Introduction

This paper deals with the construction of numerical methods for the estimation of integrals in the form

$$I(f) = \int_{U_n}f({\bf z}) d{\bf\sigma},$$

where ${\bf z}= (z_1,z_2,\ldots,z_n)$, $U_n = \{{\bf z}\vert {\bf z}\in {\cal R}^n, z_1^2+z_2^2\cdots+z_n^2 = 1\}$, and where ${\bf\sigma}$ is an element of surface on $U_n$. This is an important problem in pure and applied science, which has been studied by various authors. The books by Stroud [7] and Mysovskikh [5] both contain a number of formulas, and the paper by Keast and Diaz [4] provides a general method for constructing fully symmetric rules. Recent work by Xu [9] provides some fully symmetric rules with explicit formulas for the rule weights.

The purpose of this paper is to show how to modify a method for construction of the numerical integration rules described by Sylvester [8], for integration over an $n-1$-dimensional simplex $T_{n-1}$. The modified $T_{n-1}$ rules can then be transformed to provide a family of rules for integration over $U_n$. Sylvester's integration rules for $T_{n-1}$ are interpolatory rules, with explicit formulas for the rule weights, so the transformed rules for $U_n$ also have explicit formulas for the weights. The resulting integration rules are new for the cases where the polynomial degree of precision is greater than 7. The rule construction method also allows the construction of two types of randomized rules, using methods previously described by the present author (Genz [2]) for randomized rules over ${\cal R}^n$ with Gaussian weight.