Limits on the number of function evaluations required by certain high-dimensional integration rules of hypercubic symmetry

Abstract

Abstract. We consider an n-dimensional integration rule Rt n of degree 2t -1 and of hypercubic symmetry. We derive theorems which set a lower bound in terms of n and t on the number of function evaluations such a rule requires. These results apply to spaces of integration which have hypercubic symmetry. In certain cases this bound is very close to the number of points required by a known rule.

Symmetric integration rules for hypercubes. III. Construction of integration rules using null rules

“…(This is a direct verification of Theorem 3.4 of Lyness [3].) The precise form of this additional term is obtained if we include in *S£+72) a basic rule (R(/3)*((R(a))'~3 but not (R(a)'~2.…”

Cubature formulas for symmetric measures in higher dimensions with few points

“…Fully symmetric cubature formulas were developed by Lyness (1965aLyness ( , 1965b, Mc-Namee and Stenger (1967), Genz (1986), Cools and Haegemans (1994), Capstick and Keister (1996), Genz and Keister (1996) and other authors. The best results with respect to polynomial exactness are obtained by Genz (1986) and Genz and Keister (1996).…”

Efficient cubature rules

“…This point set is fully symmetric; i.e., closed under all coordinate permutations and sign changes. However, relaxing this symmetry requirement can allow us to find formulas with fewer points [9,10]. For example (as shown in Figure 5), in two dimensions, there is a formula of degree five with points at the vertices of a regular hexagon [11, formula V], which is closed under sign permutations but not coordinate permutations.…”