Fully symmetric interpolatory rules for multiple integrals over infinite regions with Gaussian weight (Q1919958)

This paper deals with the construction of numerical methods for the estimation of integrals of the form \[ I(f) = {1\over(2\pi)^{n/2}} \int^\infty_{-\infty} \int^\infty_{-\infty} \cdots \int^\infty_{-\infty}e^{-x^Tx/2} f(x)dx_1,dx_2 \cdots dx_n \] with \(x=(x_1,x_2,\dots,x_n)^T\). Fully symmetric interpolatory integration rules are constructed for multidimensional integrals over infinite integration regions with a Gaussian weight function. The points for these rules are determined by successive extensions of the one-dimensional three-point Gauss-Hermite rule. The new rules are shown to be efficient and only moderately unstable.