A Version of Simpson's Rule for Multiple Integrals

Abstract Let M(f) denote the midpoint rule and T(f) the trapezoidal rule for estimating ∫ a b f(x) d x . Then Simpson's rule =λM(f)+(1−λ)T(f), where λ= 2 3 . We generalize Simpson's rule to multiple integrals as follows. Let Dn be some polygonal region in R n , let P0,…,Pm denote the vertices of Dn, and let Pm+1 equal the center of mass of Dn. Define the linear functionals M(f)=Vol(Dn)f(Pn+1), which generalizes the midpoint rule, and T(f)=Vol(Dn)([1/(m+1)]∑j=0mf(Pj)), which generalizes the trapezoidal rule. Finally, our generalization of Simpson's rule is given by the cubature rule (CR) Lλ=λM(f)+(1−λ)T(f), for fixed λ, 0⩽λ⩽1 . We choose λ, depending on Dn, so that Lλ 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.