Integration rules of hypercubic symmetry over a certain spherically symmetric space

Abstract

A theory of integration rules suitable for integration over a hypercube and having hypercubic symmetry has recently been published. In this paper it is found that, with minor modification, this theory may be directly applied to obtain integration rules of hypercubic symmetry suitable for integration over a complete n n -dimensional space with the weight function exp ⁡ ( − x 1 2 − x 2 2 ⋅ ⋅ ⋅ − x n 2 ) \exp ( - {x_1}^2 - {x_2}^2 \cdot \cdot \cdot - {x_n}^2) . As in the case of integration over hypercubes, an n n -dimensional rule of degree 2 t + 1 2t + 1 may be constructed requiring a number of function evaluations of order 2 t n t / t ! {2^t}{n^t}/t! , only.