f165393
Ref: specfunc c gsl_sf_ellint_Kcomp165498
Ref: 1a0165498
Ref: specfunc c gsl_sf_ellint_Kcomp_e165567
Ref: 1a1165567
Ref: specfunc c gsl_sf_ellint_Ecomp165900
Ref: 1a2165900
Ref: specfunc c gsl_sf_ellint_Ecomp_e165969
Ref: 1a3165969
Ref: specfunc c gsl_sf_ellint_Pcomp166302
Ref: 1a4166302
Ref: specfunc c gsl_sf_ellint_Pcomp_e166391
Ref: 1a5166391
Node: Legendre Form of Incomplete Elliptic Integrals166803
Ref: specfunc legendre-form-of-incomplete-elliptic-integrals166980
Ref: 1a6166980
Ref: specfunc c gsl_sf_ellint_F167089
Ref: 1a7167089
Ref: specfunc c gsl_sf_ellint_F_e167176
Ref: 1a8167176
Ref: specfunc c gsl_sf_ellint_E167523
Ref: 1a9167523
Ref: specfunc c gsl_sf_ellint_E_e167610
Ref: 1aa167610
Ref: specfunc c gsl_sf_ellint_P167957
Ref: 1ab167957
Ref: specfunc c gsl_sf_ellint_P_e168054
Ref: 1ac168054
Ref: specfunc c gsl_sf_ellint_D168481
Ref: 1ad168481
Ref: specfunc c gsl_sf_ellint_D_e168568
Ref: 1ae168568
Node: Carlson Forms168928
Ref: specfunc carlson-forms169052
Ref: 1af169052
Ref: specfunc c gsl_sf_ellint_RC169095
Ref: 1b0169095
Ref: specfunc c gsl_sf_ellint_RC_e169181
Ref: 1b1169181
Ref: specfunc c gsl_sf_ellint_RD169429
Ref: 1b2169429
Ref: specfunc c gsl_sf_ellint_RD_e169525
Ref: 1b3169525
Ref: specfunc c gsl_sf_ellint_RF169785
Ref: 1b4169785
Ref: specfunc c gsl_sf_ellint_RF_e169881
Ref: 1b5169881
Ref: specfunc c gsl_sf_ellint_RJ170141
Ref: 1b6170141
Ref: specfunc c gsl_sf_ellint_RJ_e170247
Ref: 1b7170247
Node: Elliptic Functions Jacobi170519
Ref: specfunc elliptic-functions-jacobi170650
Ref: 1b8170650
Ref: specfunc c gsl_sf_elljac_e170868
Ref: 1b9170868
Node: Error Functions171103
Ref: specfunc error-functions171237
Ref: 1ba171237
Node: Error Function171556
Ref: specfunc error-function171660
Ref: 1bb171660
Ref: specfunc c gsl_sf_erf171705
Ref: 1bc171705
Ref: specfunc c gsl_sf_erf_e171748
Ref: 1bd171748
Node: Complementary Error Function171932
Ref: specfunc complementary-error-function172077
Ref: 1be172077
Ref: specfunc c gsl_sf_erfc172150
Ref: 1bf172150
Ref: specfunc c gsl_sf_erfc_e172194
Ref: 1c0172194
Node: Log Complementary Error Function172394
Ref: specfunc log-complementary-error-function172546
Ref: 1c1172546
Ref: specfunc c gsl_sf_log_erfc172627
Ref: 1c2172627
Ref: specfunc c gsl_sf_log_erfc_e172675
Ref: 1c3172675
Node: Probability functions172847
Ref: specfunc probability-functions172962
Ref: 1c4172962
Ref: specfunc c gsl_sf_erf_Z173140
Ref: 1c5173140
Ref: specfunc c gsl_sf_erf_Z_e173185
Ref: 1c6173185
Ref: specfunc c gsl_sf_erf_Q173366
Ref: 1c7173366
Ref: specfunc c gsl_sf_erf_Q_e173411
Ref: 1c8173411
Ref: specfunc c gsl_sf_hazard173901
Ref: 1c9173901
Ref: specfunc c gsl_sf_hazard_e173947
Ref: 1ca173947
Node: Exponential Functions174100
Ref: specfunc exponential-functions174230
Ref: 1cb174230
Node: Exponential Function174487
Ref: specfunc exponential-function174605
Ref: 1cc174605
Ref: specfunc c gsl_sf_exp174662
Ref: 1cd174662
Ref: specfunc c gsl_sf_exp_e174705
Ref: 1ce174705
Ref: specfunc c gsl_sf_exp_e10_e174878
Ref: 1cf174878
Ref: specfunc c gsl_sf_exp_mult175217
Ref: 1d0175217
Ref: specfunc c gsl_sf_exp_mult_e175275
Ref: 1d1175275
Ref: specfunc c gsl_sf_exp_mult_e10_e175482
Ref: 1d2175482
Node: Relative Exponential Functions175747
Ref: specfunc relative-exponential-functions175908
Ref: 1d3175908
Ref: specfunc c gsl_sf_expm1175985
Ref: 1d4175985
Ref: specfunc c gsl_sf_expm1_e176030
Ref: 1d5176030
Ref: specfunc c gsl_sf_exprel176205
Ref: 1d6176205
Ref: specfunc c gsl_sf_exprel_e176251
Ref: 1d7176251
Ref: specfunc c gsl_sf_exprel_2176565
Ref: 1d8176565
Ref: specfunc c gsl_sf_exprel_2_e176613
Ref: 1d9176613
Ref: specfunc c gsl_sf_exprel_n176939
Ref: 1da176939
Ref: specfunc c gsl_sf_exprel_n_e176994
Ref: 1db176994
Node: Exponentiation With Error Estimate177481
Ref: specfunc exponentiation-with-error-estimate177613
Ref: 1dc177613
Ref: specfunc c gsl_sf_exp_err_e177698
Ref: 1dd177698
Ref: specfunc c gsl_sf_exp_err_e10_e177881
Ref: 1de177881
Ref: specfunc c gsl_sf_exp_mult_err_e178171
Ref: 1df178171
Ref: specfunc c gsl_sf_exp_mult_err_e10_e178432
Ref: 1e0178432
Node: Exponential Integrals178789
Ref: specfunc exponential-integrals178924
Ref: 1e1178924
Node: Exponential Integral179287
Ref: specfunc exponential-integral179379
Ref: 1e2179379
Ref: specfunc c gsl_sf_expint_E1179436
Ref: 1e3179436
Ref: specfunc c gsl_sf_expint_E1_e179485
Ref: 1e4179485
Ref: specfunc c gsl_sf_expint_E2179675
Ref: 1e5179675
Ref: specfunc c gsl_sf_expint_E2_e179724
Ref: 1e6179724
Ref: specfunc c gsl_sf_expint_En179933
Ref: 1e7179933
Ref: specfunc c gsl_sf_expint_En_e179989
Ref: 1e8179989
Node: Ei x180220
Ref: specfunc ei-x180341
Ref: 1e9180341
Ref: specfunc c gsl_sf_expint_Ei180368
Ref: 1ea180368
Ref: specfunc c gsl_sf_expint_Ei_e180417
Ref: 1eb180417
Node: Hyperbolic Integrals180667
Ref: specfunc hyperbolic-integrals180774
Ref: 1ec180774
Ref: specfunc c gsl_sf_Shi180831
Ref: 1ed180831
Ref: specfunc c gsl_sf_Shi_e180874
Ref: 1ee180874
Ref: specfunc c gsl_sf_Chi181026
Ref: 1ef181026
Ref: specfunc c gsl_sf_Chi_e181069
Ref: 1f0181069
Node: Ei_3 x181342
Ref: specfunc ei-3-x181468
Ref: 1f1181468
Ref: specfunc c gsl_sf_expint_3181499
Ref: 1f2181499
Ref: specfunc c gsl_sf_expint_3_e181547
Ref: 1f3181547
Node: Trigonometric Integrals181749
Ref: specfunc trigonometric-integrals181874
Ref: 1f4181874
Ref: specfunc c gsl_sf_Si181937
Ref: 1f5181937
Ref: specfunc c gsl_sf_Si_e181985
Ref: 1f6181985
Ref: specfunc c gsl_sf_Ci182139
Ref: 1f7182139
Ref: specfunc c gsl_sf_Ci_e182187
Ref: 1f8182187
Node: Arctangent Integral182366
Ref: specfunc arctangent-integral182476
Ref: 1f9182476
Ref: specfunc c gsl_sf_atanint182531
Ref: 1fa182531
Ref: specfunc c gsl_sf_atanint_e182578
Ref: 1fb182578
Node: Fermi-Dirac Function182772
Ref: specfunc fermi-dirac-function182910
Ref: 1fc182910
Node: Complete Fermi-Dirac Integrals183146
Ref: specfunc complete-fermi-dirac-integrals183275
Ref: 1fd183275
Ref: specfunc c gsl_sf_fermi_dirac_m1183586
Ref: 1fe183586
Ref: specfunc c gsl_sf_fermi_dirac_m1_e183640
Ref: 1ff183640
Ref: specfunc c gsl_sf_fermi_dirac_0183878
Ref: 200183878
Ref: specfunc c gsl_sf_fermi_dirac_0_e183931
Ref: 201183931
Ref: specfunc c gsl_sf_fermi_dirac_1184156
Ref: 202184156
Ref: specfunc c gsl_sf_fermi_dirac_1_e184209
Ref: 203184209
Ref: specfunc c gsl_sf_fermi_dirac_2184430
Ref: 204184430
Ref: specfunc c gsl_sf_fermi_dirac_2_e184483
Ref: 205184483
Ref: specfunc c gsl_sf_fermi_dirac_int184712
Ref: 206184712
Ref: specfunc c gsl_sf_fermi_dirac_int_e184774
Ref: 207184774
Ref: specfunc c gsl_sf_fermi_dirac_mhalf185035
Ref: 208185035
Ref: specfunc c gsl_sf_fermi_dirac_mhalf_e185092
Ref: 209185092
Ref: specfunc c gsl_sf_fermi_dirac_half185264
Ref: 20a185264
Ref: specfunc c gsl_sf_fermi_dirac_half_e185320
Ref: 20b185320
Ref: specfunc c gsl_sf_fermi_dirac_3half185490
Ref: 20c185490
Ref: specfunc c gsl_sf_fermi_dirac_3half_e185547
Ref: 20d185547
Node: Incomplete Fermi-Dirac Integrals185718
Ref: specfunc incomplete-fermi-dirac-integrals185847
Ref: 20e185847
Ref: specfunc c gsl_sf_fermi_dirac_inc_0186062
Ref: 20f186062
Ref: specfunc c gsl_sf_fermi_dirac_inc_0_e186129
Ref: 210186129
Node: Gamma and Beta Functions186359
Ref: specfunc gamma-and-beta-functions186496
Ref: 211186496
Node: Gamma Functions186935
Ref: specfunc gamma-functions187031
Ref: 212187031
Ref: specfunc c gsl_sf_gamma187368
Ref: 213187368
Ref: specfunc c gsl_sf_gamma_e187413
Ref: 214187413
Ref: specfunc c gsl_sf_lngamma187803
Ref: 215187803
Ref: specfunc c gsl_sf_lngamma_e187850
Ref: 216187850
Ref: specfunc c gsl_sf_lngamma_sgn_e188230
Ref: 217188230
Ref: specfunc c gsl_sf_gammastar188747
Ref: 218188747
Ref: specfunc c gsl_sf_gammastar_e188796
Ref: 219188796
Ref: specfunc c gsl_sf_gammainv189173
Ref: 21a189173
Ref: specfunc c gsl_sf_gammainv_e189221
Ref: 21b189221
Ref: specfunc c gsl_sf_lngamma_complex_e189408
Ref: 21c189408
Node: Factorials190086
Ref: specfunc factorials190208
Ref: 21d190208
Ref: specfunc c gsl_sf_fact190534
Ref: 21e190534
Ref: specfunc c gsl_sf_fact_e190584
Ref: 21f190584
Ref: specfunc c gsl_sf_doublefact190913
Ref: 220190913
Ref: specfunc c gsl_sf_doublefact_e190969
Ref: 221190969
Ref: specfunc c gsl_sf_lnfact191275
Ref: 222191275
Ref: specfunc c gsl_sf_lnfact_e191327
Ref: 223191327
Ref: specfunc c gsl_sf_lndoublefact191644
Ref: 224191644
Ref: specfunc c gsl_sf_lndoublefact_e191702
Ref: 225191702
Ref: specfunc c gsl_sf_choose191889
Ref: 226191889
Ref: specfunc c gsl_sf_choose_e191957
Ref: 227191957
Ref: specfunc c gsl_sf_lnchoose192151
Ref: 228192151
Ref: specfunc c gsl_sf_lnchoose_e192221
Ref: 229192221
Ref: specfunc c gsl_sf_taylorcoeff192461
Ref: 22a192461
Ref: specfunc c gsl_sf_taylorcoeff_e192519
Ref: 22b192519
Node: Pochhammer Symbol192698
Ref: specfunc id3192831
Ref: 22c192831
Ref: specfunc pochhammer-symbol192831
Ref: 22d192831
Ref: specfunc c gsl_sf_poch192882
Ref: 22e192882
Ref: specfunc c gsl_sf_poch_e192936
Ref: 22f192936
Ref: specfunc c gsl_sf_lnpoch193311
Ref: 230193311
Ref: specfunc c gsl_sf_lnpoch_e193367
Ref: 231193367
Ref: specfunc c gsl_sf_lnpoch_sgn_e193576
Ref: 232193576
Ref: specfunc c gsl_sf_pochrel193944
Ref: 233193944
Ref: specfunc c gsl_sf_pochrel_e194001
Ref: 234194001
Node: Incomplete Gamma Functions194210
Ref: specfunc incomplete-gamma-functions194347
Ref: 235194347
Ref: specfunc c gsl_sf_gamma_inc194416
Ref: 236194416
Ref: specfunc c gsl_sf_gamma_inc_e194475
Ref: 237194475
Ref: specfunc c gsl_sf_gamma_inc_Q194725
Ref: 238194725
Ref: specfunc c gsl_sf_gamma_inc_Q_e194786
Ref: 239194786
Ref: specfunc c gsl_sf_gamma_inc_P195038
Ref: 23a195038
Ref: specfunc c gsl_sf_gamma_inc_P_e195099
Ref: 23b195099
Node: Beta Functions195471
Ref: specfunc beta-functions195615
Ref: 23c195615
Ref: specfunc c gsl_sf_beta195660
Ref: 23d195660
Ref: specfunc c gsl_sf_beta_e195714
Ref: 23e195714
Ref: specfunc c gsl_sf_lnbeta195948
Ref: 23f195948
Ref: specfunc c gsl_sf_lnbeta_e196004
Ref: 240196004
Node: Incomplete Beta Function196225
Ref: specfunc incomplete-beta-function196334
Ref: 241196334
Ref: specfunc c gsl_sf_beta_inc196399
Ref: 242196399
Ref: specfunc c gsl_sf_beta_inc_e196467
Ref: 243196467
Node: Gegenbauer Functions196961
Ref: specfunc gegenbauer-functions197111
Ref: 244197111
Ref: specfunc c gsl_sf_gegenpoly_1197392
Ref: 245197392
Ref: specfunc c gsl_sf_gegenpoly_2197458
Ref: 246197458
Ref: specfunc c gsl_sf_gegenpoly_3197524
Ref: 247197524
Ref: specfunc c gsl_sf_gegenpoly_1_e197590
Ref: 248197590
Ref: specfunc c gsl_sf_gegenpoly_2_e197689
Ref: 249197689
Ref: specfunc c gsl_sf_gegenpoly_3_e197788
Ref: 24a197788
Ref: specfunc c gsl_sf_gegenpoly_n198018
Ref: 24b198018
Ref: specfunc c gsl_sf_gegenpoly_n_e198091
Ref: 24c198091
Ref: specfunc c gsl_sf_gegenpoly_array198375
Ref: 24d198375
Node: Hermite Polynomials and Functions198648
Ref: specfunc hermite-polynomials-and-functions198798
Ref: 24e198798
Node: Hermite Polynomials199334
Ref: specfunc hermite-polynomials199467
Ref: 24f199467
Ref: specfunc c gsl_sf_hermite200032
Ref: 250200032
Ref: specfunc c gsl_sf_hermite_e200098
Ref: 251200098
Ref: specfunc c gsl_sf_hermite_array200410
Ref: 252200410
Ref: specfunc c gsl_sf_hermite_series200684
Ref: 253200684
Ref: specfunc c gsl_sf_hermite_series_e200785
Ref: 254200785
Ref: specfunc c gsl_sf_hermite_prob201068
Ref: 255201068
Ref: specfunc c gsl_sf_hermite_prob_e201139
Ref: 256201139
Ref: specfunc c gsl_sf_hermite_prob_array201459
Ref: 257201459
Ref: specfunc c gsl_sf_hermite_prob_series201744
Ref: 258201744
Ref: specfunc c gsl_sf_hermite_prob_series_e201850
Ref: 259201850
Node: Derivatives of Hermite Polynomials202142
Ref: specfunc derivatives-of-hermite-polynomials202301
Ref: 25a202301
Ref: specfunc c gsl_sf_hermite_deriv202386
Ref: 25b202386
Ref: specfunc c gsl_sf_hermite_deriv_e202481
Ref: 25c202481
Ref: specfunc c gsl_sf_hermite_array_deriv202742
Ref: 25d202742
Ref: specfunc c gsl_sf_hermite_deriv_array203178
Ref: 25e203178
Ref: specfunc c gsl_sf_hermite_prob_deriv203618
Ref: 25f203618
Ref: specfunc c gsl_sf_hermite_prob_deriv_e203718
Ref: 260203718
Ref: specfunc c gsl_sf_hermite_prob_array_deriv203987
Ref: 261203987
Ref: specfunc c gsl_sf_hermite_prob_deriv_array204432
Ref: 262204432
Node: Hermite Functions204881
Ref: specfunc hermite-functions205053
Ref: 263205053
Ref: specfunc c gsl_sf_hermite_func205935
Ref: 264205935
Ref: specfunc c gsl_sf_hermite_func_e206006
Ref: 265206006
Ref: specfunc c gsl_sf_hermite_func_fast206294
Ref: 266206294
Ref: specfunc c gsl_sf_hermite_func_fast_e206380
Ref: 267206380
Ref: specfunc c gsl_sf_hermite_func_array206698
Ref: 268206698
Ref: specfunc c gsl_sf_hermite_func_series207074
Ref: 269207074
Ref: specfunc c gsl_sf_hermite_func_series_e207180
Ref: 26a207180
Node: Derivatives of Hermite Functions207462
Ref: specfunc derivatives-of-hermite-functions207650
Ref: 26b207650
Ref: specfunc c gsl_sf_hermite_func_der207731
Ref: 26c207731
Ref: specfunc c gsl_sf_hermite_func_der_e207829
Ref: 26d207829
Node: Zeros of Hermite Polynomials and Hermite Functions208084
Ref: specfunc zeros-of-hermite-polynomials-and-hermite-functions208246
Ref: 26e208246
Ref: specfunc c gsl_sf_hermite_zero208672
Ref: 26f208672
Ref: specfunc c gsl_sf_hermite_zero_e208740
Ref: 270208740
Ref: specfunc c gsl_sf_hermite_prob_zero208958
Ref: 271208958
Ref: specfunc c gsl_sf_hermite_prob_zero_e209031
Ref: 272209031
Ref: specfunc c gsl_sf_hermite_func_zero209257
Ref: 273209257
Ref: specfunc c gsl_sf_hermite_func_zero_e209330
Ref: 274209330
Node: Hypergeometric Functions209544
Ref: specfunc hypergeometric-functions209692
Ref: 275209692
Ref: specfunc c gsl_sf_hyperg_0F1209909
Ref: 276209909
Ref: specfunc c gsl_sf_hyperg_0F1_e209969
Ref: 277209969
Ref: specfunc c gsl_sf_hyperg_1F1_int210140
Ref: 278210140
Ref: specfunc c gsl_sf_hyperg_1F1_int_e210208
Ref: 279210208
Ref: specfunc c gsl_sf_hyperg_1F1210457
Ref: 27a210457
Ref: specfunc c gsl_sf_hyperg_1F1_e210527
Ref: 27b210527
Ref: specfunc c gsl_sf_hyperg_U_int210778
Ref: 27c210778
Ref: specfunc c gsl_sf_hyperg_U_int_e210844
Ref: 27d210844
Ref: specfunc c gsl_sf_hyperg_U_int_e10_e211066
Ref: 27e211066
Ref: specfunc c gsl_sf_hyperg_U211383
Ref: 27f211383
Ref: specfunc c gsl_sf_hyperg_U_e211451
Ref: 280211451
Ref: specfunc c gsl_sf_hyperg_U_e10_e211635
Ref: 281211635
Ref: specfunc c gsl_sf_hyperg_2F1211914
Ref: 282211914
Ref: specfunc c gsl_sf_hyperg_2F1_e212004
Ref: 283212004
Ref: specfunc c gsl_sf_hyperg_2F1_conj212491
Ref: 284212491
Ref: specfunc c gsl_sf_hyperg_2F1_conj_e212588
Ref: 285212588
Ref: specfunc c gsl_sf_hyperg_2F1_renorm212860
Ref: 286212860
Ref: specfunc c gsl_sf_hyperg_2F1_renorm_e212957
Ref: 287212957
Ref: specfunc c gsl_sf_hyperg_2F1_conj_renorm213214
Ref: 288213214
Ref: specfunc c gsl_sf_hyperg_2F1_conj_renorm_e213318
Ref: 289213318
Ref: specfunc c gsl_sf_hyperg_2F0213605
Ref: 28a213605
Ref: specfunc c gsl_sf_hyperg_2F0_e213675
Ref: 28b213675
Node: Laguerre Functions214009
Ref: specfunc id4214143
Ref: 28c214143
Ref: specfunc laguerre-functions214143
Ref: 16e214143
Ref: specfunc c gsl_sf_laguerre_1214744
Ref: 28d214744
Ref: specfunc c gsl_sf_laguerre_2214804
Ref: 28e214804
Ref: specfunc c gsl_sf_laguerre_3214864
Ref: 28f214864
Ref: specfunc c gsl_sf_laguerre_1_e214924
Ref: 290214924
Ref: specfunc c gsl_sf_laguerre_2_e215017
Ref: 291215017
Ref: specfunc c gsl_sf_laguerre_3_e215110
Ref: 292215110
Ref: specfunc c gsl_sf_laguerre_n215337
Ref: 293215337
Ref: specfunc c gsl_sf_laguerre_n_e215432
Ref: 294215432
Node: Lambert W Functions215635
Ref: specfunc lambert-w-functions215787
Ref: 295215787
Ref: specfunc c gsl_sf_lambert_W0216246
Ref: 296216246
Ref: specfunc c gsl_sf_lambert_W0_e216296
Ref: 297216296
Ref: specfunc c gsl_sf_lambert_Wm1216451
Ref: 298216451
Ref: specfunc c gsl_sf_lambert_Wm1_e216502
Ref: 299216502
Node: Legendre Functions and Spherical Harmonics216683
Ref: specfunc legendre-functions-and-spherical-harmonics216848
Ref: 29a216848
Node: Legendre Polynomials217276
Ref: specfunc legendre-polynomials217440
Ref: 29b217440
Ref: specfunc c gsl_sf_legendre_P1217497
Ref: 29c217497
Ref: specfunc c gsl_sf_legendre_P2217548
Ref: 29d217548
Ref: specfunc c gsl_sf_legendre_P3217599
Ref: 29e217599
Ref: specfunc c gsl_sf_legendre_P1_e217650
Ref: 29f217650
Ref: specfunc c gsl_sf_legendre_P2_e217734
Ref: 2a0217734
Ref: specfunc c gsl_sf_legendre_P3_e217818
Ref: 2a1217818
Ref: specfunc c gsl_sf_legendre_Pl218019
Ref: 2a2218019
Ref: specfunc c gsl_sf_legendre_Pl_e218077
Ref: 2a3218077
Ref: specfunc c gsl_sf_legendre_Pl_array218311
Ref: 2a4218311
Ref: specfunc c gsl_sf_legendre_Pl_deriv_array218408
Ref: 2a5218408
Ref: specfunc c gsl_sf_legendre_Q0218679
Ref: 2a6218679
Ref: specfunc c gsl_sf_legendre_Q0_e218730
Ref: 2a7218730
Ref: specfunc c gsl_sf_legendre_Q1218902
Ref: 2a8218902
Ref: specfunc c gsl_sf_legendre_Q1_e218953
Ref: 2a9218953
Ref: specfunc c gsl_sf_legendre_Ql219125
Ref: 2aa219125
Ref: specfunc c gsl_sf_legendre_Ql_e219183
Ref: 2ab219183
Node: Associated Legendre Polynomials and Spherical Harmonics219371
Ref: specfunc associated-legendre-polynomials-and-spherical-harmonics219561
Ref: 2ac219561
Ref: specfunc c gsl_sf_legendre_t222109
Ref: 2ad222109
Ref: specfunc c gsl_sf_legendre_array223174
Ref: 2ae223174
Ref: specfunc c gsl_sf_legendre_array_e223313
Ref: 2af223313
Ref: specfunc c gsl_sf_legendre_deriv_array224169
Ref: 2b1224169
Ref: specfunc c gsl_sf_legendre_deriv_array_e224353
Ref: 2b2224353
Ref: specfunc c gsl_sf_legendre_deriv_alt_array225145
Ref: 2b3225145
Ref: specfunc c gsl_sf_legendre_deriv_alt_array_e225333
Ref: 2b4225333
Ref: specfunc c gsl_sf_legendre_deriv2_array226097
Ref: 2b5226097
Ref: specfunc c gsl_sf_legendre_deriv2_array_e226322
Ref: 2b6226322
Ref: specfunc c gsl_sf_legendre_deriv2_alt_array227238
Ref: 2b7227238
Ref: specfunc c gsl_sf_legendre_deriv2_alt_array_e227467
Ref: 2b8227467
Ref: specfunc c gsl_sf_legendre_array_n228436
Ref: 2b0228436
Ref: specfunc c gsl_sf_legendre_array_index228825
Ref: 2b9228825
Ref: specfunc c gsl_sf_legendre_Plm229145
Ref: 2ba229145
Ref: specfunc c gsl_sf_legendre_Plm_e229211
Ref: 2bb229211
Ref: specfunc c gsl_sf_legendre_sphPlm229426
Ref: 2bc229426
Ref: specfunc c gsl_sf_legendre_sphPlm_e229495
Ref: 2bd229495
Ref: specfunc c gsl_sf_legendre_Plm_array229938
Ref: 2be229938
Ref: specfunc c gsl_sf_legendre_Plm_deriv_array230043
Ref: 2bf230043
Ref: specfunc c gsl_sf_legendre_sphPlm_array230362
Ref: 2c0230362
Ref: specfunc c gsl_sf_legendre_sphPlm_deriv_array230470
Ref: 2c1230470
Ref: specfunc c gsl_sf_legendre_array_size230792
Ref: 2c2230792
Node: Conical Functions230962
Ref: specfunc conical-functions231169
Ref: 2c3231169
Ref: specfunc c gsl_sf_conicalP_half231350
Ref: 2c4231350
Ref: specfunc c gsl_sf_conicalP_half_e231418
Ref: 2c5231418
Ref: specfunc c gsl_sf_conicalP_mhalf231637
Ref: 2c6231637
Ref: specfunc c gsl_sf_conicalP_mhalf_e231706
Ref: 2c7231706
Ref: specfunc c gsl_sf_conicalP_0231925
Ref: 2c8231925
Ref: specfunc c gsl_sf_conicalP_0_e231990
Ref: 2c9231990
Ref: specfunc c gsl_sf_conicalP_1232182
Ref: 2ca232182
Ref: specfunc c gsl_sf_conicalP_1_e232247
Ref: 2cb232247
Ref: specfunc c gsl_sf_conicalP_sph_reg232439
Ref: 2cc232439
Ref: specfunc c gsl_sf_conicalP_sph_reg_e232527
Ref: 2cd232527
Ref: specfunc c gsl_sf_conicalP_cyl_reg232770
Ref: 2ce232770
Ref: specfunc c gsl_sf_conicalP_cyl_reg_e232858
Ref: 2cf232858
Node: Radial Functions for Hyperbolic Space233099
Ref: specfunc radial-functions-for-hyperbolic-space233242
Ref: 2d0233242
Ref: specfunc c gsl_sf_legendre_H3d_0233600
Ref: 2d1233600
Ref: specfunc c gsl_sf_legendre_H3d_0_e233671
Ref: 2d2233671
Ref: specfunc c gsl_sf_legendre_H3d_1234091
Ref: 2d3234091
Ref: specfunc c gsl_sf_legendre_H3d_1_e234162
Ref: 2d4234162
Ref: specfunc c gsl_sf_legendre_H3d234675
Ref: 2d5234675
Ref: specfunc c gsl_sf_legendre_H3d_e234761
Ref: 2d6234761
Ref: specfunc c gsl_sf_legendre_H3d_array235113
Ref: 2d7235113
Node: Logarithm and Related Functions235345
Ref: specfunc logarithm-and-related-functions235508
Ref: 2d8235508
Ref: specfunc c gsl_sf_log235781
Ref: 2d9235781
Ref: specfunc c gsl_sf_log_e235824
Ref: 2da235824
Ref: specfunc c gsl_sf_log_abs235966
Ref: 2db235966
Ref: specfunc c gsl_sf_log_abs_e236013
Ref: 2dc236013
Ref: specfunc c gsl_sf_complex_log_e236185
Ref: 2dd236185
Ref: specfunc c gsl_sf_log_1plusx236522
Ref: 2de236522
Ref: specfunc c gsl_sf_log_1plusx_e236572
Ref: 2df236572
Ref: specfunc c gsl_sf_log_1plusx_mx236758
Ref: 2e0236758
Ref: specfunc c gsl_sf_log_1plusx_mx_e236811
Ref: 2e1236811
Node: Mathieu Functions237014
Ref: specfunc mathieu-functions237149
Ref: 2e2237149
Node: Mathieu Function Workspace238372
Ref: specfunc mathieu-function-workspace238500
Ref: 2e3238500
Ref: specfunc c gsl_sf_mathieu_workspace238739
Ref: 2e4238739
Ref: specfunc c gsl_sf_mathieu_alloc238825
Ref: 2e5238825
Ref: specfunc c gsl_sf_mathieu_free239159
Ref: 2e6239159
Node: Mathieu Function Characteristic Values239285
Ref: specfunc mathieu-function-characteristic-values239447
Ref: 2e7239447
Ref: specfunc c gsl_sf_mathieu_a239540
Ref: 2e8239540
Ref: specfunc c gsl_sf_mathieu_a_e239593
Ref: 2e9239593
Ref: specfunc c gsl_sf_mathieu_b239682
Ref: 2ea239682
Ref: specfunc c gsl_sf_mathieu_b_e239735
Ref: 2eb239735
Ref: specfunc c gsl_sf_mathieu_a_array239964
Ref: 2ec239964
Ref: specfunc c gsl_sf_mathieu_b_array240122
Ref: 2ed240122
Node: Angular Mathieu Functions240487
Ref: specfunc angular-mathieu-functions240647
Ref: 2ee240647
Ref: specfunc c gsl_sf_mathieu_ce240714
Ref: 2ef240714
Ref: specfunc c gsl_sf_mathieu_ce_e240778
Ref: 2f0240778
Ref: specfunc c gsl_sf_mathieu_se240878
Ref: 2f1240878
Ref: specfunc c gsl_sf_mathieu_se_e240942
Ref: 2f2240942
Ref: specfunc c gsl_sf_mathieu_ce_array241146
Ref: 2f3241146
Ref: specfunc c gsl_sf_mathieu_se_array241305
Ref: 2f4241305
Node: Radial Mathieu Functions241675
Ref: specfunc radial-mathieu-functions241788
Ref: 2f5241788
Ref: specfunc c gsl_sf_mathieu_Mc241853
Ref: 2f6241853
Ref: specfunc c gsl_sf_mathieu_Mc_e241924
Ref: 2f7241924
Ref: specfunc c gsl_sf_mathieu_Ms242031
Ref: 2f8242031
Ref: specfunc c gsl_sf_mathieu_Ms_e242102
Ref: 2f9242102
Ref: specfunc c gsl_sf_mathieu_Mc_array242572
Ref: 2fa242572
Ref: specfunc c gsl_sf_mathieu_Ms_array242738
Ref: 2fb242738
Node: Power Function243107
Ref: specfunc power-function243231
Ref: 2fc243231
Ref: specfunc c gsl_sf_pow_int243450
Ref: 2fd243450
Ref: specfunc c gsl_sf_pow_int_e243504
Ref: 4c243504
Node: Psi Digamma Function244080
Ref: specfunc psi-digamma-function244208
Ref: 2fe244208
Node: Digamma Function244602
Ref: specfunc digamma-function244702
Ref: 2ff244702
Ref: specfunc c gsl_sf_psi_int244751
Ref: 300244751
Ref: specfunc c gsl_sf_psi_int_e244795
Ref: 301244795
Ref: specfunc c gsl_sf_psi245011
Ref: 302245011
Ref: specfunc c gsl_sf_psi_e245054
Ref: 303245054
Ref: specfunc c gsl_sf_psi_1piy245214
Ref: 304245214
Ref: specfunc c gsl_sf_psi_1piy_e245262
Ref: 305245262
Node: Trigamma Function245447
Ref: specfunc trigamma-function245574
Ref: 306245574
Ref: specfunc c gsl_sf_psi_1_int245625
Ref: 307245625
Ref: specfunc c gsl_sf_psi_1_int_e245671
Ref: 308245671
Ref: specfunc c gsl_sf_psi_1245830
Ref: 309245830
Ref: specfunc c gsl_sf_psi_1_e245875
Ref: 30a245875
Node: Polygamma Function246030
Ref: specfunc polygamma-function246132
Ref: 30b246132
Ref: specfunc c gsl_sf_psi_n246185
Ref: 30c246185
Ref: specfunc c gsl_sf_psi_n_e246237
Ref: 30d246237
Node: Synchrotron Functions246414
Ref: specfunc synchrotron-functions246547
Ref: 30e246547
Ref: specfunc c gsl_sf_synchrotron_1246703
Ref: 30f246703
Ref: specfunc c gsl_sf_synchrotron_1_e246756
Ref: 310246756
Ref: specfunc c gsl_sf_synchrotron_2246951
Ref: 311246951
Ref: specfunc c gsl_sf_synchrotron_2_e247004
Ref: 312247004
Node: Transport Functions247183
Ref: specfunc transport-functions247319
Ref: 313247319
Ref: specfunc c gsl_sf_transport_2247558
Ref: 314247558
Ref: specfunc c gsl_sf_transport_2_e247609
Ref: 315247609
Ref: specfunc c gsl_sf_transport_3247754
Ref: 316247754
Ref: specfunc c gsl_sf_transport_3_e247805
Ref: 317247805
Ref: specfunc c gsl_sf_transport_4247950
Ref: 318247950
Ref: specfunc c gsl_sf_transport_4_e248001
Ref: 319248001
Ref: specfunc c gsl_sf_transport_5248146
Ref: 31a248146
Ref: specfunc c gsl_sf_transport_5_e248197
Ref: 31b248197
Node: Trigonometric Functions248342
Ref: specfunc trigonometric-functions248471
Ref: 31c248471
Node: Circular Trigonometric Functions248969
Ref: specfunc circular-trigonometric-functions249116
Ref: 31d249116
Ref: specfunc c gsl_sf_sin249197
Ref: 31e249197
Ref: specfunc c gsl_sf_sin_e249240
Ref: 31f249240
Ref: specfunc c gsl_sf_cos249363
Ref: 320249363
Ref: specfunc c gsl_sf_cos_e249406
Ref: 321249406
Ref: specfunc c gsl_sf_hypot249531
Ref: 322249531
Ref: specfunc c gsl_sf_hypot_e249586
Ref: 323249586
Ref: specfunc c gsl_sf_sinc249783
Ref: 324249783
Ref: specfunc c gsl_sf_sinc_e249827
Ref: 325249827
Node: Trigonometric Functions for Complex Arguments249988
Ref: specfunc trigonometric-functions-for-complex-arguments250178
Ref: 326250178
Ref: specfunc c gsl_sf_complex_sin_e250285
Ref: 327250285
Ref: specfunc c gsl_sf_complex_cos_e250532
Ref: 328250532
Ref: specfunc c gsl_sf_complex_logsin_e250781
Ref: 329250781
Node: Hyperbolic Trigonometric Functions251063
Ref: specfunc hyperbolic-trigonometric-functions251241
Ref: 32a251241
Ref: specfunc c gsl_sf_lnsinh251326
Ref: 32b251326
Ref: specfunc c gsl_sf_lnsinh_e251372
Ref: 32c251372
Ref: specfunc c gsl_sf_lncosh251497
Ref: 32d251497
Ref: specfunc c gsl_sf_lncosh_e251543
Ref: 32e251543
Node: Conversion Functions251674
Ref: specfunc conversion-functions251828
Ref: 32f251828
Ref: specfunc c gsl_sf_polar_to_rect251885
Ref: 330251885
Ref: specfunc c gsl_sf_rect_to_polar252167
Ref: 331252167
Node: Restriction Functions252516
Ref: specfunc restriction-functions252680
Ref: 332252680
Ref: specfunc c gsl_sf_angle_restrict_symm252739
Ref: 333252739
Ref: specfunc c gsl_sf_angle_restrict_symm_e252802
Ref: 334252802
Ref: specfunc c gsl_sf_angle_restrict_pos253123
Ref: 335253123
Ref: specfunc c gsl_sf_angle_restrict_pos_e253185
Ref: 336253185
Node: Trigonometric Functions With Error Estimates253486
Ref: specfunc trigonometric-functions-with-error-estimates253621
Ref: 337253621
Ref: specfunc c gsl_sf_sin_err_e253726
Ref: 338253726
Ref: specfunc c gsl_sf_cos_err_e254073
Ref: 339254073
Node: Zeta Functions254422
Ref: specfunc zeta-functions254543
Ref: 33a254543
Node: Riemann Zeta Function254869
Ref: specfunc riemann-zeta-function254982
Ref: 33b254982
Ref: specfunc c gsl_sf_zeta_int255141
Ref: 33c255141
Ref: specfunc c gsl_sf_zeta_int_e255186
Ref: 33d255186
Ref: specfunc c gsl_sf_zeta255354
Ref: 33e255354
Ref: specfunc c gsl_sf_zeta_e255398
Ref: 33f255398
Node: Riemann Zeta Function Minus One255567
Ref: specfunc riemann-zeta-function-minus-one255710
Ref: 340255710
Ref: specfunc c gsl_sf_zetam1_int255972
Ref: 341255972
Ref: specfunc c gsl_sf_zetam1_int_e256019
Ref: 342256019
Ref: specfunc c gsl_sf_zetam1256162
Ref: 343256162
Ref: specfunc c gsl_sf_zetam1_e256208
Ref: 344256208
Node: Hurwitz Zeta Function256352
Ref: specfunc hurwitz-zeta-function256486
Ref: 345256486
Ref: specfunc c gsl_sf_hzeta256630
Ref: 346256630
Ref: specfunc c gsl_sf_hzeta_e256685
Ref: 347256685
Node: Eta Function256863
Ref: specfunc eta-function256957
Ref: 348256957
Ref: specfunc c gsl_sf_eta_int257067
Ref: 349257067
Ref: specfunc c gsl_sf_eta_int_e257111
Ref: 34a257111
Ref: specfunc c gsl_sf_eta257254
Ref: 34b257254
Ref: specfunc c gsl_sf_eta_e257297
Ref: 34c257297
Node: Examples<3>257441
Ref: specfunc examples257572
Ref: 34d257572
Node: References and Further Reading<3>259082
Ref: specfunc references-and-further-reading259190
Ref: 34e259190
Node: Vectors and Matrices260729
Ref: vectors doc260837
Ref: 34f260837
Ref: vectors vectors-and-matrices260837
Ref: 350260837
Node: Data types261436
Ref: vectors data-types261519
Ref: 351261519
Node: Blocks263841
Ref: vectors blocks263940
Ref: 352263940
Ref: vectors c gsl_block264187
Ref: 353264187
Node: Block allocation264814
Ref: vectors block-allocation264909
Ref: 354264909
Ref: vectors c gsl_block_alloc265427
Ref: 355265427
Ref: vectors c gsl_block_calloc265986
Ref: 356265986
Ref: vectors c gsl_block_free266150
Ref: 357266150
Node: Reading and writing blocks266362
Ref: vectors reading-and-writing-blocks266493
Ref: 358266493
Ref: vectors c gsl_block_fwrite266667
Ref: 359266667
Ref: vectors c gsl_block_fread267078
Ref: 35a267078
Ref: vectors c gsl_block_fprintf267619
Ref: 35b267619
Ref: vectors c gsl_block_fscanf268108
Ref: 35c268108
Node: Example programs for blocks268540
Ref: vectors example-programs-for-blocks268646
Ref: 35d268646
Node: Vectors269167
Ref: vectors vectors269264
Ref: 35e269264
Ref: vectors c gsl_vector269832
Ref: 35f269832
Node: Vector allocation271193
Ref: vectors vector-allocation271289
Ref: 360271289
Ref: vectors c gsl_vector_alloc271811
Ref: 361271811
Ref: vectors c gsl_vector_calloc272278
Ref: 362272278
Ref: vectors c gsl_vector_free272464
Ref: 363272464
Node: Accessing vector elements272856
Ref: vectors accessing-vector-elements272989
Ref: 364272989
Ref: vectors c GSL_RANGE_CHECK_OFF273651
Ref: 367273651
Ref: vectors c GSL_C99_INLINE274276
Ref: 368274276
Ref: vectors c gsl_check_range274628
Ref: 369274628
Ref: vectors c gsl_vector_get275194
Ref: 365275194
Ref: vectors c gsl_vector_set275565
Ref: 366275565
Ref: vectors c gsl_vector_ptr275941
Ref: 36a275941
Ref: vectors c gsl_vector_const_ptr276006
Ref: 36b276006
Ref: Accessing vector elements-Footnote-1276453
Node: Initializing vector elements276696
Ref: vectors initializing-vector-elements276839
Ref: 36c276839
Ref: vectors c gsl_vector_set_all276910
Ref: 36d276910
Ref: vectors c gsl_vector_set_zero277067
Ref: 36e277067
Ref: vectors c gsl_vector_set_basis277197
Ref: 36f277197
Node: Reading and writing vectors277428
Ref: vectors reading-and-writing-vectors277558
Ref: 370277558
Ref: vectors c gsl_vector_fwrite277735
Ref: 371277735
Ref: vectors c gsl_vector_fread278159
Ref: 372278159
Ref: vectors c gsl_vector_fprintf278704
Ref: 373278704
Ref: vectors c gsl_vector_fscanf279196
Ref: 374279196
Node: Vector views279632
Ref: vectors vector-views279749
Ref: 375279749
Ref: vectors c gsl_vector_view280067
Ref: 376280067
Ref: vectors c gsl_vector_const_view280093
Ref: 377280093
Ref: vectors c gsl_vector_subvector281086
Ref: 378281086
Ref: vectors c gsl_vector_const_subvector281189
Ref: 379281189
Ref: vectors c gsl_vector_subvector_with_stride282567
Ref: 37a282567
Ref: vectors c gsl_vector_const_subvector_with_stride282697
Ref: 37b282697
Ref: vectors c gsl_vector_complex_real284434
Ref: 37c284434
Ref: vectors c gsl_vector_complex_const_real284523
Ref: 37d284523
Ref: vectors c gsl_vector_complex_imag284916
Ref: 37e284916
Ref: vectors c gsl_vector_complex_const_imag285005
Ref: 37f285005
Ref: vectors c gsl_vector_view_array285403
Ref: 380285403
Ref: vectors c gsl_vector_const_view_array285491
Ref: 381285491
Ref: vectors c gsl_vector_view_array_with_stride286461
Ref: 382286461
Ref: vectors c gsl_vector_const_view_array_with_stride286576
Ref: 383286576
Node: Copying vectors287694
Ref: vectors copying-vectors287803
Ref: 384287803
Ref: vectors c gsl_vector_memcpy288148
Ref: 385288148
Ref: vectors c gsl_vector_swap288380
Ref: 386288380
Node: Exchanging elements288587
Ref: vectors exchanging-elements288701
Ref: 387288701
Ref: vectors c gsl_vector_swap_elements288842
Ref: 388288842
Ref: vectors c gsl_vector_reverse289039
Ref: 389289039
Node: Vector operations289172
Ref: vectors vector-operations289318
Ref: 38a289318
Ref: vectors c gsl_vector_add289367
Ref: 38b289367
Ref: vectors c gsl_vector_sub289679
Ref: 38c289679
Ref: vectors c gsl_vector_mul289998
Ref: 38d289998
Ref: vectors c gsl_vector_div290316
Ref: 38e290316
Ref: vectors c gsl_vector_scale290631
Ref: 38f290631
Ref: vectors c gsl_vector_add_constant290854
Ref: 390290854
Ref: vectors c gsl_vector_axpby291093
Ref: 391291093
Node: Finding maximum and minimum elements of vectors291358
Ref: vectors finding-maximum-and-minimum-elements-of-vectors291502
Ref: 392291502
Ref: vectors c gsl_vector_max291672
Ref: 393291672
Ref: vectors c gsl_vector_min291801
Ref: 394291801
Ref: vectors c gsl_vector_minmax291930
Ref: 395291930
Ref: vectors c gsl_vector_max_index292173
Ref: 396292173
Ref: vectors c gsl_vector_min_index292413
Ref: 397292413
Ref: vectors c gsl_vector_minmax_index292653
Ref: 398292653
Node: Vector properties293011
Ref: vectors vector-properties293166
Ref: 399293166
Ref: vectors c gsl_vector_isnull293368
Ref: 39a293368
Ref: vectors c gsl_vector_ispos293427
Ref: 39b293427
Ref: vectors c gsl_vector_isneg293485
Ref: 39c293485
Ref: vectors c gsl_vector_isnonneg293543
Ref: 39d293543
Ref: vectors c gsl_vector_equal293783
Ref: 39e293783
Node: Example programs for vectors294005
Ref: vectors example-programs-for-vectors294104
Ref: 39f294104
Node: Matrices296359
Ref: vectors matrices296441
Ref: 3a0296441
Ref: vectors c gsl_matrix296682
Ref: 3a1296682
Node: Matrix allocation299465
Ref: vectors matrix-allocation299562
Ref: 3a2299562
Ref: vectors c gsl_matrix_alloc300072
Ref: 3a3300072
Ref: vectors c gsl_matrix_calloc300607
Ref: 3a4300607
Ref: vectors c gsl_matrix_free300834
Ref: 3a5300834
Node: Accessing matrix elements301226
Ref: vectors accessing-matrix-elements301360
Ref: 3a6301360
Ref: vectors c gsl_matrix_get301976
Ref: 3a7301976
Ref: vectors c gsl_matrix_set302396
Ref: 3a8302396
Ref: vectors c gsl_matrix_ptr302822
Ref: 3a9302822
Ref: vectors c gsl_matrix_const_ptr302907
Ref: 3aa302907
Node: Initializing matrix elements303357
Ref: vectors initializing-matrix-elements303502
Ref: 3ab303502
Ref: vectors c gsl_matrix_set_all303573
Ref: 3ac303573
Ref: vectors c gsl_matrix_set_zero303730
Ref: 3ad303730
Ref: vectors c gsl_matrix_set_identity303860
Ref: 3ae303860
Node: Reading and writing matrices304186
Ref: vectors reading-and-writing-matrices304318
Ref: 3af304318
Ref: vectors c gsl_matrix_fwrite304498
Ref: 3b0304498
Ref: vectors c gsl_matrix_fread304922
Ref: 3b1304922
Ref: vectors c gsl_matrix_fprintf305471
Ref: 3b2305471
Ref: vectors c gsl_matrix_fscanf305963
Ref: 3b3305963
Node: Matrix views306408
Ref: vectors matrix-views306541
Ref: 3b4306541
Ref: vectors c gsl_matrix_view306580
Ref: 3b5306580
Ref: vectors c gsl_matrix_const_view306606
Ref: 3b6306606
Ref: vectors c gsl_matrix_submatrix307439
Ref: 3b7307439
Ref: vectors c gsl_matrix_const_submatrix307561
Ref: 3b8307561
Ref: vectors c gsl_matrix_view_array309143
Ref: 3b9309143
Ref: vectors c gsl_matrix_const_view_array309243
Ref: 3ba309243
Ref: vectors c gsl_matrix_view_array_with_tda310302
Ref: 3bb310302
Ref: vectors c gsl_matrix_const_view_array_with_tda310423
Ref: 3bc310423
Ref: vectors c gsl_matrix_view_vector311639
Ref: 3bd311639
Ref: vectors c gsl_matrix_const_view_vector311741
Ref: 3be311741
Ref: vectors c gsl_matrix_view_vector_with_tda312836
Ref: 3bf312836
Ref: vectors c gsl_matrix_const_view_vector_with_tda312959
Ref: 3c0312959
Node: Creating row and column views314215
Ref: vectors creating-row-and-column-views314336
Ref: 3c1314336
Ref: vectors c gsl_matrix_row314758
Ref: 3c2314758
Ref: vectors c gsl_matrix_const_row314830
Ref: 3c3314830
Ref: vectors c gsl_matrix_column315285
Ref: 3c4315285
Ref: vectors c gsl_matrix_const_column315370
Ref: 3c5315370
Ref: vectors c gsl_matrix_subrow315837
Ref: 3c6315837
Ref: vectors c gsl_matrix_const_subrow315947
Ref: 3c7315947
Ref: vectors c gsl_matrix_subcolumn316557
Ref: 3c8316557
Ref: vectors c gsl_matrix_const_subcolumn316670
Ref: 3c9316670
Ref: vectors c gsl_matrix_diagonal317289
Ref: 3ca317289
Ref: vectors c gsl_matrix_const_diagonal317356
Ref: 3cb317356
Ref: vectors c gsl_matrix_subdiagonal317885
Ref: 3cc317885
Ref: vectors c gsl_matrix_const_subdiagonal317975
Ref: 3cd317975
Ref: vectors c gsl_matrix_superdiagonal318477
Ref: 3ce318477
Ref: vectors c gsl_matrix_const_superdiagonal318569
Ref: 3cf318569
Node: Copying matrices319079
Ref: vectors copying-matrices319212
Ref: 3d0319212
Ref: vectors c gsl_matrix_memcpy319259
Ref: 3d1319259
Ref: vectors c gsl_matrix_swap319490
Ref: 3d2319490
Node: Copying rows and columns319701
Ref: vectors copying-rows-and-columns319832
Ref: 3d3319832
Ref: vectors c gsl_matrix_get_row320314
Ref: 3d4320314
Ref: vectors c gsl_matrix_get_col320599
Ref: 3d5320599
Ref: vectors c gsl_matrix_set_row320890
Ref: 3d6320890
Ref: vectors c gsl_matrix_set_col321175
Ref: 3d7321175
Node: Exchanging rows and columns321466
Ref: vectors exchanging-rows-and-columns321598
Ref: 3d8321598
Ref: vectors c gsl_matrix_swap_rows321750
Ref: 3d9321750
Ref: vectors c gsl_matrix_swap_columns321939
Ref: 3da321939
Ref: vectors c gsl_matrix_swap_rowcol322134
Ref: 3db322134
Ref: vectors c gsl_matrix_transpose_memcpy322398
Ref: 3dc322398
Ref: vectors c gsl_matrix_transpose322803
Ref: 3dd322803
Node: Matrix operations323045
Ref: vectors matrix-operations323201
Ref: 3de323201
Ref: vectors c gsl_matrix_add323321
Ref: 3df323321
Ref: vectors c gsl_matrix_sub323647
Ref: 3e0323647
Ref: vectors c gsl_matrix_mul_elements323980
Ref: 3e1323980
Ref: vectors c gsl_matrix_div_elements324331
Ref: 3e2324331
Ref: vectors c gsl_matrix_scale324679
Ref: 3e3324679
Ref: vectors c gsl_matrix_add_constant324913
Ref: 3e4324913
Node: Finding maximum and minimum elements of matrices325163
Ref: vectors finding-maximum-and-minimum-elements-of-matrices325309
Ref: 3e5325309
Ref: vectors c gsl_matrix_max325484
Ref: 3e6325484
Ref: vectors c gsl_matrix_min325613
Ref: 3e7325613
Ref: vectors c gsl_matrix_minmax325742
Ref: 3e8325742
Ref: vectors c gsl_matrix_max_index325985
Ref: 3e9325985
Ref: vectors c gsl_matrix_min_index326350
Ref: 3ea326350
Ref: vectors c gsl_matrix_minmax_index326715
Ref: 3eb326715
Node: Matrix properties327172
Ref: vectors matrix-properties327330
Ref: 3ec327330
Ref: vectors c gsl_matrix_isnull327534
Ref: 3ed327534
Ref: vectors c gsl_matrix_ispos327593
Ref: 3ee327593
Ref: vectors c gsl_matrix_isneg327651
Ref: 3ef327651
Ref: vectors c gsl_matrix_isnonneg327709
Ref: 3f0327709
Ref: vectors c gsl_matrix_equal328045
Ref: 3f2328045
Node: Example programs for matrices328268
Ref: vectors example-programs-for-matrices328411
Ref: 3f3328411
Node: References and Further Reading<4>332433
Ref: vectors references-and-further-reading332550
Ref: 3f4332550
Node: Permutations332921
Ref: permutation doc333024
Ref: 3f5333024
Ref: permutation permutations333024
Ref: 3f6333024
Node: The Permutation struct334230
Ref: permutation the-permutation-struct334333
Ref: 3f7334333
Ref: permutation c gsl_permutation334388
Ref: 3f8334388
Node: Permutation allocation334817
Ref: permutation permutation-allocation334959
Ref: 3f9334959
Ref: permutation c gsl_permutation_alloc335014
Ref: 3fa335014
Ref: permutation c gsl_permutation_calloc335467
Ref: 3fb335467
Ref: permutation c gsl_permutation_init335746
Ref: 3fc335746
Ref: permutation c gsl_permutation_free335917
Ref: 3fd335917
Ref: permutation c gsl_permutation_memcpy336054
Ref: 3fe336054
Node: Accessing permutation elements336314
Ref: permutation accessing-permutation-elements336456
Ref: 3ff336456
Ref: permutation c gsl_permutation_get336603
Ref: 400336603
Ref: permutation c gsl_permutation_swap336995
Ref: 401336995
Node: Permutation properties337201
Ref: permutation permutation-properties337342
Ref: 402337342
Ref: permutation c gsl_permutation_size337397
Ref: 403337397
Ref: permutation c gsl_permutation_data337533
Ref: 404337533
Ref: permutation c gsl_permutation_valid337702
Ref: 405337702
Node: Permutation functions337941
Ref: permutation permutation-functions338073
Ref: 406338073
Ref: permutation c gsl_permutation_reverse338126
Ref: 407338126
Ref: permutation c gsl_permutation_inverse338262
Ref: 408338262
Ref: permutation c gsl_permutation_next338473
Ref: 409338473
Ref: permutation c gsl_permutation_prev338936
Ref: 40a338936
Node: Applying Permutations339260
Ref: permutation applying-permutations339402
Ref: 40b339402
Ref: permutation c gsl_permute339564
Ref: 40c339564
Ref: permutation c gsl_permute_inverse339787
Ref: 40d339787
Ref: permutation c gsl_permute_vector340033
Ref: 40e340033
Ref: permutation c gsl_permute_vector_inverse340495
Ref: 40f340495
Ref: permutation c gsl_permute_matrix341078
Ref: 410341078
Ref: permutation c gsl_permutation_mul341569
Ref: 411341569
Node: Reading and writing permutations341915
Ref: permutation reading-and-writing-permutations342063
Ref: 412342063
Ref: permutation c gsl_permutation_fwrite342251
Ref: 413342251
Ref: permutation c gsl_permutation_fread342673
Ref: 414342673
Ref: permutation c gsl_permutation_fprintf343230
Ref: 415343230
Ref: permutation c gsl_permutation_fscanf343763
Ref: 416343763
Ref: Reading and writing permutations-Footnote-1344247
Node: Permutations in cyclic form344363
Ref: permutation permutations-in-cyclic-form344501
Ref: 417344501
Ref: permutation c gsl_permutation_linear_to_canonical346536
Ref: 418346536
Ref: permutation c gsl_permutation_canonical_to_linear346776
Ref: 419346776
Ref: permutation c gsl_permutation_inversions347029
Ref: 41a347029
Ref: permutation c gsl_permutation_linear_cycles347428
Ref: 41b347428
Ref: permutation c gsl_permutation_canonical_cycles347621
Ref: 41c347621
Node: Examples<4>347820
Ref: permutation examples347959
Ref: 41d347959
Node: References and Further Reading<5>350377
Ref: permutation references-and-further-reading350480
Ref: 41e350480
Node: Combinations351011
Ref: combination doc351103
Ref: 41f351103
Ref: combination combinations351103
Ref: 420351103
Node: The Combination struct351884
Ref: combination the-combination-struct351987
Ref: 421351987
Ref: combination c gsl_combination352044
Ref: 422352044
Node: Combination allocation352531
Ref: combination combination-allocation352673
Ref: 423352673
Ref: combination c gsl_combination_alloc352730
Ref: 424352730
Ref: combination c gsl_combination_calloc353254
Ref: 425353254
Ref: combination c gsl_combination_init_first353602
Ref: 426353602
Ref: combination c gsl_combination_init_last353807
Ref: 427353807
Ref: combination c gsl_combination_free354024
Ref: 428354024
Ref: combination c gsl_combination_memcpy354161
Ref: 429354161
Node: Accessing combination elements354421
Ref: combination accessing-combination-elements354563
Ref: 42a354563
Ref: combination c gsl_combination_get354713
Ref: 42b354713
Node: Combination properties355105
Ref: combination combination-properties355246
Ref: 42c355246
Ref: combination c gsl_combination_n355303
Ref: 42d355303
Ref: combination c gsl_combination_k355435
Ref: 42e355435
Ref: combination c gsl_combination_data355591
Ref: 42f355591
Ref: combination c gsl_combination_valid355760
Ref: 430355760
Node: Combination functions356021
Ref: combination combination-functions356164
Ref: 431356164
Ref: combination c gsl_combination_next356219
Ref: 432356219
Ref: combination c gsl_combination_prev356679
Ref: 433356679
Node: Reading and writing combinations357003
Ref: combination reading-and-writing-combinations357135
Ref: 434357135
Ref: combination c gsl_combination_fwrite357325
Ref: 435357325
Ref: combination c gsl_combination_fread357747
Ref: 436357747
Ref: combination c gsl_combination_fprintf358320
Ref: 437358320
Ref: combination c gsl_combination_fscanf358853
Ref: 438358853
Ref: Reading and writing combinations-Footnote-1359344
Node: Examples<5>359460
Ref: combination examples359604
Ref: 439359604
Node: References and Further Reading<6>360735
Ref: combination references-and-further-reading360838
Ref: 43a360838
Node: Multisets361118
Ref: multiset doc361205
Ref: 43b361205
Ref: multiset multisets361205
Ref: 43c361205
Node: The Multiset struct362050
Ref: multiset the-multiset-struct362144
Ref: 43d362144
Ref: multiset c gsl_multiset362195
Ref: 43e362195
Node: Multiset allocation362664
Ref: multiset multiset-allocation362794
Ref: 43f362794
Ref: multiset c gsl_multiset_alloc362845
Ref: 440362845
Ref: multiset c gsl_multiset_calloc363343
Ref: 441363343
Ref: multiset c gsl_multiset_init_first363674
Ref: 442363674
Ref: multiset c gsl_multiset_init_last363870
Ref: 443363870
Ref: multiset c gsl_multiset_free364066
Ref: 444364066
Ref: multiset c gsl_multiset_memcpy364194
Ref: 445364194
Node: Accessing multiset elements364436
Ref: multiset accessing-multiset-elements364566
Ref: 446364566
Ref: multiset c gsl_multiset_get364707
Ref: 447364707
Node: Multiset properties365090
Ref: multiset multiset-properties365219
Ref: 448365219
Ref: multiset c gsl_multiset_n365270
Ref: 449365270
Ref: multiset c gsl_multiset_k365399
Ref: 44a365399
Ref: multiset c gsl_multiset_data365546
Ref: 44b365546
Ref: multiset c gsl_multiset_valid365706
Ref: 44c365706
Node: Multiset functions365944
Ref: multiset multiset-functions366075
Ref: 44d366075
Ref: multiset c gsl_multiset_next366124
Ref: 44e366124
Ref: multiset c gsl_multiset_prev366580
Ref: 44f366580
Node: Reading and writing multisets366896
Ref: multiset reading-and-writing-multisets367019
Ref: 450367019
Ref: multiset c gsl_multiset_fwrite367200
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Ref: multiset c gsl_multiset_fread367613
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Ref: multiset c gsl_multiset_fprintf368164
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Ref: multiset c gsl_multiset_fscanf368688
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Ref: Reading and writing multisets-Footnote-1369157
Node: Examples<6>369273
Ref: multiset examples369369
Ref: 455369369
Node: Sorting371360
Ref: sort doc371447
Ref: 456371447
Ref: sort sorting371447
Ref: 457371447
Node: Sorting objects372338
Ref: sort sorting-objects372422
Ref: 458372422
Ref: sort c gsl_heapsort372960
Ref: 459372960
Ref: sort c gsl_comparison_fn_t373303
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Ref: sort c gsl_heapsort_index374526
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Node: Sorting vectors375309
Ref: sort sorting-vectors375446
Ref: 45c375446
Ref: sort c gsl_sort376481
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Ref: sort c gsl_sort2376700
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Ref: sort c gsl_sort_vector377082
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Ref: sort c gsl_sort_vector2377233
Ref: 460377233
Ref: sort c gsl_sort_index377469
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Ref: sort c gsl_sort_vector_index378073
Ref: 462378073
Node: Selecting the k smallest or largest elements378685
Ref: sort selecting-the-k-smallest-or-largest-elements378825
Ref: 463378825
Ref: sort c gsl_sort_smallest379534
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Ref: sort c gsl_sort_largest379971
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Ref: sort c gsl_sort_vector_smallest380384
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Ref: sort c gsl_sort_vector_largest380485
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Ref: sort c gsl_sort_smallest_index380880
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Ref: sort c gsl_sort_largest_index381369
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Ref: sort c gsl_sort_vector_smallest_index381857
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Ref: sort c gsl_sort_vector_largest_index381961
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Node: Computing the rank382276
Ref: sort computing-the-rank382412
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Node: Examples<7>383485
Ref: sort examples383610
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Node: References and Further Reading<7>385121
Ref: sort references-and-further-reading385219
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Node: BLAS Support385630
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Ref: BLAS Support-Footnote-1389672
Node: GSL BLAS Interface389841
Ref: blas gsl-blas-interface389929
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Node: Level 1390273
Ref: blas level-1390352
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Ref: blas c gsl_blas_sdsdot390383
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Ref: blas c gsl_blas_sdot390646
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Ref: blas c gsl_blas_srotm396927
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Ref: blas c gsl_blas_drotm397032
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Node: Level 2397189
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Ref: blas c gsl_blas_sgemv397315
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Ref: blas c gsl_blas_chemv401660
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Ref: blas c gsl_blas_cher2405993
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Ref: blas c gsl_blas_zher2406209
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Node: Level 3406907
Ref: blas level-3406986
Ref: 4c2406986
Ref: blas c gsl_blas_sgemm407017
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Ref: blas c gsl_blas_dgemm407236
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Node: Examples<8>418664
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Node: References and Further Reading<8>420066
Ref: blas references-and-further-reading420169
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Node: Linear Algebra421421
Ref: linalg doc421518
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Ref: linalg linear-algebra421518
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Node: LU Decomposition422801
Ref: linalg lu-decomposition422894
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Ref: linalg c gsl_linalg_LU_decomp423554
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Ref: linalg c gsl_linalg_complex_LU_decomp423655
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Ref: linalg c gsl_linalg_LU_solve424800
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Ref: linalg c gsl_linalg_complex_LU_solve424937
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Ref: linalg c gsl_linalg_LU_svx425343
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Ref: linalg c gsl_linalg_LU_refine425843
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Ref: linalg c gsl_linalg_complex_LU_sgndet429067
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Node: QR Decomposition429321
Ref: linalg linalg-qr429460
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Ref: linalg qr-decomposition429460
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Ref: linalg c gsl_linalg_QR_decomp431930
Ref: 4fd431930
Ref: linalg c gsl_linalg_QR_decomp_r432966
Ref: 4fe432966
Ref: linalg c gsl_linalg_QR_solve433633
Ref: 4ff433633
Ref: linalg c gsl_linalg_QR_solve_r433767
Ref: 500433767
Ref: linalg c gsl_linalg_QR_svx434340
Ref: 503434340
Ref: linalg c gsl_linalg_QR_lssolve434775
Ref: 501434775
Ref: linalg c gsl_linalg_QR_lssolve_r435440
Ref: 502435440
Ref: linalg c gsl_linalg_QR_QTvec436361
Ref: 504436361
Ref: linalg c gsl_linalg_QR_QTvec_r436473
Ref: 505436473
Ref: linalg c gsl_linalg_QR_Qvec437046
Ref: 506437046
Ref: linalg c gsl_linalg_QR_QTmat437467
Ref: 507437467
Ref: linalg c gsl_linalg_QR_QTmat_r437579
Ref: 508437579
Ref: linalg c gsl_linalg_QR_Rsolve438164
Ref: 509438164
Ref: linalg c gsl_linalg_QR_Rsvx438466
Ref: 50a438466
Ref: linalg c gsl_linalg_QR_unpack438872
Ref: 50b438872
Ref: linalg c gsl_linalg_QR_unpack_r439177
Ref: 50c439177
Ref: linalg c gsl_linalg_QR_QRsolve439675
Ref: 50d439675
Ref: linalg c gsl_linalg_QR_update439977
Ref: 50e439977
Ref: linalg c gsl_linalg_R_solve440376
Ref: 50f440376
Ref: linalg c gsl_linalg_R_svx440578
Ref: 510440578
Ref: linalg c gsl_linalg_QR_rcond440841
Ref: 511440841
Node: Triangle on Top of Rectangle441327
Ref: linalg triangle-on-top-of-rectangle441409
Ref: 512441409
Ref: linalg c gsl_linalg_QR_TR_decomp442013
Ref: 513442013
Node: QR Decomposition with Column Pivoting442323
Ref: linalg linalg-qrpt442462
Ref: 4fb442462
Ref: linalg qr-decomposition-with-column-pivoting442462
Ref: 514442462
Ref: linalg c gsl_linalg_QRPT_decomp443614
Ref: 515443614
Ref: linalg c gsl_linalg_QRPT_decomp2444938
Ref: 516444938
Ref: linalg c gsl_linalg_QRPT_solve445325
Ref: 517445325
Ref: linalg c gsl_linalg_QRPT_svx445717
Ref: 518445717
Ref: linalg c gsl_linalg_QRPT_lssolve446110
Ref: 519446110
Ref: linalg c gsl_linalg_QRPT_lssolve2446953
Ref: 51a446953
Ref: linalg c gsl_linalg_QRPT_QRsolve447900
Ref: 51c447900
Ref: linalg c gsl_linalg_QRPT_update448264
Ref: 51d448264
Ref: linalg c gsl_linalg_QRPT_Rsolve448763
Ref: 51e448763
Ref: linalg c gsl_linalg_QRPT_Rsvx449017
Ref: 51f449017
Ref: linalg c gsl_linalg_QRPT_rank449362
Ref: 51b449362
Ref: linalg c gsl_linalg_QRPT_rcond449800
Ref: 520449800
Node: LQ Decomposition450245
Ref: linalg lq-decomposition450401
Ref: 521450401
Ref: linalg c gsl_linalg_LQ_decomp451227
Ref: 522451227
Ref: linalg c gsl_linalg_LQ_lssolve452010
Ref: 523452010
Ref: linalg c gsl_linalg_LQ_unpack452601
Ref: 524452601
Ref: linalg c gsl_linalg_LQ_QTvec452906
Ref: 525452906
Node: Complete Orthogonal Decomposition453126
Ref: linalg complete-orthogonal-decomposition453273
Ref: 526453273
Ref: linalg linalg-cod453273
Ref: 4fc453273
Ref: linalg c gsl_linalg_COD_decomp454898
Ref: 527454898
Ref: linalg c gsl_linalg_COD_decomp_e455070
Ref: 528455070
Ref: linalg c gsl_linalg_COD_lssolve456071
Ref: 529456071
Ref: linalg c gsl_linalg_COD_lssolve2456948
Ref: 52a456948
Ref: linalg c gsl_linalg_COD_unpack457906
Ref: 52b457906
Ref: linalg c gsl_linalg_COD_matZ458348
Ref: 52c458348
Node: Singular Value Decomposition458816
Ref: linalg singular-value-decomposition458969
Ref: 52d458969
Ref: linalg c gsl_linalg_SV_decomp460596
Ref: 52e460596
Ref: linalg c gsl_linalg_SV_decomp_mod461355
Ref: 52f461355
Ref: linalg c gsl_linalg_SV_decomp_jacobi461729
Ref: 530461729
Ref: linalg c gsl_linalg_SV_solve462104
Ref: 531462104
Ref: linalg c gsl_linalg_SV_leverage462935
Ref: 532462935
Node: Cholesky Decomposition463389
Ref: linalg cholesky-decomposition463539
Ref: 533463539
Ref: linalg sec-cholesky-decomposition463539
Ref: 3f1463539
Ref: linalg c gsl_linalg_cholesky_decomp1464508
Ref: 534464508
Ref: linalg c gsl_linalg_complex_cholesky_decomp464571
Ref: 535464571
Ref: linalg c gsl_linalg_cholesky_decomp465529
Ref: 536465529
Ref: linalg c gsl_linalg_cholesky_solve465684
Ref: 537465684
Ref: linalg c gsl_linalg_complex_cholesky_solve465806
Ref: 538465806
Ref: linalg c gsl_linalg_cholesky_svx466242
Ref: 539466242
Ref: linalg c gsl_linalg_complex_cholesky_svx466340
Ref: 53a466340
Ref: linalg c gsl_linalg_cholesky_invert466859
Ref: 53b466859
Ref: linalg c gsl_linalg_complex_cholesky_invert466928
Ref: 53c466928
Ref: linalg c gsl_linalg_cholesky_decomp2467345
Ref: 53d467345
Ref: linalg c gsl_linalg_cholesky_solve2468345
Ref: 53e468345
Ref: linalg c gsl_linalg_cholesky_svx2468741
Ref: 53f468741
Ref: linalg c gsl_linalg_cholesky_scale469218
Ref: 540469218
Ref: linalg c gsl_linalg_cholesky_scale_apply469759
Ref: 541469759
Ref: linalg c gsl_linalg_cholesky_rcond469992
Ref: 542469992
Node: Pivoted Cholesky Decomposition470499
Ref: linalg pivoted-cholesky-decomposition470652
Ref: 543470652
Ref: linalg c gsl_linalg_pcholesky_decomp471403
Ref: 544471403
Ref: linalg c gsl_linalg_pcholesky_solve472153
Ref: 545472153
Ref: linalg c gsl_linalg_pcholesky_svx472557
Ref: 546472557
Ref: linalg c gsl_linalg_pcholesky_decomp2473055
Ref: 547473055
Ref: linalg c gsl_linalg_pcholesky_solve2474193
Ref: 548474193
Ref: linalg c gsl_linalg_pcholesky_svx2474660
Ref: 549474660
Ref: linalg c gsl_linalg_pcholesky_invert475229
Ref: 54a475229
Ref: linalg c gsl_linalg_pcholesky_rcond475552
Ref: 54b475552
Node: Modified Cholesky Decomposition476087
Ref: linalg modified-cholesky-decomposition476236
Ref: 54c476236
Ref: linalg c gsl_linalg_mcholesky_decomp477319
Ref: 54d477319
Ref: linalg c gsl_linalg_mcholesky_solve478225
Ref: 54e478225
Ref: linalg c gsl_linalg_mcholesky_svx478641
Ref: 54f478641
Ref: linalg c gsl_linalg_mcholesky_rcond479151
Ref: 550479151
Node: LDLT Decomposition479682
Ref: linalg ldlt-decomposition479853
Ref: 551479853
Ref: linalg sec-ldlt-decomposition479853
Ref: 552479853
Ref: linalg c gsl_linalg_ldlt_decomp480650
Ref: 553480650
Ref: linalg c gsl_linalg_ldlt_solve481392
Ref: 555481392
Ref: linalg c gsl_linalg_ldlt_svx481711
Ref: 556481711
Ref: linalg c gsl_linalg_ldlt_rcond482126
Ref: 554482126
Node: Tridiagonal Decomposition of Real Symmetric Matrices482609
Ref: linalg tridiagonal-decomposition-of-real-symmetric-matrices482796
Ref: 557482796
Ref: linalg c gsl_linalg_symmtd_decomp483092
Ref: 558483092
Ref: linalg c gsl_linalg_symmtd_unpack483732
Ref: 559483732
Ref: linalg c gsl_linalg_symmtd_unpack_T484203
Ref: 55a484203
Node: Tridiagonal Decomposition of Hermitian Matrices484567
Ref: linalg tridiagonal-decomposition-of-hermitian-matrices484777
Ref: 55b484777
Ref: linalg c gsl_linalg_hermtd_decomp485064
Ref: 55c485064
Ref: linalg c gsl_linalg_hermtd_unpack485770
Ref: 55d485770
Ref: linalg c gsl_linalg_hermtd_unpack_T486257
Ref: 55e486257
Node: Hessenberg Decomposition of Real Matrices486628
Ref: linalg hessenberg-decomposition-of-real-matrices486838
Ref: 55f486838
Ref: linalg c gsl_linalg_hessenberg_decomp487323
Ref: 560487323
Ref: linalg c gsl_linalg_hessenberg_unpack487994
Ref: 561487994
Ref: linalg c gsl_linalg_hessenberg_unpack_accum488350
Ref: 562488350
Ref: linalg c gsl_linalg_hessenberg_set_zero488928
Ref: 563488928
Node: Hessenberg-Triangular Decomposition of Real Matrices489220
Ref: linalg hessenberg-triangular-decomposition-of-real-matrices489400
Ref: 564489400
Ref: linalg c gsl_linalg_hesstri_decomp489887
Ref: 565489887
Node: Bidiagonalization490385
Ref: linalg bidiagonalization490540
Ref: 566490540
Ref: linalg c gsl_linalg_bidiag_decomp490892
Ref: 567490892
Ref: linalg c gsl_linalg_bidiag_unpack491585
Ref: 568491585
Ref: linalg c gsl_linalg_bidiag_unpack2492208
Ref: 569492208
Ref: linalg c gsl_linalg_bidiag_unpack_B492687
Ref: 56a492687
Node: Givens Rotations493048
Ref: linalg givens-rotations493178
Ref: 56b493178
Ref: linalg c gsl_linalg_givens493820
Ref: 56c493820
Ref: linalg c gsl_linalg_givens_gv494109
Ref: 56d494109
Node: Householder Transformations494454
Ref: linalg householder-transformations494604
Ref: 56e494604
Ref: linalg c gsl_linalg_householder_transform495107
Ref: 56f495107
Ref: linalg c gsl_linalg_complex_householder_transform495178
Ref: 570495178
Ref: linalg c gsl_linalg_householder_hm495855
Ref: 571495855
Ref: linalg c gsl_linalg_complex_householder_hm495960
Ref: 572495960
Ref: linalg c gsl_linalg_householder_mh496306
Ref: 573496306
Ref: linalg c gsl_linalg_complex_householder_mh496411
Ref: 574496411
Ref: linalg c gsl_linalg_householder_hv496758
Ref: 575496758
Ref: linalg c gsl_linalg_complex_householder_hv496863
Ref: 576496863
Node: Householder solver for linear systems497195
Ref: linalg householder-solver-for-linear-systems497348
Ref: 577497348
Ref: linalg c gsl_linalg_HH_solve497437
Ref: 578497437
Ref: linalg c gsl_linalg_HH_svx497789
Ref: 579497789
Node: Tridiagonal Systems498134
Ref: linalg tridiagonal-systems498278
Ref: 57a498278
Ref: linalg c gsl_linalg_solve_tridiag498710
Ref: 57b498710
Ref: linalg c gsl_linalg_solve_symm_tridiag499323
Ref: 57c499323
Ref: linalg c gsl_linalg_solve_cyc_tridiag499897
Ref: 57d499897
Ref: linalg c gsl_linalg_solve_symm_cyc_tridiag500541
Ref: 57e500541
Node: Triangular Systems501141
Ref: linalg triangular-systems501262
Ref: 57f501262
Ref: linalg c gsl_linalg_tri_invert501313
Ref: 580501313
Ref: linalg c gsl_linalg_complex_tri_invert501418
Ref: 581501418
Ref: linalg c gsl_linalg_tri_LTL501877
Ref: 582501877
Ref: linalg c gsl_linalg_complex_tri_LHL501931
Ref: 583501931
Ref: linalg c gsl_linalg_tri_UL502141
Ref: 584502141
Ref: linalg c gsl_linalg_complex_tri_UL502195
Ref: 585502195
Ref: linalg c gsl_linalg_tri_rcond502562
Ref: 586502562
Node: Banded Systems503156
Ref: linalg banded-systems503267
Ref: 587503267
Node: General Banded Format503884
Ref: linalg general-banded-format503989
Ref: 588503989
Node: Symmetric Banded Format505506
Ref: linalg sec-symmetric-banded505649
Ref: 589505649
Ref: linalg symmetric-banded-format505649
Ref: 58a505649
Node: Banded Cholesky Decomposition507025
Ref: linalg banded-cholesky-decomposition507172
Ref: 58b507172
Ref: linalg c gsl_linalg_cholesky_band_decomp507722
Ref: 58c507722
Ref: linalg c gsl_linalg_cholesky_band_solve508654
Ref: 58e508654
Ref: linalg c gsl_linalg_cholesky_band_svx509012
Ref: 58f509012
Ref: linalg c gsl_linalg_cholesky_band_invert509466
Ref: 590509466
Ref: linalg c gsl_linalg_cholesky_band_unpack509884
Ref: 591509884
Ref: linalg c gsl_linalg_cholesky_band_rcond510212
Ref: 58d510212
Node: Banded LDLT Decomposition510721
Ref: linalg banded-ldlt-decomposition510836
Ref: 592510836
Ref: linalg c gsl_linalg_ldlt_band_decomp511375
Ref: 593511375
Ref: linalg c gsl_linalg_ldlt_band_solve511956
Ref: 594511956
Ref: linalg c gsl_linalg_ldlt_band_svx512307
Ref: 595512307
Ref: linalg c gsl_linalg_ldlt_band_unpack512749
Ref: 596512749
Ref: linalg c gsl_linalg_ldlt_band_rcond513145
Ref: 597513145
Node: Balancing513645
Ref: linalg balancing513749
Ref: 598513749
Ref: linalg id1513749
Ref: 599513749
Ref: linalg c gsl_linalg_balance_matrix514174
Ref: 59a514174
Node: Examples<9>514441
Ref: linalg examples514564
Ref: 59b514564
Node: References and Further Reading<9>516411
Ref: linalg references-and-further-reading516516
Ref: 59c516516
Node: Eigensystems518906
Ref: eigen doc519019
Ref: 59d519019
Ref: eigen eigensystems519019
Ref: 59e519019
Node: Real Symmetric Matrices520142
Ref: eigen real-symmetric-matrices520250
Ref: 59f520250
Ref: eigen c gsl_eigen_symm_workspace520591
Ref: 5a0520591
Ref: eigen c gsl_eigen_symm_alloc520730
Ref: 5a1520730
Ref: eigen c gsl_eigen_symm_free520982
Ref: 5a2520982
Ref: eigen c gsl_eigen_symm521129
Ref: 5a3521129
Ref: eigen c gsl_eigen_symmv_workspace521639
Ref: 5a4521639
Ref: eigen c gsl_eigen_symmv_alloc521795
Ref: 5a5521795
Ref: eigen c gsl_eigen_symmv_free522066
Ref: 5a6522066
Ref: eigen c gsl_eigen_symmv522215
Ref: 5a7522215
Node: Complex Hermitian Matrices523047
Ref: eigen complex-hermitian-matrices523190
Ref: 5a8523190
Ref: eigen c gsl_eigen_herm_workspace523374
Ref: 5a9523374
Ref: eigen c gsl_eigen_herm_alloc523513
Ref: 5aa523513
Ref: eigen c gsl_eigen_herm_free523768
Ref: 5ab523768
Ref: eigen c gsl_eigen_herm523915
Ref: 5ac523915
Ref: eigen c gsl_eigen_hermv_workspace524525
Ref: 5ad524525
Ref: eigen c gsl_eigen_hermv_alloc524681
Ref: 5ae524681
Ref: eigen c gsl_eigen_hermv_free524955
Ref: 5af524955
Ref: eigen c gsl_eigen_hermv525104
Ref: 5b0525104
Node: Real Nonsymmetric Matrices526067
Ref: eigen real-nonsymmetric-matrices526235
Ref: 5b1526235
Ref: eigen c gsl_eigen_nonsymm_workspace526735
Ref: 5b2526735
Ref: eigen c gsl_eigen_nonsymm_alloc526880
Ref: 5b3526880
Ref: eigen c gsl_eigen_nonsymm_free527141
Ref: 5b4527141
Ref: eigen c gsl_eigen_nonsymm_params527304
Ref: 5b5527304
Ref: eigen c gsl_eigen_nonsymm528999
Ref: 5b6528999
Ref: eigen c gsl_eigen_nonsymm_Z529793
Ref: 5b7529793
Ref: eigen c gsl_eigen_nonsymmv_workspace530100
Ref: 5b8530100
Ref: eigen c gsl_eigen_nonsymmv_alloc530262
Ref: 5b9530262
Ref: eigen c gsl_eigen_nonsymmv_free530542
Ref: 5ba530542
Ref: eigen c gsl_eigen_nonsymmv_params530707
Ref: 5bb530707
Ref: eigen c gsl_eigen_nonsymmv531244
Ref: 5bc531244
Ref: eigen c gsl_eigen_nonsymmv_Z532114
Ref: 5bd532114
Node: Real Generalized Symmetric-Definite Eigensystems532427
Ref: eigen real-generalized-symmetric-definite-eigensystems532620
Ref: 5be532620
Ref: eigen c gsl_eigen_gensymm_workspace533515
Ref: 5bf533515
Ref: eigen c gsl_eigen_gensymm_alloc533669
Ref: 5c0533669
Ref: eigen c gsl_eigen_gensymm_free533952
Ref: 5c1533952
Ref: eigen c gsl_eigen_gensymm534115
Ref: 5c2534115
Ref: eigen c gsl_eigen_gensymmv_workspace534528
Ref: 5c3534528
Ref: eigen c gsl_eigen_gensymmv_alloc534699
Ref: 5c4534699
Ref: eigen c gsl_eigen_gensymmv_free535001
Ref: 5c5535001
Ref: eigen c gsl_eigen_gensymmv535166
Ref: 5c6535166
Node: Complex Generalized Hermitian-Definite Eigensystems535693
Ref: eigen complex-generalized-hermitian-definite-eigensystems535902
Ref: 5c7535902
Ref: eigen c gsl_eigen_genherm_workspace536612
Ref: 5c8536612
Ref: eigen c gsl_eigen_genherm_alloc536766
Ref: 5c9536766
Ref: eigen c gsl_eigen_genherm_free537051
Ref: 5ca537051
Ref: eigen c gsl_eigen_genherm537214
Ref: 5cb537214
Ref: eigen c gsl_eigen_genhermv_workspace537656
Ref: 5cc537656
Ref: eigen c gsl_eigen_genhermv_alloc537827
Ref: 5cd537827
Ref: eigen c gsl_eigen_genhermv_free538132
Ref: 5ce538132
Ref: eigen c gsl_eigen_genhermv538297
Ref: 5cf538297
Node: Real Generalized Nonsymmetric Eigensystems538851
Ref: eigen real-generalized-nonsymmetric-eigensystems539048
Ref: 5d0539048
Ref: eigen c gsl_eigen_gen_workspace540840
Ref: 5d1540840
Ref: eigen c gsl_eigen_gen_alloc540980
Ref: 5d2540980
Ref: eigen c gsl_eigen_gen_free541248
Ref: 5d3541248
Ref: eigen c gsl_eigen_gen_params541393
Ref: 5d4541393
Ref: eigen c gsl_eigen_gen542546
Ref: 5d5542546
Ref: eigen c gsl_eigen_gen_QZ543617
Ref: 5d6543617
Ref: eigen c gsl_eigen_genv_workspace544005
Ref: 5d7544005
Ref: eigen c gsl_eigen_genv_alloc544162
Ref: 5d8544162
Ref: eigen c gsl_eigen_genv_free544450
Ref: 5d9544450
Ref: eigen c gsl_eigen_genv544597
Ref: 5da544597
Ref: eigen c gsl_eigen_genv_QZ545650
Ref: 5db545650
Node: Sorting Eigenvalues and Eigenvectors546078
Ref: eigen sorting-eigenvalues-and-eigenvectors546236
Ref: 5dc546236
Ref: eigen c gsl_eigen_symmv_sort546321
Ref: 5dd546321
Ref: eigen c gsl_eigen_sort_t546721
Ref: 5de546721
Ref: eigen c gsl_eigen_hermv_sort547436
Ref: 5df547436
Ref: eigen c gsl_eigen_nonsymmv_sort547862
Ref: 5e0547862
Ref: eigen c gsl_eigen_gensymmv_sort548429
Ref: 5e1548429
Ref: eigen c gsl_eigen_genhermv_sort548847
Ref: 5e2548847
Ref: eigen c gsl_eigen_genv_sort549276
Ref: 5e3549276
Node: Examples<10>549896
Ref: eigen examples550046
Ref: 5e4550046
Node: References and Further Reading<10>555255
Ref: eigen references-and-further-reading555360
Ref: 5e5555360
Node: Fast Fourier Transforms FFTs556255
Ref: fft doc556375
Ref: 5e6556375
Ref: fft fast-fourier-transforms-ffts556375
Ref: 5e7556375
Node: Mathematical Definitions557507
Ref: fft mathematical-definitions557635
Ref: 5e9557635
Node: Overview of complex data FFTs559949
Ref: fft overview-of-complex-data-ffts560123
Ref: 5ec560123
Node: Radix-2 FFT routines for complex data562910
Ref: fft radix-2-fft-routines-for-complex-data563101
Ref: 5ed563101
Ref: fft c gsl_fft_complex_radix2_forward563683
Ref: 35563683
Ref: fft c gsl_fft_complex_radix2_transform563799
Ref: 5ee563799
Ref: fft c gsl_fft_complex_radix2_backward563951
Ref: 5ef563951
Ref: fft c gsl_fft_complex_radix2_inverse564068
Ref: 5f0564068
Ref: fft c gsl_fft_complex_radix2_dif_forward564787
Ref: 5f1564787
Ref: fft c gsl_fft_complex_radix2_dif_transform564907
Ref: 5f2564907
Ref: fft c gsl_fft_complex_radix2_dif_backward565063
Ref: 5f3565063
Ref: fft c gsl_fft_complex_radix2_dif_inverse565184
Ref: 5f4565184
Ref: fft fig-fft-complex-radix2567333
Ref: 5f5567333
Node: Mixed-radix FFT routines for complex data567463
Ref: fft mixed-radix-fft-routines-for-complex-data567651
Ref: 5f6567651
Ref: fft c gsl_fft_complex_wavetable_alloc569914
Ref: 5f7569914
Ref: fft c gsl_fft_complex_wavetable_free571073
Ref: 5f9571073
Ref: fft c gsl_fft_complex_wavetable571804
Ref: 5f8571804
Ref: fft c gsl_fft_complex_workspace573010
Ref: 5fa573010
Ref: fft c gsl_fft_complex_workspace_alloc573167
Ref: 5fb573167
Ref: fft c gsl_fft_complex_workspace_free573353
Ref: 5fc573353
Ref: fft c gsl_fft_complex_forward573678
Ref: 5ea573678
Ref: fft c gsl_fft_complex_transform573886
Ref: 5fd573886
Ref: fft c gsl_fft_complex_backward574120
Ref: 5fe574120
Ref: fft c gsl_fft_complex_inverse574329
Ref: 5eb574329
Node: Overview of real data FFTs577657
Ref: fft overview-of-real-data-ffts577842
Ref: 5ff577842
Node: Radix-2 FFT routines for real data580114
Ref: fft radix-2-fft-routines-for-real-data580296
Ref: 600580296
Ref: fft c gsl_fft_real_radix2_transform580632
Ref: 601580632
Ref: fft c gsl_fft_halfcomplex_radix2_inverse582703
Ref: 603582703
Ref: fft c gsl_fft_halfcomplex_radix2_backward582807
Ref: 604582807
Ref: fft c gsl_fft_halfcomplex_radix2_unpack583216
Ref: 602583216
Node: Mixed-radix FFT routines for real data584567
Ref: fft mixed-radix-fft-routines-for-real-data584757
Ref: 605584757
Ref: fft c gsl_fft_real_wavetable587986
Ref: 607587986
Ref: fft c gsl_fft_halfcomplex_wavetable588019
Ref: 608588019
Ref: fft c gsl_fft_real_wavetable_alloc588143
Ref: 609588143
Ref: fft c gsl_fft_halfcomplex_wavetable_alloc588232
Ref: 60a588232
Ref: fft c gsl_fft_real_wavetable_free589369
Ref: 60b589369
Ref: fft c gsl_fft_halfcomplex_wavetable_free589463
Ref: 60c589463
Ref: fft c gsl_fft_real_workspace589859
Ref: 60d589859
Ref: fft c gsl_fft_real_workspace_alloc589964
Ref: 60e589964
Ref: fft c gsl_fft_real_workspace_free590236
Ref: 60f590236
Ref: fft c gsl_fft_real_transform590586
Ref: 606590586
Ref: fft c gsl_fft_halfcomplex_transform590761
Ref: 610590761
Ref: fft c gsl_fft_real_unpack591692
Ref: 611591692
Ref: fft c gsl_fft_halfcomplex_unpack592328
Ref: 612592328
Ref: fft fig-fft-real-mixedradix595703
Ref: 613595703
Node: References and Further Reading<11>595833
Ref: fft fft-references595980
Ref: 5e8595980
Ref: fft references-and-further-reading595980
Ref: 614595980
Node: Numerical Integration598663
Ref: integration doc598795
Ref: 615598795
Ref: integration numerical-integration598795
Ref: 616598795
Node: Introduction<2>600520
Ref: integration introduction600645
Ref: 617600645
Node: Integrands without weight functions602745
Ref: integration integrands-without-weight-functions602874
Ref: 618602874
Node: Integrands with weight functions603552
Ref: integration integrands-with-weight-functions603731
Ref: 619603731
Node: Integrands with singular weight functions604241
Ref: integration integrands-with-singular-weight-functions604376
Ref: 61a604376
Node: QNG non-adaptive Gauss-Kronrod integration605138
Ref: integration qng-non-adaptive-gauss-kronrod-integration605296
Ref: 61b605296
Ref: integration c gsl_integration_qng605600
Ref: 61c605600
Node: QAG adaptive integration606430
Ref: integration qag-adaptive-integration606617
Ref: 61d606617
Ref: integration c gsl_integration_workspace607061
Ref: 61e607061
Ref: integration c gsl_integration_workspace_alloc607200
Ref: 61f607200
Ref: integration c gsl_integration_workspace_free607591
Ref: 620607591
Ref: integration c gsl_integration_qag607760
Ref: 621607760
Ref: integration c GSL_INTEG_GAUSS15608481
Ref: 622608481
Ref: integration c GSL_INTEG_GAUSS21608531
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Ref: integration c GSL_INTEG_GAUSS31608581
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Ref: integration c GSL_INTEG_GAUSS41608631
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Ref: integration c GSL_INTEG_GAUSS51608681
Ref: 626608681
Ref: integration c GSL_INTEG_GAUSS61608731
Ref: 627608731
Node: QAGS adaptive integration with singularities609379
Ref: integration qags-adaptive-integration-with-singularities609576
Ref: 628609576
Ref: integration c gsl_integration_qags610177
Ref: 629610177
Node: QAGP adaptive integration with known singular points611215
Ref: integration qagp-adaptive-integration-with-known-singular-points611435
Ref: 62a611435
Ref: integration c gsl_integration_qagp611552
Ref: 62b611552
Node: QAGI adaptive integration on infinite intervals612586
Ref: integration qagi-adaptive-integration-on-infinite-intervals612815
Ref: 62c612815
Ref: integration c gsl_integration_qagi612922
Ref: 62d612922
Ref: integration c gsl_integration_qagiu613708
Ref: 62e613708
Ref: integration c gsl_integration_qagil614274
Ref: 62f614274
Node: QAWC adaptive integration for Cauchy principal values614840
Ref: integration qawc-adaptive-integration-for-cauchy-principal-values615065
Ref: 630615065
Ref: integration c gsl_integration_qawc615184
Ref: 631615184
Node: QAWS adaptive integration for singular functions616016
Ref: integration qaws-adaptive-integration-for-singular-functions616245
Ref: 632616245
Ref: integration c gsl_integration_qaws_table616581
Ref: 633616581
Ref: integration c gsl_integration_qaws_table_alloc616701
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Ref: integration c gsl_integration_qaws_table_set618722
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Ref: integration c gsl_integration_qaws_table_free619013
Ref: 636619013
Ref: integration c gsl_integration_qaws619229
Ref: 637619229
Node: QAWO adaptive integration for oscillatory functions620267
Ref: integration qawo-adaptive-integration-for-oscillatory-functions620490
Ref: 638620490
Ref: integration c gsl_integration_qawo_table_alloc620858
Ref: 639620858
Ref: integration c GSL_INTEG_COSINE621563
Ref: 63a621563
Ref: integration c GSL_INTEG_SINE621597
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Ref: integration c gsl_integration_qawo_table_set622183
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Ref: integration c gsl_integration_qawo_table_set_length622466
Ref: 63e622466
Ref: integration c gsl_integration_qawo_table_free622677
Ref: 63f622677
Ref: integration c gsl_integration_qawo622857
Ref: 63c622857
Node: QAWF adaptive integration for Fourier integrals624208
Ref: integration qawf-adaptive-integration-for-fourier-integrals624416
Ref: 640624416
Ref: integration c gsl_integration_qawf624525
Ref: 641624525
Node: CQUAD doubly-adaptive integration626886
Ref: integration cquad-doubly-adaptive-integration627062
Ref: 642627062
Ref: integration c gsl_integration_cquad_workspace_alloc627933
Ref: 643627933
Ref: integration c gsl_integration_cquad_workspace_free628428
Ref: 644628428
Ref: integration c gsl_integration_cquad628609
Ref: 645628609
Node: Romberg integration630208
Ref: integration romberg-integration630363
Ref: 646630363
Ref: integration c gsl_integration_romberg_alloc630945
Ref: 647630945
Ref: integration c gsl_integration_romberg_free631396
Ref: 648631396
Ref: integration c gsl_integration_romberg631571
Ref: 649631571
Node: Gauss-Legendre integration632545
Ref: integration gauss-legendre-integration632690
Ref: 64a632690
Ref: integration c gsl_integration_glfixed_table_alloc633222
Ref: 64b633222
Ref: integration c gsl_integration_glfixed633636
Ref: 64c633636
Ref: integration c gsl_integration_glfixed_point633898
Ref: 64d633898
Ref: integration c gsl_integration_glfixed_table_free634401
Ref: 64e634401
Node: Fixed point quadratures634574
Ref: integration fixed-point-quadratures634711
Ref: 64f634711
Ref: integration tab-fixed-quadratures635654
Ref: 650635654
Ref: integration c gsl_integration_fixed_workspace639016
Ref: 651639016
Ref: integration c gsl_integration_fixed_alloc639573
Ref: 652639573
Ref: integration c gsl_integration_fixed_type640227
Ref: 653640227
Ref: integration c gsl_integration_fixed_legendre640386
Ref: 654640386
Ref: integration c gsl_integration_fixed_chebyshev640612
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Ref: integration c gsl_integration_fixed_gegenbauer640846
Ref: 656640846
Ref: integration c gsl_integration_fixed_jacobi641037
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Ref: integration c gsl_integration_fixed_laguerre641153
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Ref: integration c gsl_integration_fixed_hermite641340
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Ref: integration c gsl_integration_fixed_exponential641525
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Ref: integration c gsl_integration_fixed_rational641718
Ref: 65b641718
Ref: integration c gsl_integration_fixed_chebyshev2641838
Ref: 65c641838
Ref: integration c gsl_integration_fixed_free642073
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Ref: integration c gsl_integration_fixed_n642242
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Ref: integration c gsl_integration_fixed_nodes642414
Ref: 65f642414
Ref: integration c gsl_integration_fixed_weights642628
Ref: 660642628
Ref: integration c gsl_integration_fixed642846
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Node: Error codes643382
Ref: integration error-codes643505
Ref: 662643505
Node: Examples<11>644425
Ref: integration examples644559
Ref: 663644559
Node: Adaptive integration example644668
Ref: integration adaptive-integration-example644785
Ref: 664644785
Node: Fixed-point quadrature example646692
Ref: integration fixed-point-quadrature-example646809
Ref: 665646809
Node: References and Further Reading<12>649483
Ref: integration references-and-further-reading649597
Ref: 666649597
Node: Random Number Generation650924
Ref: rng doc651050
Ref: 667651050
Ref: rng random-number-generation651050
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Node: General comments on random numbers652399
Ref: rng general-comments-on-random-numbers652541
Ref: 669652541
Node: The Random Number Generator Interface654368
Ref: rng the-random-number-generator-interface654557
Ref: 66a654557
Ref: rng c gsl_rng_type655410
Ref: 66b655410
Ref: rng c gsl_rng655433
Ref: 66c655433
Node: Random number generator initialization655823
Ref: rng random-number-generator-initialization656017
Ref: 66d656017
Ref: rng c gsl_rng_alloc656106
Ref: 66e656106
Ref: rng c gsl_rng_set656954
Ref: 671656954
Ref: rng c gsl_rng_free658101
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Node: Sampling from a random number generator658233
Ref: rng sampling-from-a-random-number-generator658433
Ref: 674658433
Ref: rng c gsl_rng_get658819
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Ref: rng c gsl_rng_uniform659255
Ref: 679659255
Ref: rng c gsl_rng_uniform_pos659863
Ref: 67a659863
Ref: rng c gsl_rng_uniform_int660306
Ref: 67b660306
Node: Auxiliary random number generator functions661637
Ref: rng auxiliary-random-number-generator-functions661834
Ref: 67f661834
Ref: rng c gsl_rng_name662101
Ref: 680662101
Ref: rng c gsl_rng_max662383
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Ref: rng c gsl_rng_min662542
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Ref: rng c gsl_rng_state662858
Ref: 681662858
Ref: rng c gsl_rng_size662913
Ref: 682662913
Ref: rng c gsl_rng_types_setup663336
Ref: 683663336
Node: Random number environment variables663983
Ref: rng random-number-environment-variables664178
Ref: 684664178
Ref: rng c GSL_RNG_TYPE664559
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Ref: rng c GSL_RNG_SEED664749
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Ref: rng c gsl_rng_default664870
Ref: 687664870
Ref: rng c gsl_rng_default_seed665165
Ref: 66f665165
Ref: rng c gsl_rng_env_setup665516
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Node: Copying random number generator state667264
Ref: rng copying-random-number-generator-state667465
Ref: 688667465
Ref: rng c gsl_rng_memcpy667801
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Ref: rng c gsl_rng_clone668089
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Node: Reading and writing random number generator state668269
Ref: rng reading-and-writing-random-number-generator-state668469
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Ref: rng c gsl_rng_fwrite668686
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Ref: rng c gsl_rng_fread669127
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Node: Random number generator algorithms669703
Ref: rng random-number-generator-algorithms669895
Ref: 68e669895
Ref: rng c gsl_rng_mt19937670667
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Ref: rng c gsl_rng_ranlxs0672153
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Ref: rng c gsl_rng_ranlxs1672183
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Ref: rng c gsl_rng_ranlxs2672213
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Ref: rng c gsl_rng_ranlxd1673194
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Ref: rng c gsl_rng_ranlxd2673224
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Ref: rng c gsl_rng_ranlux673463
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Ref: rng c gsl_rng_ranlux389673492
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Ref: rng c gsl_rng_cmrg674954
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Ref: rng c gsl_rng_mrg675811
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Ref: rng c gsl_rng_taus676461
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Ref: rng c gsl_rng_taus2676488
Ref: 696676488
Ref: rng c gsl_rng_gfsr4677925
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Node: Unix random number generators679898
Ref: rng unix-random-number-generators680071
Ref: 698680071
Ref: rng c gsl_rng_rand680812
Ref: 699680812
Ref: rng c gsl_rng_random_bsd681134
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Ref: rng c gsl_rng_random_libc5681167
Ref: 69b681167
Ref: rng c gsl_rng_random_glibc2681202
Ref: 69c681202
Ref: rng c gsl_rng_rand48682725
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Node: Other random number generators683745
Ref: rng other-random-number-generators683895
Ref: 69e683895
Ref: rng c gsl_rng_ranf684924
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Ref: rng c gsl_rng_ranmar686375
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Ref: rng c gsl_rng_r250686661
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Ref: rng c gsl_rng_tt800687229
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Ref: rng c gsl_rng_vax687795
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Ref: rng c gsl_rng_transputer688113
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Ref: rng c gsl_rng_randu688382
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Ref: rng c gsl_rng_minstd688702
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Ref: rng c gsl_rng_uni689553
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Ref: rng c gsl_rng_uni32689579
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Ref: rng c gsl_rng_slatec689846
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Ref: rng c gsl_rng_zuf690002
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Ref: rng c gsl_rng_knuthran2690442
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Ref: rng c gsl_rng_knuthran2002690743
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Ref: rng c gsl_rng_knuthran690778
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Ref: rng c gsl_rng_borosh13691184
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Ref: rng c gsl_rng_fishman18691215
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Ref: rng c gsl_rng_fishman20691247
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Ref: rng c gsl_rng_lecuyer21691279
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Ref: rng c gsl_rng_waterman14691311
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Ref: rng c gsl_rng_fishman2x691842
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Ref: rng c gsl_rng_coveyou692232
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Node: Performance692531
Ref: rng performance692664
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Node: Examples<12>693620
Ref: rng examples693757
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Node: References and Further Reading<13>695087
Ref: rng references-and-further-reading695229
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Node: Acknowledgements696368
Ref: rng acknowledgements696489
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Node: Quasi-Random Sequences696829
Ref: qrng doc696961
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Ref: qrng quasi-random-sequences696961
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Node: Quasi-random number generator initialization697900
Ref: qrng quasi-random-number-generator-initialization698058
Ref: 6bb698058
Ref: qrng c gsl_qrng698159
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Ref: qrng c gsl_qrng_alloc698243
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Ref: qrng c gsl_qrng_free698673
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Ref: qrng c gsl_qrng_init698807
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Node: Sampling from a quasi-random number generator699040
Ref: qrng sampling-from-a-quasi-random-number-generator699256
Ref: 6c0699256
Ref: qrng c gsl_qrng_get699359
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Node: Auxiliary quasi-random number generator functions699766
Ref: qrng auxiliary-quasi-random-number-generator-functions699994
Ref: 6c2699994
Ref: qrng c gsl_qrng_name700105
Ref: 6c3700105
Ref: qrng c gsl_qrng_size700236
Ref: 6c4700236
Ref: qrng c gsl_qrng_state700292
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Node: Saving and restoring quasi-random number generator state700720
Ref: qrng saving-and-restoring-quasi-random-number-generator-state700943
Ref: 6c6700943
Ref: qrng c gsl_qrng_memcpy701068
Ref: 6c7701068
Ref: qrng c gsl_qrng_clone701377
Ref: 6c8701377
Node: Quasi-random number generator algorithms701560
Ref: qrng quasi-random-number-generator-algorithms701746
Ref: 6c9701746
Ref: qrng c gsl_qrng_type701902
Ref: 6ca701902
Ref: qrng c gsl_qrng_niederreiter_2701927
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Ref: qrng c gsl_qrng_sobol702158
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Ref: qrng c gsl_qrng_halton702378
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Ref: qrng c gsl_qrng_reversehalton702413
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Node: Examples<13>702771
Ref: qrng examples702911
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Ref: qrng fig-qrng703808
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Node: References703922
Ref: qrng references704013
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Node: Random Number Distributions704411
Ref: randist doc704529
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Ref: randist chap-random-number-distributions704529
Ref: 675704529
Ref: randist random-number-distributions704529
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Node: Introduction<3>707701
Ref: randist introduction707815
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Node: The Gaussian Distribution709473
Ref: randist the-gaussian-distribution709626
Ref: 6d5709626
Ref: randist c gsl_ran_gaussian709689
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Ref: randist c gsl_ran_gaussian_pdf710339
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Ref: randist c gsl_ran_gaussian_ziggurat710619
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Ref: randist c gsl_ran_gaussian_ratio_method710710
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Ref: randist c gsl_ran_ugaussian711039
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Ref: randist c gsl_ran_ugaussian_pdf711098
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Ref: randist c gsl_ran_ugaussian_ratio_method711152
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Ref: randist c gsl_cdf_gaussian_P711403
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Ref: randist c gsl_cdf_gaussian_Q711468
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Ref: randist c gsl_cdf_gaussian_Pinv711533
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Ref: randist c gsl_cdf_gaussian_Qinv711601
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Ref: randist c gsl_cdf_ugaussian_P711845
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Ref: randist c gsl_cdf_ugaussian_Q711897
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Ref: randist c gsl_cdf_ugaussian_Pinv711949
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Ref: randist c gsl_cdf_ugaussian_Qinv712004
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Node: The Gaussian Tail Distribution712199
Ref: randist the-gaussian-tail-distribution712372
Ref: 6e5712372
Ref: randist c gsl_ran_gaussian_tail712445
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Ref: randist c gsl_ran_gaussian_tail_pdf713270
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Ref: randist c gsl_ran_ugaussian_tail713609
Ref: 6e8713609
Ref: randist c gsl_ran_ugaussian_tail_pdf713693
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Node: The Bivariate Gaussian Distribution713952
Ref: randist the-bivariate-gaussian-distribution714138
Ref: 6ea714138
Ref: randist c gsl_ran_bivariate_gaussian714221
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Ref: randist c gsl_ran_bivariate_gaussian_pdf714955
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Node: The Multivariate Gaussian Distribution715389
Ref: randist the-multivariate-gaussian-distribution715573
Ref: 6ed715573
Ref: randist c gsl_ran_multivariate_gaussian715662
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Ref: randist c gsl_ran_multivariate_gaussian_pdf716490
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Ref: randist c gsl_ran_multivariate_gaussian_log_pdf716666
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Ref: randist c gsl_ran_multivariate_gaussian_mean717118
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Ref: randist c gsl_ran_multivariate_gaussian_vcov717648
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Node: The Exponential Distribution718269
Ref: randist the-exponential-distribution718442
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Ref: randist c gsl_ran_exponential718511
Ref: 6f4718511
Ref: randist c gsl_ran_exponential_pdf718780
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Ref: randist c gsl_cdf_exponential_P719050
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Ref: randist c gsl_cdf_exponential_Q719115
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Ref: randist c gsl_cdf_exponential_Pinv719180
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Ref: randist c gsl_cdf_exponential_Qinv719248
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Node: The Laplace Distribution719478
Ref: randist the-laplace-distribution719647
Ref: 6fa719647
Ref: randist c gsl_ran_laplace719708
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Ref: randist c gsl_ran_laplace_pdf719981
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Ref: randist c gsl_cdf_laplace_P720232
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Ref: randist c gsl_cdf_laplace_Q720292
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Ref: randist c gsl_cdf_laplace_Pinv720352
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Ref: randist c gsl_cdf_laplace_Qinv720415
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Node: The Exponential Power Distribution720636
Ref: randist the-exponential-power-distribution720800
Ref: 701720800
Ref: randist c gsl_ran_exppow720881
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Ref: randist c gsl_ran_exppow_pdf721374
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Ref: randist c gsl_cdf_exppow_P721680
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Ref: randist c gsl_cdf_exppow_Q721749
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Node: The Cauchy Distribution721984
Ref: randist the-cauchy-distribution722149
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Ref: randist c gsl_ran_cauchy722208
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Ref: randist c gsl_ran_cauchy_pdf722622
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Ref: randist c gsl_cdf_cauchy_P722885
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Ref: randist c gsl_cdf_cauchy_Pinv723003
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Ref: randist c gsl_cdf_cauchy_Qinv723065
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Node: The Rayleigh Distribution723294
Ref: randist the-rayleigh-distribution723455
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Ref: randist c gsl_ran_rayleigh723520
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Ref: randist c gsl_ran_rayleigh_pdf723815
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Ref: randist c gsl_cdf_rayleigh_P724092
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Ref: randist c gsl_cdf_rayleigh_Pinv724222
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Ref: randist c gsl_cdf_rayleigh_Qinv724290
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Node: The Rayleigh Tail Distribution724531
Ref: randist the-rayleigh-tail-distribution724692
Ref: 714724692
Ref: randist c gsl_ran_rayleigh_tail724767
Ref: 715724767
Ref: randist c gsl_ran_rayleigh_tail_pdf725141
Ref: 716725141
Node: The Landau Distribution725477
Ref: randist the-landau-distribution725648
Ref: 717725648
Ref: randist c gsl_ran_landau725709
Ref: 718725709
Ref: randist c gsl_ran_landau_pdf726230
Ref: 719726230
Node: The Levy alpha-Stable Distributions726475
Ref: randist the-levy-alpha-stable-distributions726655
Ref: 71a726655
Ref: randist c gsl_ran_levy726740
Ref: 71b726740
Node: The Levy skew alpha-Stable Distribution727589
Ref: randist the-levy-skew-alpha-stable-distribution727768
Ref: 71c727768
Ref: randist c gsl_ran_levy_skew727861
Ref: 71d727861
Node: The Gamma Distribution729261
Ref: randist the-gamma-distribution729434
Ref: 71e729434
Ref: randist c gsl_ran_gamma729493
Ref: 71f729493
Ref: randist c gsl_ran_gamma_knuth730083
Ref: 720730083
Ref: randist c gsl_ran_gamma_pdf730265
Ref: 721730265
Ref: randist c gsl_cdf_gamma_P730542
Ref: 722730542
Ref: randist c gsl_cdf_gamma_Q730610
Ref: 723730610
Ref: randist c gsl_cdf_gamma_Pinv730678
Ref: 724730678
Ref: randist c gsl_cdf_gamma_Qinv730749
Ref: 725730749
Node: The Flat Uniform Distribution730993
Ref: randist the-flat-uniform-distribution731153
Ref: 726731153
Ref: randist c gsl_ran_flat731230
Ref: 727731230
Ref: randist c gsl_ran_flat_pdf731527
Ref: 728731527
Ref: randist c gsl_cdf_flat_P731792
Ref: 729731792
Ref: randist c gsl_cdf_flat_Q731859
Ref: 72a731859
Ref: randist c gsl_cdf_flat_Pinv731926
Ref: 72b731926
Ref: randist c gsl_cdf_flat_Qinv731996
Ref: 72c731996
Node: The Lognormal Distribution732222
Ref: randist the-lognormal-distribution732388
Ref: 72d732388
Ref: randist c gsl_ran_lognormal732455
Ref: 72e732455
Ref: randist c gsl_ran_lognormal_pdf732778
Ref: 72f732778
Ref: randist c gsl_cdf_lognormal_P733091
Ref: 730733091
Ref: randist c gsl_cdf_lognormal_Q733180
Ref: 731733180
Ref: randist c gsl_cdf_lognormal_Pinv733269
Ref: 732733269
Ref: randist c gsl_cdf_lognormal_Qinv733361
Ref: 733733361
Node: The Chi-squared Distribution733637
Ref: randist the-chi-squared-distribution733792
Ref: 734733792
Ref: randist c gsl_ran_chisq734094
Ref: 735734094
Ref: randist c gsl_ran_chisq_pdf734414
Ref: 736734414
Ref: randist c gsl_cdf_chisq_P734685
Ref: 737734685
Ref: randist c gsl_cdf_chisq_Q734744
Ref: 738734744
Ref: randist c gsl_cdf_chisq_Pinv734803
Ref: 739734803
Ref: randist c gsl_cdf_chisq_Qinv734865
Ref: 73a734865
Node: The F-distribution735103
Ref: randist the-f-distribution735250
Ref: 73b735250
Ref: randist c gsl_ran_fdist735529
Ref: 73c735529
Ref: randist c gsl_ran_fdist_pdf736024
Ref: 73d736024
Ref: randist c gsl_cdf_fdist_P736324
Ref: 73e736324
Ref: randist c gsl_cdf_fdist_Q736396
Ref: 73f736396
Ref: randist c gsl_cdf_fdist_Pinv736468
Ref: 740736468
Ref: randist c gsl_cdf_fdist_Qinv736553
Ref: 741736553
Node: The t-distribution736819
Ref: randist the-t-distribution736959
Ref: 742736959
Ref: randist c gsl_ran_tdist737270
Ref: 743737270
Ref: randist c gsl_ran_tdist_pdf737585
Ref: 744737585
Ref: randist c gsl_cdf_tdist_P737846
Ref: 745737846
Ref: randist c gsl_cdf_tdist_Q737905
Ref: 746737905
Ref: randist c gsl_cdf_tdist_Pinv737964
Ref: 747737964
Ref: randist c gsl_cdf_tdist_Qinv738026
Ref: 748738026
Node: The Beta Distribution738254
Ref: randist the-beta-distribution738401
Ref: 749738401
Ref: randist c gsl_ran_beta738458
Ref: 74a738458
Ref: randist c gsl_ran_beta_pdf738760
Ref: 74b738760
Ref: randist c gsl_cdf_beta_P739034
Ref: 74c739034
Ref: randist c gsl_cdf_beta_Q739101
Ref: 74d739101
Ref: randist c gsl_cdf_beta_Pinv739168
Ref: 74e739168
Ref: randist c gsl_cdf_beta_Qinv739238
Ref: 74f739238
Node: The Logistic Distribution739480
Ref: randist the-logistic-distribution739632
Ref: 750739632
Ref: randist c gsl_ran_logistic739697
Ref: 751739697
Ref: randist c gsl_ran_logistic_pdf739978
Ref: 752739978
Ref: randist c gsl_cdf_logistic_P740247
Ref: 753740247
Ref: randist c gsl_cdf_logistic_Q740308
Ref: 754740308
Ref: randist c gsl_cdf_logistic_Pinv740369
Ref: 755740369
Ref: randist c gsl_cdf_logistic_Qinv740433
Ref: 756740433
Node: The Pareto Distribution740666
Ref: randist the-pareto-distribution740827
Ref: 757740827
Ref: randist c gsl_ran_pareto740888
Ref: 758740888
Ref: randist c gsl_ran_pareto_pdf741166
Ref: 759741166
Ref: randist c gsl_cdf_pareto_P741450
Ref: 75a741450
Ref: randist c gsl_cdf_pareto_Q741519
Ref: 75b741519
Ref: randist c gsl_cdf_pareto_Pinv741588
Ref: 75c741588
Ref: randist c gsl_cdf_pareto_Qinv741660
Ref: 75d741660
Node: Spherical Vector Distributions741910
Ref: randist spherical-vector-distributions742070
Ref: 75e742070
Ref: randist c gsl_ran_dir_2d742314
Ref: 75f742314
Ref: randist c gsl_ran_dir_2d_trig_method742402
Ref: 760742402
Ref: randist c gsl_ran_dir_3d743654
Ref: 761743654
Ref: randist c gsl_ran_dir_nd744196
Ref: 762744196
Node: The Weibull Distribution744830
Ref: randist the-weibull-distribution744997
Ref: 763744997
Ref: randist c gsl_ran_weibull745060
Ref: 764745060
Ref: randist c gsl_ran_weibull_pdf745340
Ref: 765745340
Ref: randist c gsl_cdf_weibull_P745627
Ref: 766745627
Ref: randist c gsl_cdf_weibull_Q745697
Ref: 767745697
Ref: randist c gsl_cdf_weibull_Pinv745767
Ref: 768745767
Ref: randist c gsl_cdf_weibull_Qinv745840
Ref: 769745840
Node: The Type-1 Gumbel Distribution746092
Ref: randist the-type-1-gumbel-distribution746259
Ref: 76a746259
Ref: randist c gsl_ran_gumbel1746334
Ref: 76b746334
Ref: randist c gsl_ran_gumbel1_pdf746640
Ref: 76c746640
Ref: randist c gsl_cdf_gumbel1_P746929
Ref: 76d746929
Ref: randist c gsl_cdf_gumbel1_Q746999
Ref: 76e746999
Ref: randist c gsl_cdf_gumbel1_Pinv747069
Ref: 76f747069
Ref: randist c gsl_cdf_gumbel1_Qinv747142
Ref: 770747142
Node: The Type-2 Gumbel Distribution747396
Ref: randist the-type-2-gumbel-distribution747565
Ref: 771747565
Ref: randist c gsl_ran_gumbel2747640
Ref: 772747640
Ref: randist c gsl_ran_gumbel2_pdf747939
Ref: 773747939
Ref: randist c gsl_cdf_gumbel2_P748228
Ref: 774748228
Ref: randist c gsl_cdf_gumbel2_Q748298
Ref: 775748298
Ref: randist c gsl_cdf_gumbel2_Pinv748368
Ref: 776748368
Ref: randist c gsl_cdf_gumbel2_Qinv748441
Ref: 777748441
Node: The Dirichlet Distribution748695
Ref: randist the-dirichlet-distribution748864
Ref: 778748864
Ref: randist c gsl_ran_dirichlet748931
Ref: 779748931
Ref: randist c gsl_ran_dirichlet_pdf749825
Ref: 77a749825
Ref: randist c gsl_ran_dirichlet_lnpdf750138
Ref: 77b750138
Node: General Discrete Distributions750421
Ref: randist general-discrete-distributions750584
Ref: 77c750584
Ref: randist c gsl_ran_discrete_t752933
Ref: 77d752933
Ref: randist c gsl_ran_discrete_preproc753057
Ref: 77e753057
Ref: randist c gsl_ran_discrete753664
Ref: 77f753664
Ref: randist c gsl_ran_discrete_pdf753880
Ref: 780753880
Ref: randist c gsl_ran_discrete_free754329
Ref: 781754329
Node: The Poisson Distribution754455
Ref: randist the-poisson-distribution754618
Ref: 782754618
Ref: randist c gsl_ran_poisson754681
Ref: 783754681
Ref: randist c gsl_ran_poisson_pdf754991
Ref: 784754991
Ref: randist c gsl_cdf_poisson_P755252
Ref: 785755252
Ref: randist c gsl_cdf_poisson_Q755319
Ref: 786755319
Node: The Bernoulli Distribution755525
Ref: randist the-bernoulli-distribution755683
Ref: 787755683
Ref: randist c gsl_ran_bernoulli755750
Ref: 788755750
Ref: randist c gsl_ran_bernoulli_pdf756046
Ref: 789756046
Node: The Binomial Distribution756333
Ref: randist the-binomial-distribution756495
Ref: 78a756495
Ref: randist c gsl_ran_binomial756560
Ref: 78b756560
Ref: randist c gsl_ran_binomial_pdf756969
Ref: 78c756969
Ref: randist c gsl_cdf_binomial_P757280
Ref: 78d757280
Ref: randist c gsl_cdf_binomial_Q757373
Ref: 78e757373
Node: The Multinomial Distribution757618
Ref: randist the-multinomial-distribution757788
Ref: 78f757788
Ref: randist c gsl_ran_multinomial757859
Ref: 790757859
Ref: randist c gsl_ran_multinomial_pdf758898
Ref: 791758898
Ref: randist c gsl_ran_multinomial_lnpdf759201
Ref: 792759201
Node: The Negative Binomial Distribution759469
Ref: randist the-negative-binomial-distribution759637
Ref: 793759637
Ref: randist c gsl_ran_negative_binomial759720
Ref: 794759720
Ref: randist c gsl_ran_negative_binomial_pdf760234
Ref: 795760234
Ref: randist c gsl_cdf_negative_binomial_P760559
Ref: 796760559
Ref: randist c gsl_cdf_negative_binomial_Q760655
Ref: 797760655
Node: The Pascal Distribution760917
Ref: randist the-pascal-distribution761083
Ref: 798761083
Ref: randist c gsl_ran_pascal761144
Ref: 799761144
Ref: randist c gsl_ran_pascal_pdf761507
Ref: 79a761507
Ref: randist c gsl_cdf_pascal_P761812
Ref: 79b761812
Ref: randist c gsl_cdf_pascal_Q761903
Ref: 79c761903
Node: The Geometric Distribution762144
Ref: randist the-geometric-distribution762307
Ref: 79d762307
Ref: randist c gsl_ran_geometric762374
Ref: 79e762374
Ref: randist c gsl_ran_geometric_pdf762897
Ref: 79f762897
Ref: randist c gsl_cdf_geometric_P763184
Ref: 7a0763184
Ref: randist c gsl_cdf_geometric_Q763252
Ref: 7a1763252
Node: The Hypergeometric Distribution763460
Ref: randist the-hypergeometric-distribution763628
Ref: 7a2763628
Ref: randist c gsl_ran_hypergeometric763705
Ref: 7a3763705
Ref: randist c gsl_ran_hypergeometric_pdf764440
Ref: 7a4764440
Ref: randist c gsl_cdf_hypergeometric_P764801
Ref: 7a5764801
Ref: randist c gsl_cdf_hypergeometric_Q764924
Ref: 7a6764924
Node: The Logarithmic Distribution765221
Ref: randist the-logarithmic-distribution765387
Ref: 7a7765387
Ref: randist c gsl_ran_logarithmic765458
Ref: 7a8765458
Ref: randist c gsl_ran_logarithmic_pdf765776
Ref: 7a9765776
Node: The Wishart Distribution766069
Ref: randist the-wishart-distribution766226
Ref: 7aa766226
Ref: randist c gsl_ran_wishart766289
Ref: 7ab766289
Ref: randist c gsl_ran_wishart_pdf766991
Ref: 7ac766991
Ref: randist c gsl_ran_wishart_log_pdf767170
Ref: 7ad767170
Node: Shuffling and Sampling767708
Ref: randist shuffling-and-sampling767849
Ref: 7ae767849
Ref: randist c gsl_ran_shuffle768301
Ref: 7af768301
Ref: randist c gsl_ran_choose769000
Ref: 7b0769000
Ref: randist c gsl_ran_sample770202
Ref: 7b1770202
Node: Examples<14>770650
Ref: randist examples770801
Ref: 7b2770801
Ref: randist fig-rand-walk772978
Ref: 7b3772978
Node: References and Further Reading<14>773852
Ref: randist references-and-further-reading773972
Ref: 7b4773972
Node: Statistics775969
Ref: statistics doc776083
Ref: 7b5776083
Ref: statistics statistics776083
Ref: 7b6776083
Node: Mean Standard Deviation and Variance777561
Ref: statistics mean-standard-deviation-and-variance777672
Ref: 7b7777672
Ref: statistics c gsl_stats_mean777759
Ref: 1d777759
Ref: statistics c gsl_stats_variance778259
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Ref: statistics c gsl_stats_variance_m779163
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Ref: statistics c gsl_stats_sd779539
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Ref: statistics c gsl_stats_sd_m779630
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Ref: statistics c gsl_stats_tss779910
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Ref: statistics c gsl_stats_tss_m780002
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Ref: statistics c gsl_stats_variance_with_fixed_mean780425
Ref: 7be780425
Ref: statistics c gsl_stats_sd_with_fixed_mean780930
Ref: 7bf780930
Node: Absolute deviation781240
Ref: statistics absolute-deviation781396
Ref: 7c0781396
Ref: statistics c gsl_stats_absdev781445
Ref: 7c1781445
Ref: statistics c gsl_stats_absdev_m782070
Ref: 7c2782070
Node: Higher moments skewness and kurtosis782586
Ref: statistics higher-moments-skewness-and-kurtosis782721
Ref: 7c3782721
Ref: statistics c gsl_stats_skew782810
Ref: 7c4782810
Ref: statistics c gsl_stats_skew_m_sd783409
Ref: 7c5783409
Ref: statistics c gsl_stats_kurtosis783887
Ref: 7c6783887
Ref: statistics c gsl_stats_kurtosis_m_sd784365
Ref: 7c7784365
Node: Autocorrelation784849
Ref: statistics autocorrelation784976
Ref: 7c8784976
Ref: statistics c gsl_stats_lag1_autocorrelation785019
Ref: 7c9785019
Ref: statistics c gsl_stats_lag1_autocorrelation_m785395
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Node: Covariance785680
Ref: statistics covariance785782
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Ref: statistics c gsl_stats_covariance785815
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Ref: statistics c gsl_stats_covariance_m786201
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Node: Correlation786701
Ref: statistics correlation786804
Ref: 7ce786804
Ref: statistics c gsl_stats_correlation786839
Ref: 7cf786839
Ref: statistics c gsl_stats_spearman787430
Ref: 7d0787430
Node: Weighted Samples788124
Ref: statistics weighted-samples788243
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Ref: statistics c gsl_stats_wmean788737
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Ref: statistics c gsl_stats_wvariance789155
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Ref: statistics c gsl_stats_wvariance_m789840
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Ref: statistics c gsl_stats_wsd790133
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Ref: statistics c gsl_stats_wsd_m790470
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Ref: statistics c gsl_stats_wvariance_with_fixed_mean790744
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Ref: statistics c gsl_stats_wsd_with_fixed_mean791307
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Ref: statistics c gsl_stats_wtss791650
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Ref: statistics c gsl_stats_wtss_m791783
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Ref: statistics c gsl_stats_wabsdev792286
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Ref: statistics c gsl_stats_wabsdev_m792639
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Ref: statistics c gsl_stats_wskew792931
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Ref: statistics c gsl_stats_wskew_m_sd793211
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Ref: statistics c gsl_stats_wkurtosis793572
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Ref: statistics c gsl_stats_wkurtosis_m_sd793866
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Node: Maximum and Minimum values794231
Ref: statistics maximum-and-minimum-values794361
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Ref: statistics c gsl_stats_max794738
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Ref: statistics c gsl_stats_min795243
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Ref: statistics c gsl_stats_min_index796416
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Ref: statistics c gsl_stats_minmax_index796841
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Node: Median and Percentiles797125
Ref: statistics median-and-percentiles797255
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Ref: statistics c gsl_stats_median_from_sorted_data797667
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Ref: statistics c gsl_stats_median798536
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Ref: statistics c gsl_stats_quantile_from_sorted_data798965
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Node: Order Statistics800218
Ref: statistics order-statistics800347
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Ref: statistics c gsl_stats_select800876
Ref: 7ed800876
Node: Robust Location Estimates801266
Ref: statistics robust-location-estimates801395
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Node: Trimmed Mean802074
Ref: statistics trimmed-mean802177
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Ref: statistics c gsl_stats_trmean_from_sorted_data802688
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Node: Gastwirth Estimator803311
Ref: statistics gastwirth-estimator803414
Ref: 7f1803414
Ref: statistics c gsl_stats_gastwirth_from_sorted_data803769
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Node: Robust Scale Estimates804251
Ref: statistics robust-scale-estimates804376
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Node: Median Absolute Deviation MAD804865
Ref: statistics median-absolute-deviation-mad804976
Ref: 7f4804976
Ref: statistics sec-mad-statistic804976
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Ref: statistics c gsl_stats_mad0805552
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Ref: statistics c gsl_stats_mad805673
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Node: S_n Statistic806222
Ref: statistics s-n-statistic806355
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Ref: statistics sec-sn-statistic806355
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Ref: statistics c gsl_stats_Sn0_from_sorted_data806936
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Ref: statistics c gsl_stats_Sn_from_sorted_data807090
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Node: Q_n Statistic807879
Ref: statistics q-n-statistic807974
Ref: 7fc807974
Ref: statistics sec-qn-statistic807974
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Ref: statistics c gsl_stats_Qn0_from_sorted_data808441
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Ref: statistics c gsl_stats_Qn_from_sorted_data808610
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Node: Examples<15>809430
Ref: statistics examples809564
Ref: 800809564
Node: References and Further Reading<15>812059
Ref: statistics references-and-further-reading812162
Ref: 801812162
Node: Running Statistics813632
Ref: rstat doc813743
Ref: 802813743
Ref: rstat running-statistics813743
Ref: 803813743
Node: Initializing the Accumulator814865
Ref: rstat initializing-the-accumulator814988
Ref: 804814988
Ref: rstat c gsl_rstat_workspace815057
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Ref: rstat c gsl_rstat_alloc815243
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Ref: rstat c gsl_rstat_free815420
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Ref: rstat c gsl_rstat_reset815557
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Node: Adding Data to the Accumulator815740
Ref: rstat adding-data-to-the-accumulator815890
Ref: 809815890
Ref: rstat c gsl_rstat_add815963
Ref: 80a815963
Ref: rstat c gsl_rstat_n816235
Ref: 80b816235
Node: Current Statistics816386
Ref: rstat current-statistics816517
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Ref: rstat c gsl_rstat_min816566
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Ref: rstat c gsl_rstat_max816706
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Ref: rstat c gsl_rstat_mean816846
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Ref: rstat c gsl_rstat_variance817042
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Ref: rstat c gsl_rstat_sd817269
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Ref: rstat c gsl_rstat_sd_mean817491
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Ref: rstat c gsl_rstat_rms817686
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Ref: rstat c gsl_rstat_skew817904
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Ref: rstat c gsl_rstat_kurtosis818128
Ref: 815818128
Ref: rstat c gsl_rstat_median818367
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Node: Quantiles818529
Ref: rstat quantiles818642
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Ref: rstat c gsl_rstat_quantile_workspace819064
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Ref: rstat c gsl_rstat_quantile_alloc819198
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Ref: rstat c gsl_rstat_quantile_free819503
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Ref: rstat c gsl_rstat_quantile_reset819668
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Ref: rstat c gsl_rstat_quantile_add819879
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Ref: rstat c gsl_rstat_quantile_get820079
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Node: Examples<16>820237
Ref: rstat examples820366
Ref: 81e820366
Node: References and Further Reading<16>825724
Ref: rstat references-and-further-reading825835
Ref: 81f825835
Node: Moving Window Statistics826220
Ref: movstat doc826338
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Ref: movstat moving-window-statistics826338
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Node: Introduction<4>827238
Ref: movstat introduction827342
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Node: Handling Endpoints828251
Ref: movstat handling-endpoints828403
Ref: 823828403
Ref: movstat c gsl_movstat_end_t828753
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Ref: movstat c GSL_MOVSTAT_END_PADZERO828907
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Ref: movstat c GSL_MOVSTAT_END_PADVALUE829281
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Ref: movstat c GSL_MOVSTAT_END_TRUNCATE829629
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Node: Allocation for Moving Window Statistics829806
Ref: movstat allocation-for-moving-window-statistics829954
Ref: 828829954
Ref: movstat c gsl_movstat_workspace830045
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Ref: movstat c gsl_movstat_alloc830147
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Ref: movstat c gsl_movstat_alloc2830457
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Ref: movstat c gsl_movstat_free830824
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Node: Moving Mean830953
Ref: movstat moving-mean831121
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Ref: movstat c gsl_movstat_mean831556
Ref: 82e831556
Node: Moving Variance and Standard Deviation832005
Ref: movstat moving-variance-and-standard-deviation832160
Ref: 82f832160
Ref: movstat c gsl_movstat_variance832589
Ref: 830832589
Ref: movstat c gsl_movstat_sd833049
Ref: 831833049
Node: Moving Minimum and Maximum833519
Ref: movstat moving-minimum-and-maximum833673
Ref: 832833673
Ref: movstat c gsl_movstat_min833899
Ref: 833833899
Ref: movstat c gsl_movstat_max834346
Ref: 834834346
Ref: movstat c gsl_movstat_minmax834793
Ref: 835834793
Node: Moving Sum835258
Ref: movstat moving-sum835387
Ref: 836835387
Ref: movstat c gsl_movstat_sum835543
Ref: 837835543
Node: Moving Median835989
Ref: movstat moving-median836115
Ref: 838836115
Ref: movstat c gsl_movstat_median836270
Ref: 839836270
Node: Robust Scale Estimation836721
Ref: movstat robust-scale-estimation836867
Ref: 83a836867
Node: Moving MAD837543
Ref: movstat moving-mad837633
Ref: 83b837633
Ref: movstat c gsl_movstat_mad0838062
Ref: 83c838062
Ref: movstat c gsl_movstat_mad838241
Ref: 83d838241
Node: Moving QQR838937
Ref: movstat moving-qqr839046
Ref: 83e839046
Ref: movstat c gsl_movstat_qqr839683
Ref: 83f839683
Node: Moving S_n840284
Ref: movstat moving-s-n840393
Ref: 840840393
Ref: movstat c gsl_movstat_Sn840657
Ref: 841840657
Node: Moving Q_n841182
Ref: movstat moving-q-n841272
Ref: 842841272
Ref: movstat c gsl_movstat_Qn841508
Ref: 843841508
Node: User-defined Moving Statistics842033
Ref: movstat user-defined-moving-statistics842178
Ref: 844842178
Ref: movstat c gsl_movstat_function842830
Ref: 846842830
Ref: movstat c gsl_movstat_function function843249
Ref: 848843249
Ref: movstat c gsl_movstat_function params843569
Ref: 849843569
Ref: movstat c gsl_movstat_apply843670
Ref: 847843670
Ref: movstat c gsl_movstat_fill844205
Ref: 845844205
Node: Accumulators844876
Ref: movstat accumulators845010
Ref: 84a845010
Ref: movstat c gsl_movstat_accum845305
Ref: 84b845305
Ref: movstat c gsl_movstat_accum size845875
Ref: 84c845875
Ref: movstat c gsl_movstat_accum init846071
Ref: 84d846071
Ref: movstat c gsl_movstat_accum insert846247
Ref: 84e846247
Ref: movstat c gsl_movstat_accum delete846618
Ref: 84f846618
Ref: movstat c gsl_movstat_accum get846807
Ref: 850846807
Ref: movstat c gsl_movstat_accum_min847240
Ref: 851847240
Ref: movstat c gsl_movstat_accum_max847276
Ref: 852847276
Ref: movstat c gsl_movstat_accum_minmax847312
Ref: 853847312
Ref: movstat c gsl_movstat_accum_mean847470
Ref: 854847470
Ref: movstat c gsl_movstat_accum_sd847507
Ref: 855847507
Ref: movstat c gsl_movstat_accum_variance847542
Ref: 856847542
Ref: movstat c gsl_movstat_accum_Sn847875
Ref: 857847875
Ref: movstat c gsl_movstat_accum_Qn847910
Ref: 858847910
Ref: movstat c gsl_movstat_accum_sum848061
Ref: 859848061
Ref: movstat c gsl_movstat_accum_qqr848155
Ref: 85a848155
Node: Examples<17>848262
Ref: movstat examples848400
Ref: 85b848400
Node: Example 1848585
Ref: movstat example-1848675
Ref: 85c848675
Ref: movstat fig-movstat1848879
Ref: 85d848879
Node: Example 2 Robust Scale850594
Ref: movstat example-2-robust-scale850729
Ref: 85e850729
Ref: movstat fig-movstat2851679
Ref: 85f851679
Node: Example 3 User-defined Moving Window854901
Ref: movstat example-3-user-defined-moving-window855018
Ref: 860855018
Ref: movstat fig-movstat3855585
Ref: 861855585
Node: References and Further Reading<17>857594
Ref: movstat references-and-further-reading857711
Ref: 862857711
Node: Digital Filtering858291
Ref: filter doc858401
Ref: 863858401
Ref: filter digital-filtering858401
Ref: 864858401
Node: Introduction<5>858685
Ref: filter introduction858785
Ref: 865858785
Node: Handling Endpoints<2>859166
Ref: filter handling-endpoints859297
Ref: 866859297
Ref: filter c gsl_filter_end_t859642
Ref: 867859642
Ref: filter c GSL_FILTER_END_PADZERO859795
Ref: 868859795
Ref: filter c GSL_FILTER_END_PADVALUE860170
Ref: 869860170
Ref: filter c GSL_FILTER_END_TRUNCATE860519
Ref: 86a860519
Node: Linear Digital Filters860695
Ref: filter linear-digital-filters860836
Ref: 86b860836
Node: Gaussian Filter860923
Ref: filter gaussian-filter860998
Ref: 86c860998
Ref: filter c gsl_filter_gaussian_alloc862440
Ref: 86d862440
Ref: filter c gsl_filter_gaussian_free862793
Ref: 86e862793
Ref: filter c gsl_filter_gaussian862946
Ref: 86f862946
Ref: filter c gsl_filter_gaussian_kernel863618
Ref: 870863618
Node: Nonlinear Digital Filters864245
Ref: filter nonlinear-digital-filters864377
Ref: 871864377
Node: Standard Median Filter864901
Ref: filter standard-median-filter865018
Ref: 872865018
Ref: filter c gsl_filter_median_alloc865356
Ref: 873865356
Ref: filter c gsl_filter_median_free865740
Ref: 874865740
Ref: filter c gsl_filter_median865889
Ref: 875865889
Node: Recursive Median Filter866314
Ref: filter recursive-median-filter866464
Ref: 876866464
Ref: filter c gsl_filter_rmedian_alloc867185
Ref: 877867185
Ref: filter c gsl_filter_rmedian_free867575
Ref: 878867575
Ref: filter c gsl_filter_rmedian867726
Ref: 879867726
Node: Impulse Detection Filter868154
Ref: filter impulse-detection-filter868273
Ref: 87a868273
Ref: filter c gsl_filter_scale_t870598
Ref: 87b870598
Ref: filter c GSL_FILTER_SCALE_MAD870721
Ref: 87c870721
Ref: filter c GSL_FILTER_SCALE_IQR871094
Ref: 87d871094
Ref: filter c GSL_FILTER_SCALE_SN871713
Ref: 87e871713
Ref: filter c GSL_FILTER_SCALE_QN871908
Ref: 87f871908
Ref: filter c gsl_filter_impulse_alloc872972
Ref: 880872972
Ref: filter c gsl_filter_impulse_free873351
Ref: 881873351
Ref: filter c gsl_filter_impulse873502
Ref: 882873502
Node: Examples<18>874422
Ref: filter examples874566
Ref: 883874566
Node: Gaussian Example 1874712
Ref: filter gaussian-example-1874807
Ref: 884874807
Ref: filter fig-filt-gaussian875094
Ref: 885875094
Node: Gaussian Example 2878320
Ref: filter gaussian-example-2878450
Ref: 886878450
Ref: filter fig-filt-gaussian2879173
Ref: 887879173
Node: Square Wave Signal Example882482
Ref: filter square-wave-signal-example882619
Ref: 888882619
Ref: filter fig-filt-edge883121
Ref: 889883121
Node: Impulse Detection Example885774
Ref: filter impulse-detection-example885884
Ref: 88a885884
Ref: filter fig-impulse886366
Ref: 88b886366
Node: References and Further Reading<18>889088
Ref: filter references-and-further-reading889198
Ref: 88c889198
Node: Histograms889813
Ref: histogram doc889907
Ref: 88d889907
Ref: histogram histograms889907
Ref: 88e889907
Node: The histogram struct891582
Ref: histogram the-histogram-struct891679
Ref: 88f891679
Ref: histogram c gsl_histogram891781
Ref: 890891781
Node: Histogram allocation893864
Ref: histogram histogram-allocation893988
Ref: 891893988
Ref: histogram c gsl_histogram_alloc894530
Ref: 892894530
Ref: histogram c gsl_histogram_set_ranges895068
Ref: 893895068
Ref: histogram c gsl_histogram_set_ranges_uniform896190
Ref: 894896190
Ref: histogram c gsl_histogram_free896780
Ref: 895896780
Node: Copying Histograms896934
Ref: histogram copying-histograms897079
Ref: 896897079
Ref: histogram c gsl_histogram_memcpy897128
Ref: 897897128
Ref: histogram c gsl_histogram_clone897430
Ref: 898897430
Node: Updating and accessing histogram elements897642
Ref: histogram updating-and-accessing-histogram-elements897793
Ref: 899897793
Ref: histogram c gsl_histogram_increment898148
Ref: 89a898148
Ref: histogram c gsl_histogram_accumulate898998
Ref: 89b898998
Ref: histogram c gsl_histogram_get899294
Ref: 89c899294
Ref: histogram c gsl_histogram_get_range899657
Ref: 89d899657
Ref: histogram c gsl_histogram_max900494
Ref: 89e900494
Ref: histogram c gsl_histogram_min900559
Ref: 89f900559
Ref: histogram c gsl_histogram_bins900624
Ref: 8a0900624
Ref: histogram c gsl_histogram_reset900946
Ref: 8a1900946
Node: Searching histogram ranges901080
Ref: histogram searching-histogram-ranges901233
Ref: 8a2901233
Ref: histogram c gsl_histogram_find901427
Ref: 8a3901427
Node: Histogram Statistics902130
Ref: histogram histogram-statistics902262
Ref: 8a4902262
Ref: histogram c gsl_histogram_max_val902315
Ref: 8a5902315
Ref: histogram c gsl_histogram_max_bin902469
Ref: 8a6902469
Ref: histogram c gsl_histogram_min_val902723
Ref: 8a7902723
Ref: histogram c gsl_histogram_min_bin902877
Ref: 8a8902877
Ref: histogram c gsl_histogram_mean903131
Ref: 8a9903131
Ref: histogram c gsl_histogram_sigma903471
Ref: 8aa903471
Ref: histogram c gsl_histogram_sum903831
Ref: 8ab903831
Node: Histogram Operations904003
Ref: histogram histogram-operations904139
Ref: 8ac904139
Ref: histogram c gsl_histogram_equal_bins_p904192
Ref: 8ad904192
Ref: histogram c gsl_histogram_add904430
Ref: 8ae904430
Ref: histogram c gsl_histogram_sub904740
Ref: 8af904740
Ref: histogram c gsl_histogram_mul905062
Ref: 8b0905062
Ref: histogram c gsl_histogram_div905398
Ref: 8b1905398
Ref: histogram c gsl_histogram_scale905731
Ref: 8b2905731
Ref: histogram c gsl_histogram_shift905956
Ref: 8b3905956
Node: Reading and writing histograms906180
Ref: histogram reading-and-writing-histograms906322
Ref: 8b4906322
Ref: histogram c gsl_histogram_fwrite906506
Ref: 8b5906506
Ref: histogram c gsl_histogram_fread906946
Ref: 8b6906946
Ref: histogram c gsl_histogram_fprintf907511
Ref: 8b7907511
Ref: histogram c gsl_histogram_fscanf908796
Ref: 8b8908796
Node: Resampling from histograms909362
Ref: histogram resampling-from-histograms909529
Ref: 8b9909529
Node: The histogram probability distribution struct910162
Ref: histogram the-histogram-probability-distribution-struct910330
Ref: 8ba910330
Ref: histogram c gsl_histogram_pdf911169
Ref: 8bb911169
Ref: histogram c gsl_histogram_pdf_alloc912130
Ref: 8bc912130
Ref: histogram c gsl_histogram_pdf_init912530
Ref: 8bd912530
Ref: histogram c gsl_histogram_pdf_free912951
Ref: 8be912951
Ref: histogram c gsl_histogram_pdf_sample913137
Ref: 8bf913137
Node: Example programs for histograms913662
Ref: histogram example-programs-for-histograms913830
Ref: 8c0913830
Ref: histogram fig-histogram915641
Ref: 8c1915641
Node: Two dimensional histograms915723
Ref: histogram two-dimensional-histograms915869
Ref: 8c2915869
Node: The 2D histogram struct916443
Ref: histogram the-2d-histogram-struct916581
Ref: 8c3916581
Ref: histogram c gsl_histogram2d916707
Ref: 8c4916707
Node: 2D Histogram allocation918770
Ref: histogram d-histogram-allocation918903
Ref: 8c5918903
Ref: histogram c gsl_histogram2d_alloc919459
Ref: 8c6919459
Ref: histogram c gsl_histogram2d_set_ranges920067
Ref: 8c7920067
Ref: histogram c gsl_histogram2d_set_ranges_uniform920459
Ref: 8c8920459
Ref: histogram c gsl_histogram2d_free920814
Ref: 8c9920814
Node: Copying 2D Histograms920975
Ref: histogram copying-2d-histograms921129
Ref: 8ca921129
Ref: histogram c gsl_histogram2d_memcpy921186
Ref: 8cb921186
Ref: histogram c gsl_histogram2d_clone921494
Ref: 8cc921494
Node: Updating and accessing 2D histogram elements921712
Ref: histogram updating-and-accessing-2d-histogram-elements921872
Ref: 8cd921872
Ref: histogram c gsl_histogram2d_increment922295
Ref: 8ce922295
Ref: histogram c gsl_histogram2d_accumulate923007
Ref: 8cf923007
Ref: histogram c gsl_histogram2d_get923319
Ref: 8d0923319
Ref: histogram c gsl_histogram2d_get_xrange923718
Ref: 8d1923718
Ref: histogram c gsl_histogram2d_get_yrange923845
Ref: 8d2923845
Ref: histogram c gsl_histogram2d_xmax924712
Ref: 8d3924712
Ref: histogram c gsl_histogram2d_xmin924782
Ref: 8d4924782
Ref: histogram c gsl_histogram2d_nx924852
Ref: 8d5924852
Ref: histogram c gsl_histogram2d_ymax924920
Ref: 8d6924920
Ref: histogram c gsl_histogram2d_ymin924990
Ref: 8d7924990
Ref: histogram c gsl_histogram2d_ny925060
Ref: 8d8925060
Ref: histogram c gsl_histogram2d_reset925413
Ref: 8d9925413
Node: Searching 2D histogram ranges925551
Ref: histogram searching-2d-histogram-ranges925713
Ref: 8da925713
Ref: histogram c gsl_histogram2d_find925919
Ref: 8db925919
Node: 2D Histogram Statistics926617
Ref: histogram d-histogram-statistics926758
Ref: 8dc926758
Ref: histogram c gsl_histogram2d_max_val926819
Ref: 8dd926819
Ref: histogram c gsl_histogram2d_max_bin926977
Ref: 8de926977
Ref: histogram c gsl_histogram2d_min_val927342
Ref: 8df927342
Ref: histogram c gsl_histogram2d_min_bin927500
Ref: 8e0927500
Ref: histogram c gsl_histogram2d_xmean927865
Ref: 8e1927865
Ref: histogram c gsl_histogram2d_ymean928154
Ref: 8e2928154
Ref: histogram c gsl_histogram2d_xsigma928443
Ref: 8e3928443
Ref: histogram c gsl_histogram2d_ysigma928748
Ref: 8e4928748
Ref: histogram c gsl_histogram2d_cov929053
Ref: 8e5929053
Ref: histogram c gsl_histogram2d_sum929354
Ref: 8e6929354
Node: 2D Histogram Operations929530
Ref: histogram d-histogram-operations929675
Ref: 8e7929675
Ref: histogram c gsl_histogram2d_equal_bins_p929736
Ref: 8e8929736
Ref: histogram c gsl_histogram2d_add929973
Ref: 8e9929973
Ref: histogram c gsl_histogram2d_sub930300
Ref: 8ea930300
Ref: histogram c gsl_histogram2d_mul930634
Ref: 8eb930634
Ref: histogram c gsl_histogram2d_div930982
Ref: 8ec930982
Ref: histogram c gsl_histogram2d_scale931327
Ref: 8ed931327
Ref: histogram c gsl_histogram2d_shift931568
Ref: 8ee931568
Node: Reading and writing 2D histograms931810
Ref: histogram reading-and-writing-2d-histograms931961
Ref: 8ef931961
Ref: histogram c gsl_histogram2d_fwrite932169
Ref: 8f0932169
Ref: histogram c gsl_histogram2d_fread932613
Ref: 8f1932613
Ref: histogram c gsl_histogram2d_fprintf933195
Ref: 8f2933195
Ref: histogram c gsl_histogram2d_fscanf934994
Ref: 8f3934994
Node: Resampling from 2D histograms935567
Ref: histogram resampling-from-2d-histograms935729
Ref: 8f4935729
Ref: histogram c gsl_histogram2d_pdf936442
Ref: 8f5936442
Ref: histogram c gsl_histogram2d_pdf_alloc937733
Ref: 8f6937733
Ref: histogram c gsl_histogram2d_pdf_init938193
Ref: 8f7938193
Ref: histogram c gsl_histogram2d_pdf_free938631
Ref: 8f8938631
Ref: histogram c gsl_histogram2d_pdf_sample938837
Ref: 8f9938837
Node: Example programs for 2D histograms939171
Ref: histogram example-programs-for-2d-histograms939291
Ref: 8fa939291
Node: N-tuples941363
Ref: ntuple doc941463
Ref: 8fb941463
Ref: ntuple n-tuples941463
Ref: 8fc941463
Node: The ntuple struct942643
Ref: ntuple the-ntuple-struct942731
Ref: 8fd942731
Ref: ntuple c gsl_ntuple942778
Ref: 8fe942778
Node: Creating ntuples943212
Ref: ntuple creating-ntuples943340
Ref: 8ff943340
Ref: ntuple c gsl_ntuple_create943385
Ref: 900943385
Node: Opening an existing ntuple file943899
Ref: ntuple opening-an-existing-ntuple-file944025
Ref: 901944025
Ref: ntuple c gsl_ntuple_open944100
Ref: 902944100
Node: Writing ntuples944548
Ref: ntuple writing-ntuples944673
Ref: 903944673
Ref: ntuple c gsl_ntuple_write944716
Ref: 904944716
Ref: ntuple c gsl_ntuple_bookdata944905
Ref: 905944905
Node: Reading ntuples945034
Ref: ntuple reading-ntuples945150
Ref: 906945150
Ref: ntuple c gsl_ntuple_read945193
Ref: 907945193
Node: Closing an ntuple file945378
Ref: ntuple closing-an-ntuple-file945506
Ref: 908945506
Ref: ntuple c gsl_ntuple_close945563
Ref: 909945563
Node: Histogramming ntuple values945725
Ref: ntuple histogramming-ntuple-values945850
Ref: 90a945850
Ref: ntuple c gsl_ntuple_select_fn946303
Ref: 90c946303
Ref: ntuple c gsl_ntuple_value_fn946784
Ref: 90d946784
Ref: ntuple c gsl_ntuple_project947236
Ref: 90b947236
Node: Examples<19>947963
Ref: ntuple examples948100
Ref: 90e948100
Ref: ntuple fig-ntuples951336
Ref: 90f951336
Node: References and Further Reading<19>951409
Ref: ntuple references-and-further-reading951510
Ref: 910951510
Node: Monte Carlo Integration951716
Ref: montecarlo doc951825
Ref: 911951825
Ref: montecarlo monte-carlo-integration951825
Ref: 912951825
Node: Interface952982
Ref: montecarlo interface953078
Ref: 913953078
Ref: montecarlo c gsl_monte_function954086
Ref: 914954086
Node: PLAIN Monte Carlo956327
Ref: montecarlo plain-monte-carlo956437
Ref: 915956437
Ref: montecarlo c gsl_monte_plain_state957451
Ref: 916957451
Ref: montecarlo c gsl_monte_plain_alloc957544
Ref: 917957544
Ref: montecarlo c gsl_monte_plain_init957734
Ref: 918957734
Ref: montecarlo c gsl_monte_plain_integrate957960
Ref: 919957960
Ref: montecarlo c gsl_monte_plain_free958785
Ref: 91a958785
Node: MISER958942
Ref: montecarlo miser959048
Ref: 91b959048
Ref: montecarlo c gsl_monte_miser_state960968
Ref: 91c960968
Ref: montecarlo c gsl_monte_miser_alloc961064
Ref: 91d961064
Ref: montecarlo c gsl_monte_miser_init961324
Ref: 91e961324
Ref: montecarlo c gsl_monte_miser_integrate961550
Ref: 91f961550
Ref: montecarlo c gsl_monte_miser_free962375
Ref: 920962375
Ref: montecarlo c gsl_monte_miser_params_get962649
Ref: 921962649
Ref: montecarlo c gsl_monte_miser_params_set962891
Ref: 922962891
Ref: montecarlo c gsl_monte_miser_params963514
Ref: 923963514
Ref: montecarlo c estimate_frac963548
Ref: 924963548
Ref: montecarlo c min_calls963818
Ref: 925963818
Ref: montecarlo c min_calls_per_bisection964340
Ref: 926964340
Ref: montecarlo c alpha964830
Ref: 49a964830
Ref: montecarlo c dither965706
Ref: 927965706
Ref: MISER-Footnote-1966214
Node: VEGAS966349
Ref: montecarlo vegas966450
Ref: 928966450
Ref: montecarlo c gsl_monte_vegas_state968725
Ref: 929968725
Ref: montecarlo c gsl_monte_vegas_alloc968821
Ref: 92a968821
Ref: montecarlo c gsl_monte_vegas_init969081
Ref: 92b969081
Ref: montecarlo c gsl_monte_vegas_integrate969307
Ref: 92c969307
Ref: montecarlo c gsl_monte_vegas_free970426
Ref: 92d970426
Ref: montecarlo c gsl_monte_vegas_chisq972099
Ref: 92e972099
Ref: montecarlo c gsl_monte_vegas_runval972614
Ref: 92f972614
Ref: montecarlo c gsl_monte_vegas_params_get973019
Ref: 930973019
Ref: montecarlo c gsl_monte_vegas_params_set973261
Ref: 931973261
Ref: montecarlo c gsl_monte_vegas_params973884
Ref: 932973884
Ref: montecarlo c iterations974192
Ref: 933974192
Ref: montecarlo c stage974353
Ref: 934974353
Ref: montecarlo c mode975274
Ref: d8975274
Ref: montecarlo c verbose975766
Ref: 935975766
Ref: montecarlo c ostream975797
Ref: 936975797
Node: Examples<20>976527
Ref: montecarlo examples976657
Ref: 937976657
Node: References and Further Reading<20>982578
Ref: montecarlo references-and-further-reading982694
Ref: 938982694
Node: Simulated Annealing983336
Ref: siman doc983468
Ref: 939983468
Ref: siman simulated-annealing983468
Ref: 93a983468
Node: Simulated Annealing algorithm984547
Ref: siman simulated-annealing-algorithm984671
Ref: 93b984671
Node: Simulated Annealing functions985701
Ref: siman simulated-annealing-functions985846
Ref: 93c985846
Ref: siman c gsl_siman_solve985917
Ref: 93d985917
Ref: siman c gsl_siman_Efunc_t988386
Ref: 93e988386
Ref: siman c gsl_siman_step_t988549
Ref: 93f988549
Ref: siman c gsl_siman_metric_t988871
Ref: 940988871
Ref: siman c gsl_siman_print_t989069
Ref: 941989069
Ref: siman c gsl_siman_copy_t989233
Ref: 942989233
Ref: siman c gsl_siman_copy_construct_t989414
Ref: 943989414
Ref: siman c gsl_siman_destroy_t989597
Ref: 944989597
Ref: siman c gsl_siman_params_t989771
Ref: 945989771
Node: Examples<21>990877
Ref: siman examples991027
Ref: 946991027
Node: Trivial example991394
Ref: siman trivial-example991494
Ref: 947991494
Ref: siman fig-siman-test994056
Ref: 948994056
Ref: siman fig-siman-energy994262
Ref: 949994262
Node: Traveling Salesman Problem994348
Ref: siman traveling-salesman-problem994448
Ref: 94a994448
Node: References and Further Reading<21>1008887
Ref: siman references-and-further-reading1008999
Ref: 94b1008999
Node: Ordinary Differential Equations1009261
Ref: ode-initval doc1009383
Ref: 94c1009383
Ref: ode-initval ordinary-differential-equations1009383
Ref: 94d1009383
Node: Defining the ODE System1010500
Ref: ode-initval defining-the-ode-system1010619
Ref: 94e1010619
Ref: ode-initval c gsl_odeiv2_system1011019
Ref: 94f1011019
Node: Stepping Functions1013786
Ref: ode-initval stepping-functions1013940
Ref: 9531013940
Ref: ode-initval c gsl_odeiv2_step1014155
Ref: 9541014155
Ref: ode-initval c gsl_odeiv2_step_alloc1014247
Ref: 9551014247
Ref: ode-initval c gsl_odeiv2_step_reset1014705
Ref: 9521014705
Ref: ode-initval c gsl_odeiv2_step_free1014935
Ref: 9561014935
Ref: ode-initval c gsl_odeiv2_step_name1015091
Ref: 9571015091
Ref: ode-initval c gsl_odeiv2_step_order1015407
Ref: 9581015407
Ref: ode-initval c gsl_odeiv2_step_set_driver1015655
Ref: 9591015655
Ref: ode-initval c gsl_odeiv2_step_apply1016132
Ref: 95a1016132
Ref: ode-initval c gsl_odeiv2_step_type1018268
Ref: 95b1018268
Ref: ode-initval c gsl_odeiv2_step_rk21018300
Ref: 95c1018300
Ref: ode-initval c gsl_odeiv2_step_rk41018396
Ref: 95d1018396
Ref: ode-initval c gsl_odeiv2_step_rkf451018668
Ref: 95e1018668
Ref: ode-initval c gsl_odeiv2_step_rkck1018836
Ref: 95f1018836
Ref: ode-initval c gsl_odeiv2_step_rk8pd1018943
Ref: 9601018943
Ref: ode-initval c gsl_odeiv2_step_rk1imp1019056
Ref: 9611019056
Ref: ode-initval c gsl_odeiv2_step_rk2imp1019426
Ref: 9621019426
Ref: ode-initval c gsl_odeiv2_step_rk4imp1019779
Ref: 9631019779
Ref: ode-initval c gsl_odeiv2_step_bsimp1020081
Ref: 9641020081
Ref: ode-initval c gsl_odeiv2_step_msadams1020300
Ref: 9651020300
Ref: ode-initval c gsl_odeiv2_step_msbdf1020779
Ref: 9661020779
Node: Adaptive Step-size Control1021422
Ref: ode-initval adaptive-step-size-control1021562
Ref: 9671021562
Ref: ode-initval c gsl_odeiv2_control1021811
Ref: 9681021811
Ref: ode-initval c gsl_odeiv2_control_type1021894
Ref: 9691021894
Ref: ode-initval c gsl_odeiv2_control_standard_new1021968
Ref: 96a1021968
Ref: ode-initval c gsl_odeiv2_control_y_new1023537
Ref: 96b1023537
Ref: ode-initval c gsl_odeiv2_control_yp_new1023961
Ref: 96c1023961
Ref: ode-initval c gsl_odeiv2_control_scaled_new1024407
Ref: 96d1024407
Ref: ode-initval c gsl_odeiv2_control_alloc1025093
Ref: 96e1025093
Ref: ode-initval c gsl_odeiv2_control_init1025485
Ref: 96f1025485
Ref: ode-initval c gsl_odeiv2_control_free1025866
Ref: 9701025866
Ref: ode-initval c gsl_odeiv2_control_hadjust1026027
Ref: 9711026027
Ref: ode-initval c gsl_odeiv2_control_name1026980
Ref: 9721026980
Ref: ode-initval c gsl_odeiv2_control_errlevel1027312
Ref: 9731027312
Ref: ode-initval c gsl_odeiv2_control_set_driver1027734
Ref: 9741027734
Node: Evolution1027944
Ref: ode-initval evolution1028072
Ref: 9751028072
Ref: ode-initval c gsl_odeiv2_evolve1028273
Ref: 9761028273
Ref: ode-initval c gsl_odeiv2_evolve_alloc1028387
Ref: 9771028387
Ref: ode-initval c gsl_odeiv2_evolve_apply1028595
Ref: 9781028595
Ref: ode-initval c gsl_odeiv2_evolve_apply_fixed_step1030790
Ref: 9791030790
Ref: ode-initval c gsl_odeiv2_evolve_reset1031446
Ref: 9511031446
Ref: ode-initval c gsl_odeiv2_evolve_free1031681
Ref: 97a1031681
Ref: ode-initval c gsl_odeiv2_evolve_set_driver1031842
Ref: 97b1031842
Node: Driver1032448
Ref: ode-initval driver1032562
Ref: 97c1032562
Ref: ode-initval c gsl_odeiv2_driver_alloc_y_new1032701
Ref: 97d1032701
Ref: ode-initval c gsl_odeiv2_driver_alloc_yp_new1032912
Ref: 97e1032912
Ref: ode-initval c gsl_odeiv2_driver_alloc_standard_new1033124
Ref: 97f1033124
Ref: ode-initval c gsl_odeiv2_driver_alloc_scaled_new1033391
Ref: 9801033391
Ref: ode-initval c gsl_odeiv2_driver_set_hmin1034135
Ref: 9811034135
Ref: ode-initval c gsl_odeiv2_driver_set_hmax1034347
Ref: 9821034347
Ref: ode-initval c gsl_odeiv2_driver_set_nmax1034575
Ref: 9831034575
Ref: ode-initval c gsl_odeiv2_driver_apply1034828
Ref: 9841034828
Ref: ode-initval c gsl_odeiv2_driver_apply_fixed_step1035806
Ref: 9851035806
Ref: ode-initval c gsl_odeiv2_driver_reset1036301
Ref: 9501036301
Ref: ode-initval c gsl_odeiv2_driver_reset_hstart1036430
Ref: 9861036430
Ref: ode-initval c gsl_odeiv2_driver_free1036725
Ref: 9871036725
Node: Examples<22>1036897
Ref: ode-initval examples1037036
Ref: 9881037036
Node: References and Further Reading<22>1042292
Ref: ode-initval references-and-further-reading1042416
Ref: 9891042416
Node: Interpolation1044083
Ref: interp doc1044211
Ref: 98a1044211
Ref: interp interpolation1044211
Ref: 98b1044211
Ref: interp sec-interpolation1044211
Ref: 98c1044211
Node: Introduction to 1D Interpolation1045619
Ref: interp introduction-to-1d-interpolation1045737
Ref: 98e1045737
Node: 1D Interpolation Functions1046106
Ref: interp d-interpolation-functions1046255
Ref: 98f1046255
Ref: interp c gsl_interp1046461
Ref: 9901046461
Ref: interp c gsl_interp_alloc1046520
Ref: 9911046520
Ref: interp c gsl_interp_init1046745
Ref: 9921046745
Ref: interp c gsl_interp_free1047360
Ref: 9931047360
Node: 1D Interpolation Types1047483
Ref: interp d-interpolation-types1047633
Ref: 9941047633
Ref: interp c gsl_interp_type1047761
Ref: 9951047761
Ref: interp c gsl_interp_linear1047788
Ref: 9961047788
Ref: interp c gsl_interp_polynomial1047936
Ref: 9971047936
Ref: interp c gsl_interp_cspline1048313
Ref: 9981048313
Ref: interp c gsl_interp_cspline_periodic1048659
Ref: 9991048659
Ref: interp c gsl_interp_akima1049194
Ref: 99a1049194
Ref: interp c gsl_interp_akima_periodic1049373
Ref: 99b1049373
Ref: interp c gsl_interp_steffen1049562
Ref: 99c1049562
Ref: interp c gsl_interp_name1050167
Ref: 99d1050167
Ref: interp c gsl_interp_min_size1050508
Ref: 99e1050508
Ref: interp c gsl_interp_type_min_size1050593
Ref: 99f1050593
Node: 1D Index Look-up and Acceleration1050906
Ref: interp d-index-look-up-and-acceleration1051070
Ref: 9a01051070
Ref: interp c gsl_interp_accel1051283
Ref: 9a11051283
Ref: interp c gsl_interp_bsearch1051559
Ref: 9a21051559
Ref: interp c gsl_interp_accel_alloc1051962
Ref: 9a31051962
Ref: interp c gsl_interp_accel_find1052259
Ref: 9a41052259
Ref: interp c gsl_interp_accel_free1053041
Ref: 9a51053041
Node: 1D Evaluation of Interpolating Functions1053167
Ref: interp d-evaluation-of-interpolating-functions1053334
Ref: 9a61053334
Ref: interp c gsl_interp_eval1053427
Ref: 9a71053427
Ref: interp c gsl_interp_eval_e1053584
Ref: 9a81053584
Ref: interp c gsl_interp_eval_deriv1054116
Ref: 9a91054116
Ref: interp c gsl_interp_eval_deriv_e1054279
Ref: 9aa1054279
Ref: interp c gsl_interp_eval_deriv21054681
Ref: 9ab1054681
Ref: interp c gsl_interp_eval_deriv2_e1054845
Ref: 9ac1054845
Ref: interp c gsl_interp_eval_integ1055262
Ref: 9ad1055262
Ref: interp c gsl_interp_eval_integ_e1055435
Ref: 9ae1055435
Node: 1D Higher-level Interface1055878
Ref: interp d-higher-level-interface1056045
Ref: 9af1056045
Ref: interp c gsl_spline1056555
Ref: 9b01056555
Ref: interp c gsl_spline_alloc1056675
Ref: 9b11056675
Ref: interp c gsl_spline_init1056771
Ref: 9b21056771
Ref: interp c gsl_spline_free1056889
Ref: 9b31056889
Ref: interp c gsl_spline_name1056947
Ref: 9b41056947
Ref: interp c gsl_spline_min_size1057019
Ref: 9b51057019
Ref: interp c gsl_spline_eval1057105
Ref: 9b61057105
Ref: interp c gsl_spline_eval_e1057214
Ref: 9b71057214
Ref: interp c gsl_spline_eval_deriv1057335
Ref: 9b81057335
Ref: interp c gsl_spline_eval_deriv_e1057450
Ref: 9b91057450
Ref: interp c gsl_spline_eval_deriv21057577
Ref: 9ba1057577
Ref: interp c gsl_spline_eval_deriv2_e1057693
Ref: 9bb1057693
Ref: interp c gsl_spline_eval_integ1057822
Ref: 9bc1057822
Ref: interp c gsl_spline_eval_integ_e1057947
Ref: 9bd1057947
Node: 1D Interpolation Example Programs1058089
Ref: interp d-interpolation-example-programs1058248
Ref: 9be1058248
Ref: interp fig-interp1059652
Ref: 9bf1059652
Ref: interp fig-interpp1061247
Ref: 9c01061247
Ref: interp fig-interp-compare1063416
Ref: 9c11063416
Node: Introduction to 2D Interpolation1063919
Ref: interp introduction-to-2d-interpolation1064079
Ref: 9c21064079
Node: 2D Interpolation Functions1064453
Ref: interp id11064602
Ref: 9c31064602
Ref: interp c gsl_interp2d1064810
Ref: 9c41064810
Ref: interp c gsl_interp2d_alloc1064871
Ref: 9c51064871
Ref: interp c gsl_interp2d_init1065201
Ref: 9c61065201
Ref: interp c gsl_interp2d_free1066063
Ref: 9c71066063
Node: 2D Interpolation Grids1066190
Ref: interp d-interpolation-grids1066329
Ref: 9c81066329
Ref: interp c gsl_interp2d_set1066730
Ref: 9c91066730
Ref: interp c gsl_interp2d_get1066982
Ref: 9ca1066982
Ref: interp c gsl_interp2d_idx1067226
Ref: 9cb1067226
Node: 2D Interpolation Types1067468
Ref: interp id21067621
Ref: 9cc1067621
Ref: interp c gsl_interp2d_type1067680
Ref: 9cd1067680
Ref: interp c gsl_interp2d_bilinear1067793
Ref: 9ce1067793
Ref: interp c gsl_interp2d_bicubic1067947
Ref: 9cf1067947
Ref: interp c gsl_interp2d_name1068022
Ref: 9d01068022
Ref: interp c gsl_interp2d_min_size1068380
Ref: 9d11068380
Ref: interp c gsl_interp2d_type_min_size1068469
Ref: 9d21068469
Node: 2D Evaluation of Interpolating Functions1068781
Ref: interp id31068937
Ref: 9d31068937
Ref: interp c gsl_interp2d_eval1069032
Ref: 9d41069032
Ref: interp c gsl_interp2d_eval_e1069270
Ref: 9d51069270
Ref: interp c gsl_interp2d_eval_extrap1069918
Ref: 9d61069918
Ref: interp c gsl_interp2d_eval_extrap_e1070163
Ref: 9d71070163
Ref: interp c gsl_interp2d_eval_deriv_x1070848
Ref: 9d81070848
Ref: interp c gsl_interp2d_eval_deriv_x_e1071094
Ref: 9d91071094
Ref: interp c gsl_interp2d_eval_deriv_y1071779
Ref: 9da1071779
Ref: interp c gsl_interp2d_eval_deriv_y_e1072025
Ref: 9db1072025
Ref: interp c gsl_interp2d_eval_deriv_xx1072710
Ref: 9dc1072710
Ref: interp c gsl_interp2d_eval_deriv_xx_e1072957
Ref: 9dd1072957
Ref: interp c gsl_interp2d_eval_deriv_yy1073647
Ref: 9de1073647
Ref: interp c gsl_interp2d_eval_deriv_yy_e1073894
Ref: 9df1073894
Ref: interp c gsl_interp2d_eval_deriv_xy1074584
Ref: 9e01074584
Ref: interp c gsl_interp2d_eval_deriv_xy_e1074831
Ref: 9e11074831
Node: 2D Higher-level Interface1075530
Ref: interp id41075697
Ref: 9e21075697
Ref: interp c gsl_spline2d1076218
Ref: 9e31076218
Ref: interp c gsl_spline2d_alloc1076342
Ref: 9e41076342
Ref: interp c gsl_spline2d_init1076459
Ref: 9e51076459
Ref: interp c gsl_spline2d_free1076625
Ref: 9e61076625
Ref: interp c gsl_spline2d_name1076687
Ref: 9e71076687
Ref: interp c gsl_spline2d_min_size1076773
Ref: 9e81076773
Ref: interp c gsl_spline2d_eval1076863
Ref: 9e91076863
Ref: interp c gsl_spline2d_eval_e1077034
Ref: 9ea1077034
Ref: interp c gsl_spline2d_eval_deriv_x1077217
Ref: 9eb1077217
Ref: interp c gsl_spline2d_eval_deriv_x_e1077396
Ref: 9ec1077396
Ref: interp c gsl_spline2d_eval_deriv_y1077587
Ref: 9ed1077587
Ref: interp c gsl_spline2d_eval_deriv_y_e1077766
Ref: 9ee1077766
Ref: interp c gsl_spline2d_eval_deriv_xx1077957
Ref: 9ef1077957
Ref: interp c gsl_spline2d_eval_deriv_xx_e1078137
Ref: 9f01078137
Ref: interp c gsl_spline2d_eval_deriv_yy1078329
Ref: 9f11078329
Ref: interp c gsl_spline2d_eval_deriv_yy_e1078509
Ref: 9f21078509
Ref: interp c gsl_spline2d_eval_deriv_xy1078701
Ref: 9f31078701
Ref: interp c gsl_spline2d_eval_deriv_xy_e1078881
Ref: 9f41078881
Ref: interp c gsl_spline2d_set1079073
Ref: 9f51079073
Ref: interp c gsl_spline2d_get1079210
Ref: 9f61079210
Node: 2D Interpolation Example programs1079454
Ref: interp id51079615
Ref: 9f71079615
Ref: interp fig-interp2d1081667
Ref: 9f81081667
Node: References and Further Reading<23>1081735
Ref: interp references-and-further-reading1081862
Ref: 9f91081862
Node: Numerical Differentiation1082442
Ref: diff doc1082563
Ref: 9fa1082563
Ref: diff numerical-differentiation1082563
Ref: 9fb1082563
Node: Functions1083025
Ref: diff functions1083118
Ref: 9fc1083118
Ref: diff c gsl_deriv_central1083149
Ref: 9fd1083149
Ref: diff c gsl_deriv_forward1084148
Ref: 9fe1084148
Ref: diff c gsl_deriv_backward1085252
Ref: 9ff1085252
Node: Examples<23>1086014
Ref: diff examples1086150
Ref: a001086150
Node: References and Further Reading<24>1087439
Ref: diff references-and-further-reading1087557
Ref: a011087557
Node: Chebyshev Approximations1087966
Ref: cheb doc1088093
Ref: a021088093
Ref: cheb chebyshev-approximations1088093
Ref: a031088093
Node: Definitions1088997
Ref: cheb definitions1089123
Ref: a041089123
Ref: cheb c gsl_cheb_series1089158
Ref: a051089158
Node: Creation and Calculation of Chebyshev Series1089856
Ref: cheb creation-and-calculation-of-chebyshev-series1090010
Ref: a061090010
Ref: cheb c gsl_cheb_alloc1090111
Ref: a071090111
Ref: cheb c gsl_cheb_free1090324
Ref: a081090324
Ref: cheb c gsl_cheb_init1090457
Ref: a091090457
Node: Auxiliary Functions1090848
Ref: cheb auxiliary-functions1091018
Ref: a0a1091018
Ref: cheb c gsl_cheb_order1091150
Ref: a0b1091150
Ref: cheb c gsl_cheb_size1091284
Ref: a0c1091284
Ref: cheb c gsl_cheb_coeffs1091348
Ref: a0d1091348
Node: Chebyshev Series Evaluation1091585
Ref: cheb chebyshev-series-evaluation1091736
Ref: a0e1091736
Ref: cheb c gsl_cheb_eval1091803
Ref: a0f1091803
Ref: cheb c gsl_cheb_eval_err1091979
Ref: a101091979
Ref: cheb c gsl_cheb_eval_n1092355
Ref: a111092355
Ref: cheb c gsl_cheb_eval_n_err1092589
Ref: a121092589
Node: Derivatives and Integrals1093038
Ref: cheb derivatives-and-integrals1093182
Ref: a131093182
Ref: cheb c gsl_cheb_calc_deriv1093526
Ref: a141093526
Ref: cheb c gsl_cheb_calc_integ1093875
Ref: a151093875
Node: Examples<24>1094317
Ref: cheb examples1094468
Ref: a161094468
Ref: cheb fig-cheb1095837
Ref: a171095837
Node: References and Further Reading<25>1095920
Ref: cheb references-and-further-reading1096037
Ref: a181096037
Node: Series Acceleration1096336
Ref: sum doc1096456
Ref: a191096456
Ref: sum series-acceleration1096456
Ref: a1a1096456
Node: Acceleration functions1097126
Ref: sum acceleration-functions1097261
Ref: a1b1097261
Ref: sum c gsl_sum_levin_u_workspace1098219
Ref: a1c1098219
Ref: sum c gsl_sum_levin_u_alloc1098297
Ref: a1d1098297
Ref: sum c gsl_sum_levin_u_free1098516
Ref: a1e1098516
Ref: sum c gsl_sum_levin_u_accel1098665
Ref: a1f1098665
Node: Acceleration functions without error estimation1099585
Ref: sum acceleration-functions-without-error-estimation1099741
Ref: a201099741
Ref: sum c gsl_sum_levin_utrunc_workspace1100831
Ref: a211100831
Ref: sum c gsl_sum_levin_utrunc_alloc1100938
Ref: a221100938
Ref: sum c gsl_sum_levin_utrunc_free1101191
Ref: a231101191
Ref: sum c gsl_sum_levin_utrunc_accel1101360
Ref: a241101360
Node: Examples<25>1102348
Ref: sum examples1102516
Ref: a251102516
Node: References and Further Reading<26>1104677
Ref: sum references-and-further-reading1104789
Ref: a261104789
Node: Wavelet Transforms1105561
Ref: dwt doc1105683
Ref: a271105683
Ref: dwt wavelet-transforms1105683
Ref: a281105683
Node: Definitions<2>1106173
Ref: dwt definitions1106266
Ref: a291106266
Node: Initialization1107143
Ref: dwt initialization1107264
Ref: a2a1107264
Ref: dwt c gsl_wavelet1107305
Ref: a2b1107305
Ref: dwt c gsl_wavelet_alloc1107446
Ref: a2c1107446
Ref: dwt c gsl_wavelet_type1107861
Ref: a2d1107861
Ref: dwt c gsl_wavelet_daubechies1107889
Ref: a2e1107889
Ref: dwt c gsl_wavelet_daubechies_centered1107931
Ref: a2f1107931
Ref: dwt c gsl_wavelet_haar1108165
Ref: a301108165
Ref: dwt c gsl_wavelet_haar_centered1108201
Ref: a311108201
Ref: dwt c gsl_wavelet_bspline1108351
Ref: a321108351
Ref: dwt c gsl_wavelet_bspline_centered1108390
Ref: a331108390
Ref: dwt c gsl_wavelet_name1108851
Ref: a341108851
Ref: dwt c gsl_wavelet_free1109010
Ref: a351109010
Ref: dwt c gsl_wavelet_workspace1109119
Ref: a361109119
Ref: dwt c gsl_wavelet_workspace_alloc1109299
Ref: a371109299
Ref: dwt c gsl_wavelet_workspace_free1109849
Ref: a381109849
Node: Transform Functions1109999
Ref: dwt transform-functions1110118
Ref: a391110118
Node: Wavelet transforms in one dimension1110556
Ref: dwt wavelet-transforms-in-one-dimension1110692
Ref: a3a1110692
Ref: dwt c gsl_wavelet_transform1110779
Ref: a3b1110779
Ref: dwt c gsl_wavelet_transform_forward1110960
Ref: a3c1110960
Ref: dwt c gsl_wavelet_transform_inverse1111122
Ref: a3d1111122
Node: Wavelet transforms in two dimension1112695
Ref: dwt wavelet-transforms-in-two-dimension1112831
Ref: a3e1112831
Ref: dwt c gsl_wavelet2d_transform1114142
Ref: a3f1114142
Ref: dwt c gsl_wavelet2d_transform_forward1114340
Ref: a401114340
Ref: dwt c gsl_wavelet2d_transform_inverse1114519
Ref: a411114519
Ref: dwt c gsl_wavelet2d_transform_matrix1115633
Ref: a421115633
Ref: dwt c gsl_wavelet2d_transform_matrix_forward1115799
Ref: a431115799
Ref: dwt c gsl_wavelet2d_transform_matrix_inverse1115936
Ref: a441115936
Ref: dwt c gsl_wavelet2d_nstransform1116177
Ref: a451116177
Ref: dwt c gsl_wavelet2d_nstransform_forward1116377
Ref: a461116377
Ref: dwt c gsl_wavelet2d_nstransform_inverse1116558
Ref: a471116558
Ref: dwt c gsl_wavelet2d_nstransform_matrix1116835
Ref: a481116835
Ref: dwt c gsl_wavelet2d_nstransform_matrix_forward1117003
Ref: a491117003
Ref: dwt c gsl_wavelet2d_nstransform_matrix_inverse1117142
Ref: a4a1117142
Node: Examples<26>1117410
Ref: dwt examples1117549
Ref: a4b1117549
Ref: dwt fig-dwt1119625
Ref: a4c1119625
Node: References and Further Reading<27>1119805
Ref: dwt references-and-further-reading1119916
Ref: a4d1119916
Node: Discrete Hankel Transforms1122002
Ref: dht doc1122133
Ref: a4e1122133
Ref: dht discrete-hankel-transforms1122133
Ref: a4f1122133
Node: Definitions<3>1122480
Ref: dht definitions1122579
Ref: a501122579
Node: Functions<2>1125790
Ref: dht functions1125932
Ref: a551125932
Ref: dht c gsl_dht1125963
Ref: a511125963
Ref: dht c gsl_dht_alloc1126039
Ref: a521126039
Ref: dht c gsl_dht_init1126182
Ref: a531126182
Ref: dht c gsl_dht_new1126360
Ref: a561126360
Ref: dht c gsl_dht_free1126602
Ref: a571126602
Ref: dht c gsl_dht_apply1126698
Ref: a541126698
Ref: dht c gsl_dht_x_sample1127136
Ref: a581127136
Ref: dht c gsl_dht_k_sample1127409
Ref: a591127409
Node: References and Further Reading<28>1127588
Ref: dht references-and-further-reading1127707
Ref: a5a1127707
Node: One Dimensional Root-Finding1127988
Ref: roots doc1128129
Ref: a5b1128129
Ref: roots one-dimensional-root-finding1128129
Ref: a5c1128129
Node: Overview1129232
Ref: roots overview1129322
Ref: a5d1129322
Node: Caveats1131116
Ref: roots caveats1131238
Ref: a601131238
Node: Initializing the Solver1132852
Ref: roots initializing-the-solver1132997
Ref: a611132997
Ref: roots c gsl_root_fsolver1133056
Ref: a5e1133056
Ref: roots c gsl_root_fdfsolver1133181
Ref: a5f1133181
Ref: roots c gsl_root_fsolver_alloc1133301
Ref: a621133301
Ref: roots c gsl_root_fdfsolver_alloc1133906
Ref: a631133906
Ref: roots c gsl_root_fsolver_set1134544
Ref: a641134544
Ref: roots c gsl_root_fdfsolver_set1134847
Ref: a651134847
Ref: roots c gsl_root_fsolver_free1135128
Ref: a661135128
Ref: roots c gsl_root_fdfsolver_free1135192
Ref: a671135192
Ref: roots c gsl_root_fsolver_name1135339
Ref: a681135339
Ref: roots c gsl_root_fdfsolver_name1135427
Ref: a691135427
Node: Providing the function to solve1135742
Ref: roots providing-function-to-solve1135905
Ref: a6a1135905
Ref: roots providing-the-function-to-solve1135905
Ref: a6b1135905
Ref: roots c gsl_function1136207
Ref: a6c1136207
Ref: roots c gsl_function_fdf1137345
Ref: a6d1137345
Node: Search Bounds and Guesses1139439
Ref: roots search-bounds-and-guesses1139588
Ref: a6e1139588
Node: Iteration1140286
Ref: roots iteration1140430
Ref: a6f1140430
Ref: roots c gsl_root_fsolver_iterate1140756
Ref: a701140756
Ref: roots c gsl_root_fdfsolver_iterate1140822
Ref: a711140822
Ref: roots c gsl_root_fsolver_root1141648
Ref: a721141648
Ref: roots c gsl_root_fdfsolver_root1141720
Ref: a731141720
Ref: roots c gsl_root_fsolver_x_lower1141898
Ref: a741141898
Ref: roots c gsl_root_fsolver_x_upper1141983
Ref: a751141983
Node: Search Stopping Parameters1142159
Ref: roots search-stopping-parameters1142304
Ref: a761142304
Ref: roots c gsl_root_test_interval1142783
Ref: a771142783
Ref: roots c gsl_root_test_delta1143792
Ref: a781143792
Ref: roots c gsl_root_test_residual1144222
Ref: a791144222
Node: Root Bracketing Algorithms1144739
Ref: roots root-bracketing-algorithms1144916
Ref: a7a1144916
Ref: roots c gsl_root_fsolver_type1145483
Ref: a7b1145483
Ref: roots c gsl_root_fsolver_bisection1145516
Ref: a7c1145516
Ref: roots c gsl_root_fsolver_falsepos1146297
Ref: a7d1146297
Ref: roots c gsl_root_fsolver_brent1147207
Ref: a7e1147207
Node: Root Finding Algorithms using Derivatives1148318
Ref: roots root-finding-algorithms-using-derivatives1148481
Ref: a7f1148481
Ref: roots c gsl_root_fdfsolver_type1148999
Ref: a801148999
Ref: roots c gsl_root_fdfsolver_newton1149034
Ref: a811149034
Ref: roots c gsl_root_fdfsolver_secant1149638
Ref: a821149638
Ref: roots c gsl_root_fdfsolver_steffenson1151042
Ref: a831151042
Ref: Root Finding Algorithms using Derivatives-Footnote-11152121
Node: Examples<27>1152279
Ref: roots examples1152450
Ref: a841152450
Node: References and Further Reading<29>1159595
Ref: roots references-and-further-reading1159716
Ref: a851159716
Node: One Dimensional Minimization1160235
Ref: min doc1160379
Ref: a861160379
Ref: min one-dimensional-minimization1160379
Ref: a871160379
Node: Overview<2>1161544
Ref: min overview1161640
Ref: a881161640
Ref: min fig-min-interval1161891
Ref: a891161891
Node: Caveats<2>1163405
Ref: min caveats1163536
Ref: a8b1163536
Node: Initializing the Minimizer1164720
Ref: min initializing-the-minimizer1164874
Ref: a8c1164874
Ref: min c gsl_min_fminimizer1164939
Ref: a8a1164939
Ref: min c gsl_min_fminimizer_alloc1165021
Ref: a8d1165021
Ref: min c gsl_min_fminimizer_set1165658
Ref: a8e1165658
Ref: min c gsl_min_fminimizer_set_with_values1166175
Ref: a8f1166175
Ref: min c gsl_min_fminimizer_free1166635
Ref: a901166635
Ref: min c gsl_min_fminimizer_name1166789
Ref: a911166789
Node: Providing the function to minimize1167110
Ref: min providing-the-function-to-minimize1167266
Ref: a921167266
Node: Iteration<2>1167592
Ref: min iteration1167741
Ref: a931167741
Ref: min c gsl_min_fminimizer_iterate1168073
Ref: a941168073
Ref: min c gsl_min_fminimizer_x_minimum1168805
Ref: a951168805
Ref: min c gsl_min_fminimizer_x_upper1169009
Ref: a961169009
Ref: min c gsl_min_fminimizer_x_lower1169098
Ref: a971169098
Ref: min c gsl_min_fminimizer_f_minimum1169299
Ref: a981169299
Ref: min c gsl_min_fminimizer_f_upper1169390
Ref: a991169390
Ref: min c gsl_min_fminimizer_f_lower1169479
Ref: a9a1169479
Node: Stopping Parameters1169750
Ref: min stopping-parameters1169888
Ref: a9b1169888
Ref: min c gsl_min_test_interval1170331
Ref: a9c1170331
Node: Minimization Algorithms1171366
Ref: min minimization-algorithms1171504
Ref: a9d1171504
Ref: min c gsl_min_fminimizer_type1171989
Ref: a9e1171989
Ref: min c gsl_min_fminimizer_goldensection1172024
Ref: a9f1172024
Ref: min c gsl_min_fminimizer_brent1173043
Ref: aa01173043
Ref: min c gsl_min_fminimizer_quad_golden1173968
Ref: aa11173968
Node: Examples<28>1174150
Ref: min examples1174303
Ref: aa21174303
Node: References and Further Reading<30>1176868
Ref: min references-and-further-reading1176989
Ref: aa31176989
Node: Multidimensional Root-Finding1177308
Ref: multiroots doc1177453
Ref: aa41177453
Ref: multiroots multidimensional-root-finding1177453
Ref: aa51177453
Node: Overview<3>1178747
Ref: multiroots overview1178860
Ref: aa61178860
Node: Initializing the Solver<2>1180946
Ref: multiroots initializing-the-solver1181102
Ref: aa91181102
Ref: multiroots c gsl_multiroot_fsolver1181384
Ref: aa81181384
Ref: multiroots c gsl_multiroot_fdfsolver1181503
Ref: aa71181503
Ref: multiroots c gsl_multiroot_fsolver_alloc1181621
Ref: aaa1181621
Ref: multiroots c gsl_multiroot_fdfsolver_alloc1182354
Ref: aab1182354
Ref: multiroots c gsl_multiroot_fsolver_set1183115
Ref: aac1183115
Ref: multiroots c gsl_multiroot_fdfsolver_set1183247
Ref: aad1183247
Ref: multiroots c gsl_multiroot_fsolver_free1183678
Ref: aae1183678
Ref: multiroots c gsl_multiroot_fdfsolver_free1183762
Ref: aaf1183762
Ref: multiroots c gsl_multiroot_fsolver_name1183929
Ref: ab01183929
Ref: multiroots c gsl_multiroot_fdfsolver_name1184027
Ref: ab11184027
Node: Providing the function to solve<2>1184356
Ref: multiroots providing-the-function-to-solve1184513
Ref: ab21184513
Ref: multiroots c gsl_multiroot_function1184766
Ref: ab31184766
Ref: multiroots c gsl_multiroot_function_fdf1186300
Ref: ab41186300
Node: Iteration<3>1189382
Ref: multiroots iteration1189542
Ref: ab51189542
Ref: multiroots c gsl_multiroot_fsolver_iterate1189868
Ref: ab61189868
Ref: multiroots c gsl_multiroot_fdfsolver_iterate1189954
Ref: ab71189954
Ref: multiroots c gsl_multiroot_fsolver_root1190698
Ref: ab81190698
Ref: multiroots c gsl_multiroot_fdfsolver_root1190796
Ref: ab91190796
Ref: multiroots c gsl_multiroot_fsolver_f1191011
Ref: aba1191011
Ref: multiroots c gsl_multiroot_fdfsolver_f1191106
Ref: abb1191106
Ref: multiroots c gsl_multiroot_fsolver_dx1191345
Ref: abc1191345
Ref: multiroots c gsl_multiroot_fdfsolver_dx1191441
Ref: abd1191441
Node: Search Stopping Parameters<2>1191644
Ref: multiroots search-stopping-parameters1191798
Ref: abe1191798
Ref: multiroots c gsl_multiroot_test_delta1192299
Ref: abf1192299
Ref: multiroots c gsl_multiroot_test_residual1192840
Ref: ac01192840
Node: Algorithms using Derivatives1193384
Ref: multiroots algorithms-using-derivatives1193556
Ref: ac11193556
Ref: multiroots c gsl_multiroot_fdfsolver_type1194034
Ref: ac21194034
Ref: multiroots c gsl_multiroot_fdfsolver_hybridsj1194168
Ref: ac31194168
Ref: multiroots c gsl_multiroot_fdfsolver_hybridj1196971
Ref: ac41196971
Ref: multiroots c gsl_multiroot_fdfsolver_newton1197306
Ref: ac51197306
Ref: multiroots c gsl_multiroot_fdfsolver_gnewton1198131
Ref: ac61198131
Node: Algorithms without Derivatives1198706
Ref: multiroots algorithms-without-derivatives1198861
Ref: ac71198861
Ref: multiroots c gsl_multiroot_fsolver_type1199322
Ref: ac81199322
Ref: multiroots c gsl_multiroot_fsolver_hybrids1199456
Ref: ac91199456
Ref: multiroots c gsl_multiroot_fsolver_hybrid1199901
Ref: aca1199901
Ref: multiroots c gsl_multiroot_fsolver_dnewton1200057
Ref: acb1200057
Ref: multiroots c gsl_multiroot_fsolver_broyden1201015
Ref: acc1201015
Node: Examples<29>1202147
Ref: multiroots examples1202308
Ref: acd1202308
Node: References and Further Reading<31>1210871
Ref: multiroots references-and-further-reading1210993
Ref: ace1210993
Node: Multidimensional Minimization1212072
Ref: multimin doc1212217
Ref: acf1212217
Ref: multimin multidimensional-minimization1212217
Ref: ad01212217
Node: Overview<4>1213480
Ref: multimin overview1213577
Ref: ad11213577
Node: Caveats<3>1215591
Ref: multimin caveats1215740
Ref: ad41215740
Node: Initializing the Multidimensional Minimizer1216325
Ref: multimin initializing-the-multidimensional-minimizer1216495
Ref: ad51216495
Ref: multimin c gsl_multimin_fdfminimizer1216787
Ref: ad21216787
Ref: multimin c gsl_multimin_fminimizer1216894
Ref: ad31216894
Ref: multimin c gsl_multimin_fdfminimizer_alloc1217001
Ref: ad61217001
Ref: multimin c gsl_multimin_fminimizer_alloc1217148
Ref: ad71217148
Ref: multimin c gsl_multimin_fdfminimizer_set1217609
Ref: ad81217609
Ref: multimin c gsl_multimin_fminimizer_set1217794
Ref: ad91217794
Ref: multimin c gsl_multimin_fdfminimizer_free1219179
Ref: ada1219179
Ref: multimin c gsl_multimin_fminimizer_free1219271
Ref: adb1219271
Ref: multimin c gsl_multimin_fdfminimizer_name1219445
Ref: adc1219445
Ref: multimin c gsl_multimin_fminimizer_name1219551
Ref: add1219551
Node: Providing a function to minimize1219896
Ref: multimin providing-a-function-to-minimize1220068
Ref: ade1220068
Ref: multimin c gsl_multimin_function_fdf1220505
Ref: adf1220505
Ref: multimin c gsl_multimin_function1221993
Ref: ae01221993
Ref: multimin multimin-paraboloid1222617
Ref: ae11222617
Node: Iteration<4>1224128
Ref: multimin iteration1224274
Ref: ae21224274
Ref: multimin c gsl_multimin_fdfminimizer_iterate1224578
Ref: ae31224578
Ref: multimin c gsl_multimin_fminimizer_iterate1224672
Ref: ae41224672
Ref: multimin c gsl_multimin_fdfminimizer_x1225299
Ref: ae51225299
Ref: multimin c gsl_multimin_fminimizer_x1225402
Ref: ae61225402
Ref: multimin c gsl_multimin_fdfminimizer_minimum1225501
Ref: ae71225501
Ref: multimin c gsl_multimin_fminimizer_minimum1225604
Ref: ae81225604
Ref: multimin c gsl_multimin_fdfminimizer_gradient1225703
Ref: ae91225703
Ref: multimin c gsl_multimin_fdfminimizer_dx1225813
Ref: aea1225813
Ref: multimin c gsl_multimin_fminimizer_size1225917
Ref: aeb1225917
Ref: multimin c gsl_multimin_fdfminimizer_restart1226282
Ref: aec1226282
Node: Stopping Criteria1226481
Ref: multimin stopping-criteria1226622
Ref: aed1226622
Ref: multimin c gsl_multimin_test_gradient1227061
Ref: aee1227061
Ref: multimin c gsl_multimin_test_size1227706
Ref: aef1227706
Node: Algorithms with Derivatives1228073
Ref: multimin algorithms-with-derivatives1228235
Ref: af01228235
Ref: multimin c gsl_multimin_fdfminimizer_type1228522
Ref: af11228522
Ref: multimin c gsl_multimin_fdfminimizer_conjugate_fr1228632
Ref: af21228632
Ref: multimin c gsl_multimin_fdfminimizer_conjugate_pr1229694
Ref: af31229694
Ref: multimin c gsl_multimin_fdfminimizer_vector_bfgs21230136
Ref: af41230136
Ref: multimin c gsl_multimin_fdfminimizer_vector_bfgs1230194
Ref: af51230194
Ref: multimin c gsl_multimin_fdfminimizer_steepest_descent1231391
Ref: af61231391
Node: Algorithms without Derivatives<2>1232010
Ref: multimin algorithms-without-derivatives1232167
Ref: af71232167
Ref: multimin c gsl_multimin_fminimizer_type1232343
Ref: af81232343
Ref: multimin c gsl_multimin_fminimizer_nmsimplex21232466
Ref: af91232466
Ref: multimin c gsl_multimin_fminimizer_nmsimplex1232520
Ref: afa1232520
Ref: multimin c gsl_multimin_fminimizer_nmsimplex2rand1234850
Ref: afb1234850
Node: Examples<30>1235538
Ref: multimin examples1235702
Ref: afc1235702
Ref: multimin fig-multimin1238452
Ref: afd1238452
Node: References and Further Reading<32>1242213
Ref: multimin references-and-further-reading1242335
Ref: afe1242335
Node: Linear Least-Squares Fitting1243074
Ref: lls doc1243221
Ref: aff1243221
Ref: lls linear-least-squares-fitting1243221
Ref: b001243221
Node: Overview<5>1244135
Ref: lls overview1244238
Ref: b011244238
Ref: lls sec-lls-overview1244238
Ref: b021244238
Node: Linear regression1246492
Ref: lls linear-regression1246630
Ref: b031246630
Node: Linear regression with a constant term1246941
Ref: lls linear-regression-with-a-constant-term1247084
Ref: b041247084
Ref: lls c gsl_fit_linear1247308
Ref: b051247308
Ref: lls c gsl_fit_wlinear1248321
Ref: b061248321
Ref: lls c gsl_fit_linear_est1249360
Ref: b071249360
Node: Linear regression without a constant term1249813
Ref: lls linear-regression-without-a-constant-term1249956
Ref: b081249956
Ref: lls c gsl_fit_mul1250199
Ref: b091250199
Ref: lls c gsl_fit_wmul1250924
Ref: b0a1250924
Ref: lls c gsl_fit_mul_est1251834
Ref: b0b1251834
Node: Multi-parameter regression1252190
Ref: lls multi-parameter-regression1252339
Ref: b0c1252339
Ref: lls c gsl_multifit_linear_workspace1253735
Ref: b0d1253735
Ref: lls c gsl_multifit_linear_alloc1253866
Ref: b0e1253866
Ref: lls c gsl_multifit_linear_free1254247
Ref: b0f1254247
Ref: lls c gsl_multifit_linear_svd1254417
Ref: b101254417
Ref: lls c gsl_multifit_linear_bsvd1254671
Ref: b111254671
Ref: lls c gsl_multifit_linear1255057
Ref: b121255057
Ref: lls c gsl_multifit_linear_tsvd1256315
Ref: b131256315
Ref: lls c gsl_multifit_wlinear1257511
Ref: b141257511
Ref: lls c gsl_multifit_wlinear_tsvd1258460
Ref: b151258460
Ref: lls c gsl_multifit_linear_est1259670
Ref: b161259670
Ref: lls c gsl_multifit_linear_residuals1260093
Ref: b171260093
Ref: lls c gsl_multifit_linear_rank1260404
Ref: b181260404
Node: Regularized regression1260827
Ref: lls regularized-regression1260983
Ref: b191260983
Ref: lls sec-regularized-regression1260983
Ref: b1a1260983
Ref: lls c gsl_multifit_linear_stdform11266290
Ref: b1b1266290
Ref: lls c gsl_multifit_linear_wstdform11266496
Ref: b1f1266496
Ref: lls c gsl_multifit_linear_L_decomp1267818
Ref: b211267818
Ref: lls c gsl_multifit_linear_stdform21268512
Ref: b1c1268512
Ref: lls c gsl_multifit_linear_wstdform21268771
Ref: b221268771
Ref: lls c gsl_multifit_linear_solve1270014
Ref: b231270014
Ref: lls c gsl_multifit_linear_genform11271178
Ref: b1d1271178
Ref: lls c gsl_multifit_linear_genform21271735
Ref: b1e1271735
Ref: lls c gsl_multifit_linear_wgenform21272005
Ref: b241272005
Ref: lls c gsl_multifit_linear_applyW1272885
Ref: b201272885
Ref: lls c gsl_multifit_linear_lcurve1273473
Ref: b251273473
Ref: lls c gsl_multifit_linear_lcorner1274558
Ref: b261274558
Ref: lls c gsl_multifit_linear_lcorner21275579
Ref: b271275579
Ref: lls c gsl_multifit_linear_gcv_init1276880
Ref: b281276880
Ref: lls c gsl_multifit_linear_gcv_curve1277511
Ref: b291277511
Ref: lls c gsl_multifit_linear_gcv_min1277979
Ref: b2a1277979
Ref: lls c gsl_multifit_linear_gcv_calc1278574
Ref: b2b1278574
Ref: lls c gsl_multifit_linear_gcv1278862
Ref: b2c1278862
Ref: lls c gsl_multifit_linear_Lk1279568
Ref: b2d1279568
Ref: lls c gsl_multifit_linear_Lsobolev1279888
Ref: b2e1279888
Ref: lls c gsl_multifit_linear_rcond1280796
Ref: b2f1280796
Node: Robust linear regression1281208
Ref: lls robust-linear-regression1281364
Ref: b301281364
Ref: lls c gsl_multifit_robust_workspace1284493
Ref: b311284493
Ref: lls c gsl_multifit_robust_alloc1284597
Ref: b321284597
Ref: lls c gsl_multifit_robust_type1285023
Ref: b331285023
Ref: lls c gsl_multifit_robust_default1285064
Ref: b341285064
Ref: lls c gsl_multifit_robust_bisquare1285300
Ref: b351285300
Ref: lls c gsl_multifit_robust_cauchy1285704
Ref: b361285704
Ref: lls c gsl_multifit_robust_fair1286252
Ref: b371286252
Ref: lls c gsl_multifit_robust_huber1286599
Ref: b381286599
Ref: lls c gsl_multifit_robust_ols1287227
Ref: b391287227
Ref: lls c gsl_multifit_robust_welsch1287612
Ref: b3a1287612
Ref: lls c gsl_multifit_robust_free1287967
Ref: b3b1287967
Ref: lls c gsl_multifit_robust_name1288134
Ref: b3c1288134
Ref: lls c gsl_multifit_robust_tune1288361
Ref: b3d1288361
Ref: lls c gsl_multifit_robust_maxiter1288776
Ref: b3e1288776
Ref: lls c gsl_multifit_robust_weights1289117
Ref: b3f1289117
Ref: lls c gsl_multifit_robust1289870
Ref: b401289870
Ref: lls c gsl_multifit_robust_est1291079
Ref: b421291079
Ref: lls c gsl_multifit_robust_residuals1291503
Ref: b431291503
Ref: lls c gsl_multifit_robust_statistics1292103
Ref: b411292103
Ref: lls c gsl_multifit_robust_stats1292569
Ref: b441292569
Node: Large dense linear systems1294503
Ref: lls large-dense-linear-systems1294652
Ref: b451294652
Node: Normal Equations Approach1296761
Ref: lls normal-equations-approach1296887
Ref: b461296887
Node: Tall Skinny QR TSQR Approach1298196
Ref: lls tall-skinny-qr-tsqr-approach1298372
Ref: b471298372
Node: Large Dense Linear Systems Solution Steps1299816
Ref: lls large-dense-linear-systems-solution-steps1300008
Ref: b481300008
Node: Large Dense Linear Least Squares Routines1300963
Ref: lls large-dense-linear-least-squares-routines1301118
Ref: b491301118
Ref: lls c gsl_multilarge_linear_workspace1301217
Ref: b4a1301217
Ref: lls c gsl_multilarge_linear_alloc1301355
Ref: b4b1301355
Ref: lls c gsl_multilarge_linear_type1301689
Ref: b4c1301689
Ref: lls c gsl_multilarge_linear_normal1301907
Ref: b4d1301907
Ref: lls c gsl_multilarge_linear_tsqr1302292
Ref: b4e1302292
Ref: lls c gsl_multilarge_linear_free1302725
Ref: b4f1302725
Ref: lls c gsl_multilarge_linear_name1302896
Ref: b501302896
Ref: lls c gsl_multilarge_linear_reset1303087
Ref: b511303087
Ref: lls c gsl_multilarge_linear_stdform11303294
Ref: b521303294
Ref: lls c gsl_multilarge_linear_wstdform11303504
Ref: b531303504
Ref: lls c gsl_multilarge_linear_L_decomp1304590
Ref: b551304590
Ref: lls c gsl_multilarge_linear_stdform21305042
Ref: b571305042
Ref: lls c gsl_multilarge_linear_wstdform21305289
Ref: b561305289
Ref: lls c gsl_multilarge_linear_accumulate1306301
Ref: b591306301
Ref: lls c gsl_multilarge_linear_solve1306785
Ref: b5a1306785
Ref: lls c gsl_multilarge_linear_genform11307345
Ref: b541307345
Ref: lls c gsl_multilarge_linear_genform21307906
Ref: b581307906
Ref: lls c gsl_multilarge_linear_lcurve1308406
Ref: b5b1308406
Ref: lls c gsl_multilarge_linear_rcond1309407
Ref: b5c1309407
Node: Troubleshooting1310006
Ref: lls troubleshooting1310143
Ref: b5d1310143
Node: Examples<31>1310633
Ref: lls examples1310778
Ref: b5e1310778
Node: Simple Linear Regression Example1311156
Ref: lls simple-linear-regression-example1311288
Ref: b5f1311288
Ref: lls fig-fit-wlinear1313381
Ref: b601313381
Node: Multi-parameter Linear Regression Example1313470
Ref: lls multi-parameter-linear-regression-example1313650
Ref: b611313650
Ref: lls fig-fit-wlinear21318520
Ref: b621318520
Node: Regularized Linear Regression Example 11318604
Ref: lls regularized-linear-regression-example-11318791
Ref: b631318791
Ref: lls fig-regularized1320489
Ref: b641320489
Node: Regularized Linear Regression Example 21326265
Ref: lls regularized-linear-regression-example-21326443
Ref: b651326443
Ref: lls fig-regularized21327919
Ref: b661327919
Node: Robust Linear Regression Example1333261
Ref: lls robust-linear-regression-example1333437
Ref: b671333437
Ref: lls fig-robust1337184
Ref: b681337184
Node: Large Dense Linear Regression Example1337262
Ref: lls large-dense-linear-regression-example1337390
Ref: b691337390
Ref: lls fig-multilarge1339312
Ref: b6a1339312
Node: References and Further Reading<33>1344758
Ref: lls references-and-further-reading1344879
Ref: b6b1344879
Node: Nonlinear Least-Squares Fitting1347053
Ref: nls doc1347184
Ref: b6c1347184
Ref: nls nonlinear-least-squares-fitting1347184
Ref: b6d1347184
Node: Overview<6>1348961
Ref: nls overview1349089
Ref: b6e1349089
Node: Solving the Trust Region Subproblem TRS1353341
Ref: nls solving-the-trust-region-subproblem-trs1353510
Ref: b6f1353510
Node: Levenberg-Marquardt1355301
Ref: nls levenberg-marquardt1355452
Ref: b701355452
Node: Levenberg-Marquardt with Geodesic Acceleration1356852
Ref: nls levenberg-marquardt-with-geodesic-acceleration1357018
Ref: b711357018
Node: Dogleg1358529
Ref: nls dogleg1358689
Ref: b721358689
Node: Double Dogleg1359923
Ref: nls double-dogleg1360061
Ref: b731360061
Node: Two Dimensional Subspace1360653
Ref: nls two-dimensional-subspace1360818
Ref: b741360818
Node: Steihaug-Toint Conjugate Gradient1361557
Ref: nls steihaug-toint-conjugate-gradient1361700
Ref: b751361700
Node: Weighted Nonlinear Least-Squares1362261
Ref: nls weighted-nonlinear-least-squares1362437
Ref: b761362437
Node: Tunable Parameters1363460
Ref: nls sec-tunable-parameters1363623
Ref: b781363623
Ref: nls tunable-parameters1363623
Ref: b791363623
Ref: nls c gsl_multifit_nlinear_parameters1363902
Ref: b7a1363902
Ref: nls c gsl_multilarge_nlinear_parameters1364997
Ref: b7b1364997
Ref: nls c gsl_multifit_nlinear_trs1366196
Ref: b7c1366196
Ref: nls c gsl_multilarge_nlinear_trs1366231
Ref: b7d1366231
Ref: nls c gsl_multifit_nlinear_trs_lm1366417
Ref: b7e1366417
Ref: nls c gsl_multilarge_nlinear_trs_lm1366464
Ref: b7f1366464
Ref: nls c gsl_multifit_nlinear_trs_lmaccel1366573
Ref: b801366573
Ref: nls c gsl_multilarge_nlinear_trs_lmaccel1366625
Ref: b811366625
Ref: nls c gsl_multifit_nlinear_trs_dogleg1366776
Ref: b821366776
Ref: nls c gsl_multilarge_nlinear_trs_dogleg1366827
Ref: b831366827
Ref: nls c gsl_multifit_nlinear_trs_ddogleg1366927
Ref: b841366927
Ref: nls c gsl_multilarge_nlinear_trs_ddogleg1366979
Ref: b851366979
Ref: nls c gsl_multifit_nlinear_trs_subspace2D1367087
Ref: b861367087
Ref: nls c gsl_multilarge_nlinear_trs_subspace2D1367142
Ref: b871367142
Ref: nls c gsl_multilarge_nlinear_trs_cgst1367251
Ref: b881367251
Ref: nls c gsl_multifit_nlinear_scale1367435
Ref: b891367435
Ref: nls c gsl_multilarge_nlinear_scale1367472
Ref: b8a1367472
Ref: nls c gsl_multifit_nlinear_scale_more1367638
Ref: b8b1367638
Ref: nls c gsl_multilarge_nlinear_scale_more1367689
Ref: b8c1367689
Ref: nls c gsl_multifit_nlinear_scale_levenberg1368649
Ref: b8d1368649
Ref: nls c gsl_multilarge_nlinear_scale_levenberg1368705
Ref: b8e1368705
Ref: nls c gsl_multifit_nlinear_scale_marquardt1369204
Ref: b8f1369204
Ref: nls c gsl_multilarge_nlinear_scale_marquardt1369260
Ref: b901369260
Ref: nls c gsl_multifit_nlinear_solver1369630
Ref: b911369630
Ref: nls c gsl_multilarge_nlinear_solver1369668
Ref: b921369668
Ref: nls c gsl_multifit_nlinear_solver_qr1370022
Ref: b931370022
Ref: nls c gsl_multifit_nlinear_solver_cholesky1370408
Ref: b941370408
Ref: nls c gsl_multilarge_nlinear_solver_cholesky1370464
Ref: b951370464
Ref: nls c gsl_multifit_nlinear_solver_mcholesky1371270
Ref: b961371270
Ref: nls c gsl_multilarge_nlinear_solver_mcholesky1371327
Ref: b971371327
Ref: nls c gsl_multifit_nlinear_solver_svd1371973
Ref: b981371973
Ref: nls c gsl_multifit_nlinear_fdtype1372274
Ref: b991372274
Ref: nls c GSL_MULTIFIT_NLINEAR_FWDIFF1372587
Ref: b9a1372587
Ref: nls c GSL_MULTIFIT_NLINEAR_CTRDIFF1373483
Ref: b9b1373483
Node: Initializing the Solver<3>1375736
Ref: nls initializing-the-solver1375905
Ref: b9c1375905
Ref: nls c gsl_multifit_nlinear_type1375964
Ref: b9d1375964
Ref: nls c gsl_multifit_nlinear_trust1376175
Ref: b9e1376175
Ref: nls c gsl_multifit_nlinear_alloc1376351
Ref: b9f1376351
Ref: nls c gsl_multilarge_nlinear_alloc1376573
Ref: ba01376573
Ref: nls c gsl_multifit_nlinear_default_parameters1378223
Ref: ba11378223
Ref: nls c gsl_multilarge_nlinear_default_parameters1378326
Ref: ba21378326
Ref: nls c gsl_multifit_nlinear_init1378696
Ref: ba31378696
Ref: nls c gsl_multifit_nlinear_winit1378851
Ref: b771378851
Ref: nls c gsl_multilarge_nlinear_init1379031
Ref: ba41379031
Ref: nls c gsl_multifit_nlinear_free1379620
Ref: ba61379620
Ref: nls c gsl_multilarge_nlinear_free1379712
Ref: ba71379712
Ref: nls c gsl_multifit_nlinear_name1379895
Ref: ba81379895
Ref: nls c gsl_multilarge_nlinear_name1380001
Ref: ba91380001
Ref: nls c gsl_multifit_nlinear_trs_name1380341
Ref: baa1380341
Ref: nls c gsl_multilarge_nlinear_trs_name1380451
Ref: bab1380451
Node: Providing the Function to be Minimized1380830
Ref: nls providing-the-function-to-be-minimized1380993
Ref: bac1380993
Ref: nls sec-providing-function-minimized1380993
Ref: ba51380993
Ref: nls c gsl_multifit_nlinear_fdf1381277
Ref: bad1381277
Ref: nls c gsl_multilarge_nlinear_fdf1384404
Ref: bae1384404
Node: Iteration<5>1388262
Ref: nls iteration1388422
Ref: baf1388422
Ref: nls c gsl_multifit_nlinear_iterate1388619
Ref: bb01388619
Ref: nls c gsl_multilarge_nlinear_iterate1388713
Ref: bb11388713
Ref: nls c gsl_multifit_nlinear_position1389730
Ref: bb21389730
Ref: nls c gsl_multilarge_nlinear_position1389840
Ref: bb31389840
Ref: nls c gsl_multifit_nlinear_residual1390063
Ref: bb41390063
Ref: nls c gsl_multilarge_nlinear_residual1390173
Ref: bb51390173
Ref: nls c gsl_multifit_nlinear_jac1390466
Ref: bb61390466
Ref: nls c gsl_multifit_nlinear_niter1390778
Ref: bb71390778
Ref: nls c gsl_multilarge_nlinear_niter1390879
Ref: bb81390879
Ref: nls c gsl_multifit_nlinear_rcond1391197
Ref: bb91391197
Ref: nls c gsl_multilarge_nlinear_rcond1391311
Ref: bba1391311
Ref: nls c gsl_multifit_nlinear_avratio1392840
Ref: bbb1392840
Ref: nls c gsl_multilarge_nlinear_avratio1392943
Ref: bbc1392943
Node: Testing for Convergence1393397
Ref: nls testing-for-convergence1393536
Ref: bbd1393536
Ref: nls c gsl_multifit_nlinear_test1394024
Ref: bbe1394024
Ref: nls c gsl_multilarge_nlinear_test1394200
Ref: bbf1394200
Node: High Level Driver1396276
Ref: nls high-level-driver1396443
Ref: bc01396443
Ref: nls c gsl_multifit_nlinear_driver1396601
Ref: bc11396601
Ref: nls c gsl_multilarge_nlinear_driver1396933
Ref: bc21396933
Node: Covariance matrix of best fit parameters1398534
Ref: nls covariance-matrix-of-best-fit-parameters1398696
Ref: bc41398696
Ref: nls c gsl_multifit_nlinear_covar1398791
Ref: bc51398791
Ref: nls c gsl_multilarge_nlinear_covar1398910
Ref: bc61398910
Node: Troubleshooting<2>1400740
Ref: nls sec-nlinear-troubleshooting1400897
Ref: bc31400897
Ref: nls troubleshooting1400897
Ref: bc71400897
Node: Examples<32>1402722
Ref: nls examples1402873
Ref: bc81402873
Node: Exponential Fitting Example1403188
Ref: nls exponential-fitting-example1403305
Ref: bc91403305
Ref: nls fig-fit-exp1407899
Ref: bca1407899
Node: Geodesic Acceleration Example 11414254
Ref: nls geodesic-acceleration-example-11414411
Ref: bcb1414411
Ref: nls fig-nlfit21416182
Ref: bcc1416182
Node: Geodesic Acceleration Example 21421602
Ref: nls geodesic-acceleration-example-21421761
Ref: bcd1421761
Ref: nls fig-nlfit2b1425552
Ref: bce1425552
Node: Comparing TRS Methods Example1433280
Ref: nls comparing-trs-methods-example1433445
Ref: bcf1433445
Ref: nls fig-nlfit31435754
Ref: bd01435754
Node: Large Nonlinear Least Squares Example1442098
Ref: nls large-nonlinear-least-squares-example1442223
Ref: bd11442223
Node: References and Further Reading<34>1451022
Ref: nls references-and-further-reading1451146
Ref: bd21451146
Node: Basis Splines1452596
Ref: bspline doc1452714
Ref: bd31452714
Ref: bspline basis-splines1452714
Ref: bd41452714
Ref: bspline chap-basis-splines1452714
Ref: 98d1452714
Node: Overview<7>1453475
Ref: bspline overview1453579
Ref: bd51453579
Node: Initializing the B-splines solver1454801
Ref: bspline initializing-the-b-splines-solver1454943
Ref: bd61454943
Ref: bspline c gsl_bspline_workspace1455022
Ref: bd71455022
Ref: bspline c gsl_bspline_alloc1455139
Ref: bd81455139
Ref: bspline c gsl_bspline_free1455546
Ref: bd91455546
Node: Constructing the knots vector1455687
Ref: bspline constructing-the-knots-vector1455841
Ref: bda1455841
Ref: bspline c gsl_bspline_knots1455912
Ref: bdb1455912
Ref: bspline c gsl_bspline_knots_uniform1456144
Ref: bdc1456144
Node: Evaluation of B-splines1456476
Ref: bspline evaluation-of-b-splines1456631
Ref: bdd1456631
Ref: bspline c gsl_bspline_eval1456690
Ref: bde1456690
Ref: bspline c gsl_bspline_eval_nonzero1457270
Ref: be01457270
Ref: bspline c gsl_bspline_ncoeffs1457994
Ref: bdf1457994
Node: Evaluation of B-spline derivatives1458162
Ref: bspline evaluation-of-b-spline-derivatives1458323
Ref: be11458323
Ref: bspline c gsl_bspline_deriv_eval1458404
Ref: be21458404
Ref: bspline c gsl_bspline_deriv_eval_nonzero1459216
Ref: be31459216
Node: Working with the Greville abscissae1460105
Ref: bspline working-with-the-greville-abscissae1460255
Ref: be41460255
Ref: bspline c gsl_bspline_greville_abscissa1460788
Ref: be51460788
Node: Examples<33>1461091
Ref: bspline examples1461241
Ref: be61461241
Ref: bspline fig-bspline1464990
Ref: be71464990
Node: References and Further Reading<35>1465068
Ref: bspline references-and-further-reading1465174
Ref: be81465174
Node: Sparse Matrices1465864
Ref: spmatrix doc1465970
Ref: be91465970
Ref: spmatrix sparse-matrices1465970
Ref: bea1465970
Node: Data types<2>1467157
Ref: spmatrix data-types1467261
Ref: beb1467261
Node: Sparse Matrix Storage Formats1469461
Ref: spmatrix sparse-matrix-storage-formats1469585
Ref: bec1469585
Node: Coordinate Storage COO1470423
Ref: spmatrix coordinate-storage-coo1470549
Ref: bed1470549
Ref: spmatrix sec-spmatrix-coo1470549
Ref: bee1470549
Node: Compressed Sparse Column CSC1472654
Ref: spmatrix compressed-sparse-column-csc1472814
Ref: bef1472814
Ref: spmatrix sec-spmatrix-csc1472814
Ref: bf01472814
Node: Compressed Sparse Row CSR1474171
Ref: spmatrix compressed-sparse-row-csr1474300
Ref: bf11474300
Ref: spmatrix sec-spmatrix-csr1474300
Ref: bf21474300
Node: Overview<8>1475587
Ref: spmatrix overview1475708
Ref: bf31475708
Ref: spmatrix c gsl_spmatrix1475930
Ref: bf41475930
Node: Allocation1478237
Ref: spmatrix allocation1478354
Ref: bf51478354
Ref: spmatrix c gsl_spmatrix_alloc1478736
Ref: bf61478736
Ref: spmatrix c gsl_spmatrix_alloc_nzmax1479598
Ref: bf71479598
Ref: spmatrix c GSL_SPMATRIX_COO1480549
Ref: bf91480549
Ref: spmatrix c GSL_SPMATRIX_CSC1480644
Ref: bfa1480644
Ref: spmatrix c GSL_SPMATRIX_CSR1480743
Ref: bfb1480743
Ref: spmatrix c gsl_spmatrix_realloc1480920
Ref: bf81480920
Ref: spmatrix c gsl_spmatrix_free1481399
Ref: bfd1481399
Node: Accessing Matrix Elements1481632
Ref: spmatrix accessing-matrix-elements1481766
Ref: bfe1481766
Ref: spmatrix c gsl_spmatrix_get1481829
Ref: bff1481829
Ref: spmatrix c gsl_spmatrix_set1482104
Ref: bfc1482104
Ref: spmatrix c gsl_spmatrix_ptr1482372
Ref: c001482372
Node: Initializing Matrix Elements1482773
Ref: spmatrix initializing-matrix-elements1482925
Ref: c011482925
Ref: spmatrix c gsl_spmatrix_set_zero1483252
Ref: c021483252
Node: Reading and Writing Matrices1483653
Ref: spmatrix reading-and-writing-matrices1483796
Ref: c031483796
Ref: spmatrix c gsl_spmatrix_fwrite1483865
Ref: c041483865
Ref: spmatrix c gsl_spmatrix_fread1484384
Ref: c051484384
Ref: spmatrix c gsl_spmatrix_fprintf1485092
Ref: c061485092
Ref: spmatrix c gsl_spmatrix_fscanf1485779
Ref: c071485779
Node: Copying Matrices1486233
Ref: spmatrix copying-matrices1486375
Ref: c081486375
Ref: spmatrix c gsl_spmatrix_memcpy1486420
Ref: c091486420
Node: Exchanging Rows and Columns1486789
Ref: spmatrix exchanging-rows-and-columns1486920
Ref: c0a1486920
Ref: spmatrix c gsl_spmatrix_transpose_memcpy1486987
Ref: c0b1486987
Ref: spmatrix c gsl_spmatrix_transpose1487431
Ref: c0c1487431
Node: Matrix Operations1488127
Ref: spmatrix matrix-operations1488259
Ref: c0d1488259
Ref: spmatrix c gsl_spmatrix_scale1488308
Ref: c0e1488308
Ref: spmatrix c gsl_spmatrix_scale_columns1488637
Ref: c0f1488637
Ref: spmatrix c gsl_spmatrix_scale_rows1489112
Ref: c101489112
Ref: spmatrix c gsl_spmatrix_add1489578
Ref: c111489578
Ref: spmatrix c gsl_spmatrix_add_to_dense1489865
Ref: c121489865
Node: Matrix Properties1490326
Ref: spmatrix matrix-properties1490467
Ref: c131490467
Ref: spmatrix c gsl_spmatrix_type1490516
Ref: c141490516
Ref: spmatrix c gsl_spmatrix_nnz1490934
Ref: c151490934
Ref: spmatrix c gsl_spmatrix_equal1491161
Ref: c161491161
Node: Finding Maximum and Minimum Elements1491578
Ref: spmatrix finding-maximum-and-minimum-elements1491719
Ref: c171491719
Ref: spmatrix c gsl_spmatrix_minmax1491806
Ref: c181491806
Ref: spmatrix c gsl_spmatrix_min_index1492190
Ref: c191492190
Node: Compressed Format1492659
Ref: spmatrix compressed-format1492827
Ref: c1a1492827
Ref: spmatrix c gsl_spmatrix_csc1492956
Ref: c1b1492956
Ref: spmatrix c gsl_spmatrix_csr1493321
Ref: c1c1493321
Ref: spmatrix c gsl_spmatrix_compress1493683
Ref: c1d1493683
Node: Conversion Between Sparse and Dense Matrices1494176
Ref: spmatrix conversion-between-sparse-and-dense-matrices1494320
Ref: c1e1494320
Ref: spmatrix c gsl_spmatrix_d2sp1494569
Ref: c1f1494569
Ref: spmatrix c gsl_spmatrix_sp2d1494827
Ref: c201494827
Node: Examples<34>1495120
Ref: spmatrix examples1495281
Ref: c211495281
Node: References and Further Reading<36>1498868
Ref: spmatrix references-and-further-reading1498976
Ref: c221498976
Node: Sparse BLAS Support1499297
Ref: spblas doc1499411
Ref: c231499411
Ref: spblas sparse-blas-support1499411
Ref: c241499411
Node: Sparse BLAS operations1499955
Ref: spblas sparse-blas-operations1500077
Ref: c251500077
Ref: spblas c gsl_spblas_dgemv1500134
Ref: c261500134
Ref: spblas c gsl_spblas_dgemm1500675
Ref: c271500675
Node: References and Further Reading<37>1500934
Ref: spblas references-and-further-reading1501056
Ref: c281501056
Node: Sparse Linear Algebra1501375
Ref: splinalg doc1501492
Ref: c291501492
Ref: splinalg sparse-linear-algebra1501492
Ref: c2a1501492
Node: Overview<9>1502017
Ref: splinalg overview1502120
Ref: c2b1502120
Node: Sparse Iterative Solvers1502746
Ref: splinalg sparse-iterative-solvers1502870
Ref: c2c1502870
Node: Overview<10>1503044
Ref: splinalg id11503160
Ref: c2d1503160
Node: Types of Sparse Iterative Solvers1503888
Ref: splinalg types-of-sparse-iterative-solvers1504047
Ref: c2e1504047
Ref: splinalg c gsl_splinalg_itersolve_type1504216
Ref: c2f1504216
Ref: splinalg c gsl_splinalg_itersolve_gmres1504255
Ref: c301504255
Node: Iterating the Sparse Linear System1506227
Ref: splinalg iterating-the-sparse-linear-system1506365
Ref: c321506365
Ref: splinalg c gsl_splinalg_itersolve_alloc1506576
Ref: c331506576
Ref: splinalg c gsl_splinalg_itersolve_free1507100
Ref: c341507100
Ref: splinalg c gsl_splinalg_itersolve_name1507263
Ref: c351507263
Ref: splinalg c gsl_splinalg_itersolve_iterate1507436
Ref: c311507436
Ref: splinalg c gsl_splinalg_itersolve_normr1508552
Ref: c361508552
Node: Examples<35>1508814
Ref: splinalg examples1508961
Ref: c371508961
Ref: splinalg fig-splinalg-poisson1510000
Ref: c381510000
Node: References and Further Reading<38>1513296
Ref: splinalg references-and-further-reading1513410
Ref: c391513410
Node: Physical Constants1513793
Ref: const doc1513921
Ref: c3a1513921
Ref: const physical-constants1513921
Ref: c3b1513921
Node: Fundamental Constants1515250
Ref: const fundamental-constants1515362
Ref: c3c1515362
Ref: const c GSL_CONST_MKSA_SPEED_OF_LIGHT1515417
Ref: c3d1515417
Ref: const c GSL_CONST_MKSA_VACUUM_PERMEABILITY1515498
Ref: c3e1515498
Ref: const c GSL_CONST_MKSA_VACUUM_PERMITTIVITY1515646
Ref: c3f1515646
Ref: const c GSL_CONST_MKSA_PLANCKS_CONSTANT_H1515799
Ref: c401515799
Ref: const c GSL_CONST_MKSA_PLANCKS_CONSTANT_HBAR1515875
Ref: c411515875
Ref: const c GSL_CONST_NUM_AVOGADRO1515974
Ref: c421515974
Ref: const c GSL_CONST_MKSA_FARADAY1516041
Ref: c431516041
Ref: const c GSL_CONST_MKSA_BOLTZMANN1516113
Ref: c441516113
Ref: const c GSL_CONST_MKSA_MOLAR_GAS1516183
Ref: c451516183
Ref: const c GSL_CONST_MKSA_STANDARD_GAS_VOLUME1516255
Ref: c461516255
Ref: const c GSL_CONST_MKSA_STEFAN_BOLTZMANN_CONSTANT1516338
Ref: c471516338
Ref: const c GSL_CONST_MKSA_GAUSS1516446
Ref: c481516446
Node: Astronomy and Astrophysics1516516
Ref: const astronomy-and-astrophysics1516663
Ref: c491516663
Ref: const c GSL_CONST_MKSA_ASTRONOMICAL_UNIT1516728
Ref: c4a1516728
Ref: const c GSL_CONST_MKSA_GRAVITATIONAL_CONSTANT1516844
Ref: c4b1516844
Ref: const c GSL_CONST_MKSA_LIGHT_YEAR1516931
Ref: c4c1516931
Ref: const c GSL_CONST_MKSA_PARSEC1517009
Ref: c4d1517009
Ref: const c GSL_CONST_MKSA_GRAV_ACCEL1517079
Ref: c4e1517079
Ref: const c GSL_CONST_MKSA_SOLAR_MASS1517176
Ref: c4f1517176
Node: Atomic and Nuclear Physics1517241
Ref: const atomic-and-nuclear-physics1517386
Ref: c501517386
Ref: const c GSL_CONST_MKSA_ELECTRON_CHARGE1517451
Ref: c511517451
Ref: const c GSL_CONST_MKSA_ELECTRON_VOLT1517531
Ref: c521517531
Ref: const c GSL_CONST_MKSA_UNIFIED_ATOMIC_MASS1517613
Ref: c531517613
Ref: const c GSL_CONST_MKSA_MASS_ELECTRON1517696
Ref: c541517696
Ref: const c GSL_CONST_MKSA_MASS_MUON1517774
Ref: c551517774
Ref: const c GSL_CONST_MKSA_MASS_PROTON1517846
Ref: c561517846
Ref: const c GSL_CONST_MKSA_MASS_NEUTRON1517920
Ref: c571517920
Ref: const c GSL_CONST_NUM_FINE_STRUCTURE1517996
Ref: c581517996
Ref: const c GSL_CONST_MKSA_RYDBERG1518095
Ref: c591518095
Ref: const c GSL_CONST_MKSA_BOHR_RADIUS1518269
Ref: c5a1518269
Ref: const c GSL_CONST_MKSA_ANGSTROM1518336
Ref: c5b1518336
Ref: const c GSL_CONST_MKSA_BARN1518404
Ref: c5c1518404
Ref: const c GSL_CONST_MKSA_BOHR_MAGNETON1518462
Ref: c5d1518462
Ref: const c GSL_CONST_MKSA_NUCLEAR_MAGNETON1518535
Ref: c5e1518535
Ref: const c GSL_CONST_MKSA_ELECTRON_MAGNETIC_MOMENT1518614
Ref: c5f1518614
Ref: const c GSL_CONST_MKSA_PROTON_MAGNETIC_MOMENT1518801
Ref: c601518801
Ref: const c GSL_CONST_MKSA_THOMSON_CROSS_SECTION1518899
Ref: c611518899
Ref: const c GSL_CONST_MKSA_DEBYE1518991
Ref: c621518991
Node: Measurement of Time1519072
Ref: const measurement-of-time1519205
Ref: c631519205
Ref: const c GSL_CONST_MKSA_MINUTE1519256
Ref: c641519256
Ref: const c GSL_CONST_MKSA_HOUR1519331
Ref: c651519331
Ref: const c GSL_CONST_MKSA_DAY1519402
Ref: c661519402
Ref: const c GSL_CONST_MKSA_WEEK1519471
Ref: c671519471
Node: Imperial Units1519542
Ref: const imperial-units1519673
Ref: c681519673
Ref: const c GSL_CONST_MKSA_INCH1519714
Ref: c691519714
Ref: const c GSL_CONST_MKSA_FOOT1519774
Ref: c6a1519774
Ref: const c GSL_CONST_MKSA_YARD1519834
Ref: c6b1519834
Ref: const c GSL_CONST_MKSA_MILE1519894
Ref: c6c1519894
Ref: const c GSL_CONST_MKSA_MIL1519954
Ref: c6d1519954
Node: Speed and Nautical Units1520034
Ref: const speed-and-nautical-units1520160
Ref: c6e1520160
Ref: const c GSL_CONST_MKSA_KILOMETERS_PER_HOUR1520221
Ref: c6f1520221
Ref: const c GSL_CONST_MKSA_MILES_PER_HOUR1520309
Ref: c701520309
Ref: const c GSL_CONST_MKSA_NAUTICAL_MILE1520387
Ref: c711520387
Ref: const c GSL_CONST_MKSA_FATHOM1520465
Ref: c721520465
Ref: const c GSL_CONST_MKSA_KNOT1520529
Ref: c731520529
Node: Printers Units1520588
Ref: const printers-units1520722
Ref: c741520722
Ref: const c GSL_CONST_MKSA_POINT1520763
Ref: c751520763
Ref: const c GSL_CONST_MKSA_TEXPOINT1520849
Ref: c761520849
Node: Volume Area and Length1520933
Ref: const volume-area-and-length1521058
Ref: c771521058
Ref: const c GSL_CONST_MKSA_MICRON1521117
Ref: c781521117
Ref: const c GSL_CONST_MKSA_HECTARE1521181
Ref: c791521181
Ref: const c GSL_CONST_MKSA_ACRE1521245
Ref: c7a1521245
Ref: const c GSL_CONST_MKSA_LITER1521303
Ref: c7b1521303
Ref: const c GSL_CONST_MKSA_US_GALLON1521365
Ref: c7c1521365
Ref: const c GSL_CONST_MKSA_CANADIAN_GALLON1521435
Ref: c7d1521435
Ref: const c GSL_CONST_MKSA_UK_GALLON1521517
Ref: c7e1521517
Ref: const c GSL_CONST_MKSA_QUART1521587
Ref: c7f1521587
Ref: const c GSL_CONST_MKSA_PINT1521649
Ref: c801521649
Node: Mass and Weight1521709
Ref: const mass-and-weight1521844
Ref: c811521844
Ref: const c GSL_CONST_MKSA_POUND_MASS1521887
Ref: c821521887
Ref: const c GSL_CONST_MKSA_OUNCE_MASS1521952
Ref: c831521952
Ref: const c GSL_CONST_MKSA_TON1522017
Ref: c841522017
Ref: const c GSL_CONST_MKSA_METRIC_TON1522073
Ref: c851522073
Ref: const c GSL_CONST_MKSA_UK_TON1522153
Ref: c861522153
Ref: const c GSL_CONST_MKSA_TROY_OUNCE1522215
Ref: c871522215
Ref: const c GSL_CONST_MKSA_CARAT1522285
Ref: c881522285
Ref: const c GSL_CONST_MKSA_GRAM_FORCE1522345
Ref: c891522345
Ref: const c GSL_CONST_MKSA_POUND_FORCE1522417
Ref: c8a1522417
Ref: const c GSL_CONST_MKSA_KILOPOUND_FORCE1522491
Ref: c8b1522491
Ref: const c GSL_CONST_MKSA_POUNDAL1522573
Ref: c8c1522573
Node: Thermal Energy and Power1522638
Ref: const thermal-energy-and-power1522759
Ref: c8d1522759
Ref: const c GSL_CONST_MKSA_CALORIE1522822
Ref: c8e1522822
Ref: const c GSL_CONST_MKSA_BTU1522888
Ref: c8f1522888
Ref: const c GSL_CONST_MKSA_THERM1522968
Ref: c901522968
Ref: const c GSL_CONST_MKSA_HORSEPOWER1523030
Ref: c911523030
Node: Pressure1523101
Ref: const pressure1523216
Ref: c921523216
Ref: const c GSL_CONST_MKSA_BAR1523247
Ref: c931523247
Ref: const c GSL_CONST_MKSA_STD_ATMOSPHERE1523307
Ref: c941523307
Ref: const c GSL_CONST_MKSA_TORR1523394
Ref: c951523394
Ref: const c GSL_CONST_MKSA_METER_OF_MERCURY1523456
Ref: c961523456
Ref: const c GSL_CONST_MKSA_INCH_OF_MERCURY1523542
Ref: c971523542
Ref: const c GSL_CONST_MKSA_INCH_OF_WATER1523626
Ref: c981523626
Ref: const c GSL_CONST_MKSA_PSI1523706
Ref: c991523706
Node: Viscosity1523784
Ref: const viscosity1523897
Ref: c9a1523897
Ref: const c GSL_CONST_MKSA_POISE1523930
Ref: c9b1523930
Ref: const c GSL_CONST_MKSA_STOKES1524003
Ref: c9c1524003
Node: Light and Illumination1524080
Ref: const light-and-illumination1524198
Ref: c9d1524198
Ref: const c GSL_CONST_MKSA_STILB1524257
Ref: c9e1524257
Ref: const c GSL_CONST_MKSA_LUMEN1524322
Ref: c9f1524322
Ref: const c GSL_CONST_MKSA_LUX1524391
Ref: ca01524391
Ref: const c GSL_CONST_MKSA_PHOT1524454
Ref: ca11524454
Ref: const c GSL_CONST_MKSA_FOOTCANDLE1524519
Ref: ca21524519
Ref: const c GSL_CONST_MKSA_LAMBERT1524596
Ref: ca31524596
Ref: const c GSL_CONST_MKSA_FOOTLAMBERT1524665
Ref: ca41524665
Node: Radioactivity1524742
Ref: const radioactivity1524867
Ref: ca51524867
Ref: const c GSL_CONST_MKSA_CURIE1524908
Ref: ca61524908
Ref: const c GSL_CONST_MKSA_ROENTGEN1524972
Ref: ca71524972
Ref: const c GSL_CONST_MKSA_RAD1525042
Ref: ca81525042
Node: Force and Energy1525107
Ref: const force-and-energy1525218
Ref: ca91525218
Ref: const c GSL_CONST_MKSA_NEWTON1525265
Ref: caa1525265
Ref: const c GSL_CONST_MKSA_DYNE1525337
Ref: cab1525337
Ref: const c GSL_CONST_MKSA_JOULE1525413
Ref: cac1525413
Ref: const c GSL_CONST_MKSA_ERG1525484
Ref: cad1525484
Node: Prefixes1525555
Ref: const prefixes1525665
Ref: cae1525665
Ref: const c GSL_CONST_NUM_YOTTA1525748
Ref: caf1525748
Ref: const c GSL_CONST_NUM_ZETTA1525794
Ref: cb01525794
Ref: const c GSL_CONST_NUM_EXA1525840
Ref: cb11525840
Ref: const c GSL_CONST_NUM_PETA1525884
Ref: cb21525884
Ref: const c GSL_CONST_NUM_TERA1525929
Ref: cb31525929
Ref: const c GSL_CONST_NUM_GIGA1525974
Ref: cb41525974
Ref: const c GSL_CONST_NUM_MEGA1526016
Ref: cb51526016
Ref: const c GSL_CONST_NUM_KILO1526058
Ref: cb61526058
Ref: const c GSL_CONST_NUM_MILLI1526100
Ref: cb71526100
Ref: const c GSL_CONST_NUM_MICRO1526146
Ref: cb81526146
Ref: const c GSL_CONST_NUM_NANO1526192
Ref: cb91526192
Ref: const c GSL_CONST_NUM_PICO1526237
Ref: cba1526237
Ref: const c GSL_CONST_NUM_FEMTO1526283
Ref: cbb1526283
Ref: const c GSL_CONST_NUM_ATTO1526330
Ref: cbc1526330
Ref: const c GSL_CONST_NUM_ZEPTO1526376
Ref: cbd1526376
Ref: const c GSL_CONST_NUM_YOCTO1526423
Ref: cbe1526423
Node: Examples<36>1526470
Ref: const examples1526598
Ref: cbf1526598
Node: References and Further Reading<39>1528228
Ref: const references-and-further-reading1528339
Ref: cc01528339
Node: IEEE floating-point arithmetic1528946
Ref: ieee754 doc1529081
Ref: cc11529081
Ref: ieee754 chap-ieee1529081
Ref: cc21529081
Ref: ieee754 ieee-floating-point-arithmetic1529081
Ref: cc31529081
Node: Representation of floating point numbers1529566
Ref: ieee754 representation-of-floating-point-numbers1529715
Ref: cc41529715
Ref: ieee754 c gsl_ieee_fprintf_float1531744
Ref: cc51531744
Ref: ieee754 c gsl_ieee_fprintf_double1531829
Ref: cc61531829
Ref: ieee754 c gsl_ieee_printf_float1532700
Ref: cc71532700
Ref: ieee754 c gsl_ieee_printf_double1532759
Ref: cc81532759
Node: Setting up your IEEE environment1534123
Ref: ieee754 setting-up-your-ieee-environment1534315
Ref: cc91534315
Ref: ieee754 c GSL_IEEE_MODE1535384
Ref: cca1535384
Ref: ieee754 c gsl_ieee_env_setup1535464
Ref: ccb1535464
Node: References and Further Reading<40>1541258
Ref: ieee754 references-and-further-reading1541401
Ref: ccc1541401
Node: Debugging Numerical Programs1542428
Ref: debug doc1542564
Ref: ccd1542564
Ref: debug debugging-numerical-programs1542564
Ref: cce1542564
Node: Using gdb1542941
Ref: debug using-gdb1543059
Ref: ccf1543059
Node: Examining floating point registers1546353
Ref: debug examining-floating-point-registers1546514
Ref: cd01546514
Node: Handling floating point exceptions1547656
Ref: debug handling-floating-point-exceptions1547850
Ref: cd11547850
Node: GCC warning options for numerical programs1549112
Ref: debug gcc-warning-options-for-numerical-programs1549306
Ref: cd21549306
Node: References and Further Reading<41>1553239
Ref: debug references-and-further-reading1553390
Ref: cd31553390
Node: Contributors to GSL1554032
Ref: contrib doc1554153
Ref: cd41554153
Ref: contrib contributors-to-gsl1554153
Ref: cd51554153
Node: Autoconf Macros1558628
Ref: autoconf doc1558738
Ref: cd61558738
Ref: autoconf autoconf-macros1558738
Ref: cd71558738
Ref: autoconf chap-autoconf-macros1558738
Ref: 161558738
Node: GSL CBLAS Library1562808
Ref: cblas doc1562925
Ref: cd81562925
Ref: cblas chap-cblas1562925
Ref: 4711562925
Ref: cblas gsl-cblas-library1562925
Ref: cd91562925
Node: Level 1<2>1563307
Ref: cblas level-11563391
Ref: cda1563391
Ref: cblas c cblas_sdsdot1563418
Ref: cdb1563418
Ref: cblas c cblas_dsdot1563561
Ref: cdc1563561
Ref: cblas c cblas_sdot1563685
Ref: cdd1563685
Ref: cblas c cblas_ddot1563807
Ref: cde1563807
Ref: cblas c cblas_cdotu_sub1563932
Ref: cdf1563932
Ref: cblas c cblas_cdotc_sub1564069
Ref: ce01564069
Ref: cblas c cblas_zdotu_sub1564206
Ref: ce11564206
Ref: cblas c cblas_zdotc_sub1564343
Ref: ce21564343
Ref: cblas c cblas_snrm21564480
Ref: ce31564480
Ref: cblas c cblas_sasum1564570
Ref: ce41564570
Ref: cblas c cblas_dnrm21564660
Ref: ce51564660
Ref: cblas c cblas_dasum1564752
Ref: ce61564752
Ref: cblas c cblas_scnrm21564844
Ref: ce71564844
Ref: cblas c cblas_scasum1564934
Ref: ce81564934
Ref: cblas c cblas_dznrm21565024
Ref: ce91565024
Ref: cblas c cblas_dzasum1565115
Ref: cea1565115
Ref: cblas c cblas_isamax1565206
Ref: ceb1565206
Ref: cblas c cblas_idamax1565303
Ref: cec1565303
Ref: cblas c cblas_icamax1565401
Ref: ced1565401
Ref: cblas c cblas_izamax1565497
Ref: cee1565497
Ref: cblas c cblas_sswap1565593
Ref: cef1565593
Ref: cblas c cblas_scopy1565703
Ref: cf01565703
Ref: cblas c cblas_saxpy1565819
Ref: cf11565819
Ref: cblas c cblas_dswap1565954
Ref: cf21565954
Ref: cblas c cblas_dcopy1566066
Ref: cf31566066
Ref: cblas c cblas_daxpy1566184
Ref: cf41566184
Ref: cblas c cblas_cswap1566322
Ref: cf51566322
Ref: cblas c cblas_ccopy1566430
Ref: cf61566430
Ref: cblas c cblas_caxpy1566544
Ref: cf71566544
Ref: cblas c cblas_zswap1566678
Ref: cf81566678
Ref: cblas c cblas_zcopy1566786
Ref: cf91566786
Ref: cblas c cblas_zaxpy1566900
Ref: cfa1566900
Ref: cblas c cblas_srotg1567034
Ref: cfb1567034
Ref: cblas c cblas_srotmg1567121
Ref: cfc1567121
Ref: cblas c cblas_srot1567228
Ref: cfd1567228
Ref: cblas c cblas_srotm1567367
Ref: cfe1567367
Ref: cblas c cblas_drotg1567494
Ref: cff1567494
Ref: cblas c cblas_drotmg1567585
Ref: d001567585
Ref: cblas c cblas_drot1567697
Ref: d011567697
Ref: cblas c cblas_drotm1567840
Ref: d021567840
Ref: cblas c cblas_sscal1567970
Ref: d031567970
Ref: cblas c cblas_dscal1568072
Ref: d041568072
Ref: cblas c cblas_cscal1568176
Ref: d051568176
Ref: cblas c cblas_zscal1568278
Ref: d061568278
Ref: cblas c cblas_csscal1568380
Ref: d071568380
Ref: cblas c cblas_zdscal1568482
Ref: d081568482
Node: Level 2<2>1568585
Ref: cblas level-21568688
Ref: d091568688
Ref: cblas c cblas_sgemv1568715
Ref: d0a1568715
Ref: cblas c cblas_sgbmv1568998
Ref: d0b1568998
Ref: cblas c cblas_strmv1569319
Ref: d0c1569319
Ref: cblas c cblas_stbmv1569575
Ref: d0d1569575
Ref: cblas c cblas_stpmv1569844
Ref: d0e1569844
Ref: cblas c cblas_strsv1570086
Ref: d0f1570086
Ref: cblas c cblas_stbsv1570342
Ref: d101570342
Ref: cblas c cblas_stpsv1570611
Ref: d111570611
Ref: cblas c cblas_dgemv1570853
Ref: d121570853
Ref: cblas c cblas_dgbmv1571151
Ref: d131571151
Ref: cblas c cblas_dtrmv1571477
Ref: d141571477
Ref: cblas c cblas_dtbmv1571735
Ref: d151571735
Ref: cblas c cblas_dtpmv1572006
Ref: d161572006
Ref: cblas c cblas_dtrsv1572250
Ref: d171572250
Ref: cblas c cblas_dtbsv1572508
Ref: d181572508
Ref: cblas c cblas_dtpsv1572779
Ref: d191572779
Ref: cblas c cblas_cgemv1573023
Ref: d1a1573023
Ref: cblas c cblas_cgbmv1573305
Ref: d1b1573305
Ref: cblas c cblas_ctrmv1573625
Ref: d1c1573625
Ref: cblas c cblas_ctbmv1573879
Ref: d1d1573879
Ref: cblas c cblas_ctpmv1574146
Ref: d1e1574146
Ref: cblas c cblas_ctrsv1574386
Ref: d1f1574386
Ref: cblas c cblas_ctbsv1574640
Ref: d201574640
Ref: cblas c cblas_ctpsv1574907
Ref: d211574907
Ref: cblas c cblas_zgemv1575147
Ref: d221575147
Ref: cblas c cblas_zgbmv1575429
Ref: d231575429
Ref: cblas c cblas_ztrmv1575749
Ref: d241575749
Ref: cblas c cblas_ztbmv1576003
Ref: d251576003
Ref: cblas c cblas_ztpmv1576270
Ref: d261576270
Ref: cblas c cblas_ztrsv1576510
Ref: d271576510
Ref: cblas c cblas_ztbsv1576764
Ref: d281576764
Ref: cblas c cblas_ztpsv1577031
Ref: d291577031
Ref: cblas c cblas_ssymv1577271
Ref: d2a1577271
Ref: cblas c cblas_ssbmv1577534
Ref: d2b1577534
Ref: cblas c cblas_sspmv1577810
Ref: d2c1577810
Ref: cblas c cblas_sger1578059
Ref: d2d1578059
Ref: cblas c cblas_ssyr1578288
Ref: d2e1578288
Ref: cblas c cblas_sspr1578489
Ref: d2f1578489
Ref: cblas c cblas_ssyr21578676
Ref: d301578676
Ref: cblas c cblas_sspr21578921
Ref: d311578921
Ref: cblas c cblas_dsymv1579151
Ref: d321579151
Ref: cblas c cblas_dsbmv1579419
Ref: d331579419
Ref: cblas c cblas_dspmv1579700
Ref: d341579700
Ref: cblas c cblas_dger1579954
Ref: d351579954
Ref: cblas c cblas_dsyr1580187
Ref: d361580187
Ref: cblas c cblas_dspr1580391
Ref: d371580391
Ref: cblas c cblas_dsyr21580581
Ref: d381580581
Ref: cblas c cblas_dspr21580830
Ref: d391580830
Ref: cblas c cblas_chemv1581064
Ref: d3a1581064
Ref: cblas c cblas_chbmv1581326
Ref: d3b1581326
Ref: cblas c cblas_chpmv1581601
Ref: d3c1581601
Ref: cblas c cblas_cgeru1581849
Ref: d3d1581849
Ref: cblas c cblas_cgerc1582077
Ref: d3e1582077
Ref: cblas c cblas_cher1582305
Ref: d3f1582305
Ref: cblas c cblas_chpr1582504
Ref: d401582504
Ref: cblas c cblas_cher21582688
Ref: d411582688
Ref: cblas c cblas_chpr21582931
Ref: d421582931
Ref: cblas c cblas_zhemv1583160
Ref: d431583160
Ref: cblas c cblas_zhbmv1583422
Ref: d441583422
Ref: cblas c cblas_zhpmv1583697
Ref: d451583697
Ref: cblas c cblas_zgeru1583945
Ref: d461583945
Ref: cblas c cblas_zgerc1584173
Ref: d471584173
Ref: cblas c cblas_zher1584401
Ref: d481584401
Ref: cblas c cblas_zhpr1584601
Ref: d491584601
Ref: cblas c cblas_zher21584786
Ref: d4a1584786
Ref: cblas c cblas_zhpr21585029
Ref: d4b1585029
Node: Level 3<2>1585258
Ref: cblas level-31585363
Ref: d4c1585363
Ref: cblas c cblas_sgemm1585390
Ref: d4d1585390
Ref: cblas c cblas_ssymm1585729
Ref: d4e1585729
Ref: cblas c cblas_ssyrk1586041
Ref: d4f1586041
Ref: cblas c cblas_ssyr2k1586317
Ref: d501586317
Ref: cblas c cblas_strmm1586636
Ref: d511586636
Ref: cblas c cblas_strsm1586961
Ref: d521586961
Ref: cblas c cblas_dgemm1587286
Ref: d531587286
Ref: cblas c cblas_dsymm1587630
Ref: d541587630
Ref: cblas c cblas_dsyrk1587947
Ref: d551587947
Ref: cblas c cblas_dsyr2k1588227
Ref: d561588227
Ref: cblas c cblas_dtrmm1588551
Ref: d571588551
Ref: cblas c cblas_dtrsm1588879
Ref: d581588879
Ref: cblas c cblas_cgemm1589207
Ref: d591589207
Ref: cblas c cblas_csymm1589545
Ref: d5a1589545
Ref: cblas c cblas_csyrk1589856
Ref: d5b1589856
Ref: cblas c cblas_csyr2k1590132
Ref: d5c1590132
Ref: cblas c cblas_ctrmm1590450
Ref: d5d1590450
Ref: cblas c cblas_ctrsm1590774
Ref: d5e1590774
Ref: cblas c cblas_zgemm1591098
Ref: d5f1591098
Ref: cblas c cblas_zsymm1591436
Ref: d601591436
Ref: cblas c cblas_zsyrk1591747
Ref: d611591747
Ref: cblas c cblas_zsyr2k1592023
Ref: d621592023
Ref: cblas c cblas_ztrmm1592341
Ref: d631592341
Ref: cblas c cblas_ztrsm1592665
Ref: d641592665
Ref: cblas c cblas_chemm1592989
Ref: d651592989
Ref: cblas c cblas_cherk1593300
Ref: d661593300
Ref: cblas c cblas_cher2k1593574
Ref: d671593574
Ref: cblas c cblas_zhemm1593891
Ref: d681593891
Ref: cblas c cblas_zherk1594202
Ref: d691594202
Ref: cblas c cblas_zher2k1594478
Ref: d6a1594478
Ref: cblas c cblas_xerbla1594796
Ref: d6b1594796
Node: Examples<37>1594890
Ref: cblas examples1594976
Ref: d6c1594976
Node: GNU General Public License1596441
Ref: gpl doc1596573
Ref: d6d1596573
Ref: gpl gnu-general-public-license1596573
Ref: d6e1596573
Node: GNU Free Documentation License1634547
Ref: fdl doc1634667
Ref: d6f1634667
Ref: fdl gnu-free-documentation-license1634667
Ref: d701634667
Node: Index1659570

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