Computational results of an O(n) volume algorithm

Recently an O^∗(n⁴) volume algorithm has been presented for convex bodies by Lovász and Vempala, where n is the number of dimensions of the convex body. Essentially the algorithm is a series of Monte Carlo integrations. In this paper we describe a computer implementation of the volume algorithm, where we improved the computational aspects of the original algorithm by adding variance decreasing modifications: a stratified sampling strategy, double point integration and orthonormalised estimators. Formulas and methodology were developed so that the errors in each phase of the algorithm can be controlled. Some computational results for convex bodies in dimensions ranging from 2 to 10 are presented as well.