A Version of Simpson's Rule for Multiple Integrals
Alan Horwitz
TL;DR
This work generalizes Simpson's Rule to multidimensional cubature over polygonal domains by forming a weighted combination $L_{\lambda}=\lambda M(f)+(1-\lambda)T(f)$ of a center-based and a boundary-vertex-based estimator, with $\lambda$ chosen to maximize polynomial exactness. It provides concrete rules for the $n$-simplex (CR1), the unit $n$-cube (CR3), and the unit disc (CR6), and explores higher-degree exactness limitations on polygons with various boundary point configurations (e.g., CR4 for a square, CR5 for a trapezoid). The paper connects these cubature rules to interpolation by showing how certain $L_{\lambda}$ rules arise from integrating suitable second-degree interpolants, and employs Grobner-basis techniques to solve the resulting systems. Overall, it presents a unified framework to derive Simpson-type cubature rules across common domains, while highlighting the trade-offs between node placement, weight signs, and degree of exactness. The results offer practical, explicitly computable formulas with positive weights for many standard shapes, contributing a methodology for constructing and evaluating multidimensional quadrature schemes.
Abstract
Let M(f) denote the Midpoint Rule and T(f) the Trapezoidal Rule for estimating integral_a^b f(x) dx. Then Simpson's Rule = tM(f) + (1-t)T(f), where t = 2/3. We generalize Simpson's Rule to multiple integrals as follows. Let D be some polygonal region in R^n, let P_0,...,P_m denote the vertices of D, and let P_(m+1) = center of mass of D. Define the linear functionals M(f) = Vol(D)f(P_(m+1)), which generalizes the Midpoint Rule, and T(f) = Vol(D)(1/(m+1))sum(f(P_j), j = 0,...,m), which generalizes the Trapezoidal Rule. Finally, our generalization of Simpson's Rule is given by the cubature rule(CR) L_t = tM(f) + (1-t)T(f), for t in [0,1]. We choose t, depending on D, so that L_t is exact for polynomials of as large a degree as possible. In particular we derive CRs for the n simplex and unit n cube.