Cubature formulae for the gaussian weight. Some old and new rules.

Abstract

In this paper we review some of the main known facts about cubature rules to approximate integrals over domains in R-n, in particular with respect to the Gaussian weight w(x) = e(-xTx); where x = (x(1); ... ; x(n)) is an element of R-n. Some new rules are also presented. Taking into account the well-known issue of the "curse of dimensionality", our aim is at providing rules with a certain degree of algebraic precision and a reasonably small number of nodes as well as an acceptable stability. We think that the methods used to construct these new rules are of further applicability in the field of cubature formulas. The efficiency of new and old rules are compared by means of several numerical experiments.