volume {volesti} | R Documentation |
For the volume approximation can be used two algorithms. Either SequenceOfBalls or CoolingGaussian. A H-polytope with m facets is described by a m\times d matrix A and a m-dimensional vector b, s.t.: Ax≤q b. A V-polytope is defined as the convex hull of m d-dimensional points which correspond to the vertices of P. A zonotope is desrcibed by the Minkowski sum of m d-dimensional segments.
volume(P, walk_step = NULL, error = NULL, InnerBall = NULL, Algo = NULL, WalkType = NULL, rounding = NULL, Parameters = NULL)
P |
A convex polytope. It is an object from class (a) Hpolytope or (b) Vpolytope or (c) Zonotope. |
walk_step |
Optional. The number of the steps for the random walk. The default value is \lfloor 10 + d/10\rfloor for SequenceOfBalls and 1 for CoolingGaussian. |
error |
Optional. Declare the upper bound for the approximation error. The default value is 1 for SequenceOfBalls and 0.1 for CoolingGaussian. |
InnerBall |
Optional. A d+1 vector that contains an inner ball. The first d coordinates corresponds to the center and the last one to the radius of the ball. If it is not given then for H-polytopes the Chebychev ball is computed, for V-polytopes d+1 vertices are picked randomly and the Chebychev ball of the defined simplex is computed. For a zonotope that is defined by the Minkowski sum of m segments we compute the maximal r s.t.: re_i\in Z for all i=1,… ,d, then the ball centered at the origin with radius r/√{d} is an inscribed ball. |
Algo |
Optional. A string that declares which algorithm to use: a) |
WalkType |
Optional. A string that declares the random walk method: a) |
rounding |
Optional. A boolean parameter for rounding. The default value is |
Parameters |
Optional. A list for the parameters of the algorithms:
|
The approximation of the volume of a convex polytope.
I.Z.Emiris and V. Fisikopoulos, “Practical polytope volume approximation,” ACM Trans. Math. Soft., 2014.,
B. Cousins and S. Vempala, “A practical volume algorithm,” Springer-Verlag Berlin Heidelberg and The Mathematical Programming Society, 2015.
# calling SOB algorithm for a H-polytope (2d unit simplex) P = GenSimplex(2,'H') vol = volume(P) # calling CG algorithm for a V-polytope (3d simplex) P = GenSimplex(2,'V') vol = volume(P, Algo = "CG") # calling CG algorithm for a 2-dimensional zonotope defined as the Minkowski sum of 4 segments Z = GenZonotope(2, 4) vol = volume(Z, WalkType = "RDHR", walk_step = 5)