A Version of Simpson's Rule for Multiple Integrals

Let () denote the Midpoint Rule and () the Trapezoidal Rule for estimating . Then Simpson's Rule = λ() + (1 – λ)(), where . We generalize Simpson's Rule to multiple integrals as follows. Let be some polygonal region in , let , …, denote the vertices of , and let equal the center of mass of D. Define the linear functionals , which generalizes the Midpoint Rule, and , which generalizes the Trapezoidal Rule. Finally, our generalization of Simpson's Rule is given by the cubature rule(CR) = λ() + (1 — λ)(), for fixed λ, 0 ≤ λ ≤ 1. We choose X, depending on , so that is exact for polynomials of as large a degree as possible. In particular we derive CRs for the simplex and unit cube. We also use points , other than the vertices , to generate (). This sometimes leads to better CRs for certain regions-in particular for quadrilaterals in the plane. We use Grobner bases to solve the system of equations which yield the coordinates of the s