pcgsolve {cPCG} | R Documentation |
Preconditioned conjugate gradient method for solving system of linear equations Ax = b, where A is symmetric and positive definite, b is a column vector.
pcgsolve(A, b, preconditioner = "Jacobi", tol = 1e-6, maxIter = 1000)
A |
matrix, symmetric and positive definite. |
b |
vector, with same dimension as number of rows of A. |
preconditioner |
string, method for preconditioning: |
tol |
numeric, threshold for convergence, default is |
maxIter |
numeric, maximum iteration, default is |
When the condition number for A is large, the conjugate gradient (CG) method may fail to converge in a reasonable number of iterations. The Preconditioned Conjugate Gradient (PCG) Method applies a precondition matrix C and approaches the problem by solving:
{C}^{-1} A x = {C}^{-1} b
where the symmetric and positive-definite matrix C approximates A and {C}^{-1} A improves the condition number of A.
Common choices for the preconditioner include: Jacobi preconditioning, symmetric successive over-relaxation (SSOR), and incomplete Cholesky factorization [2].
Returns a vector representing solution x.
Jacobi
: The Jacobi preconditioner is the diagonal of the matrix A, with an assumption that all diagonal elements are non-zero.
SSOR
: The symmetric successive over-relaxation preconditioner, implemented as M = (D+L) D^{-1} (D+L)^T. [1]
ICC
: The incomplete Cholesky factorization preconditioner. [2]
Users need to check that input matrix A is symmetric and positive definite before applying the function.
[1] David Young. “Iterative methods for solving partial difference equations of elliptic type”. In: Transactions of the American Mathematical Society 76.1 (1954), pp. 92–111.
[2] David S Kershaw. “The incomplete Cholesky—conjugate gradient method for the iter- ative solution of systems of linear equations”. In: Journal of computational physics 26.1 (1978), pp. 43–65.
## Not run: test_A <- matrix(c(4,1,1,3), ncol = 2) test_b <- matrix(1:2, ncol = 1) pcgsolve(test_A, test_b, "ICC") ## End(Not run)