A new recursive formula for integration of polynomial over simplex

Abstract

The simplex integration is a convenient method for the integration over complex domains. It automatically represents the ordinary integral over an arbitrary polyhedron as the algebraic sum of integrals over the oriented simplexes. An existing recursive formula for the integration of monomials over simplex, which was deduced based on special operations of matrices and was presented by the first author of this paper, has significant advantages: not only the computation amount is small, but also the integrals of all the lower order monomials are obtained while computing the integral of the highest order monomial. The extension to a polynomial can be obtained by the linearity of integrals. In this paper, the derivation of the existing recursive formula is detailed. A new recursive formula is emphatically proposed to simplify the existing recursive formula and further increase the speed of computation. The code for computer implementation is also presented. Examples show that the accuracy and efficiency of the recursive formula are higher than numerical integration.