Technical Report No. 2014-05 : Exact Multivariate Integration on Simplices : An Explanation of The Lasserre-Avrachenkov Theorem (AU-Cas-Mathstats)

Because the traditional method for evaluating integrals over higher dimensional simplices can be computationally challenging, Lasserre and Avrachenkov established an equation for evaluating integrals of symmetric multilinear forms over simplices. Before an integral can be evaluated in this manner the starting homogeneous polynomial must be expressed as a symmetric multilinear form, by way of a polarization identity. In this paper, the Lasserre-Avrachenkov method for evaluating integrals over simplices is explained and explored, beginning with a homogenous polynomial and a simplex, and ending with an exact value. This method can be used in computer programs that provide an efficient method for precisely evaluating integrals over simplices in higher dimensions.