A version of Simpson’s rule for multiple integrals

Let M (f) denote the midpoint rule and T (f) the trapezoidal rule for estimating ∫ b a f(x) dx. Then Simpson’s rule = M (f) + (1 − )T (f), where = 3 . We generalize Simpson’s rule to multiple integrals as follows. Let Dn be some polygonal region in R; let P0; : : : ; Pm denote the vertices of Dn, and let Pm+1 equal the center of mass of Dn. De4ne the linear functionals M (f) =Vol(Dn)f(Pn+1), which generalizes the midpoint rule, and T (f) =Vol(Dn)([1=(m+ 1)] ∑m j=0 f(Pj)), which generalizes the trapezoidal rule. Finally, our generalization of Simpson’s rule is given by the cubature rule (CR) L = M (f)+ (1− )T (f), for 4xed ; 06 61. We choose , depending on Dn, so that L is exact for polynomials of as large a degree as possible. In particular, we derive CRs for the n simplex and unit n cube. We also use points Qj ∈ @(Dn), other than the vertices Pj , to generate T (f). 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 Qj’s. c © 2001 Elsevier Science B.V. All rights reserved.