Some fifth degree integration formulas for symmetric regions II

Abstract

Here Sn is a region in n-dimensional Euclidean space (n _ 2), the Ai are constants and the (vil, ***, vin) are points in the space. We discuss a general class of 5th degree formulas for a class of regions Sn which includes the n-dimensional cube and sphere; a precise description of the regions to which our discussion applies will be given in the next section. Each formula in the class contains no more than N = 2n(n + 1) points and has all positive coefficients Ai. For the n-sphere we give constants in four useful formulas which contain 2 n(n + 1), 2"n + 1, 2n+1 - 1 and 2n + 2n points. The 2n(n + 1) point formula has all coefficients equal. The 2n + 2n point formula has fewer points than any other known 5th degree formula in which all coefficients are positive (for the n-sphere, n > 4). The corresponding formulas for the n-cube are not as useful since, for most values of n, they have some points which lie outside the cube. Previously, Hammer and Stroud [3] have given formulas of 5th degree for the n-cube and n-sphere which use only 2n2 + 1 points. Those formulas, however, have the undesirable feature that, for the n-sphere (n > 5), some of the coefficients Ai are negative. The only other known general class of 5th degree formulas for the n-sphere, for arbitrary n, are the spherical product formulas; these contain 3n - 3n-1 + 1 points and have all positive coefficients. (Spherical product formulas were first described, for n = 2, 3, by Peirce [6, 7], and for arbitrary n independently by Hetherington [4] and lMlysovskih [5]; an alternative description is given by