A version of Simpsonʹs rule for multiple integrals

Let M(f) denote the midpoint rule and T(f) the trapezoidal rule for estimating ∫abf(x) dx. Then Simpsonʹs rule =λM(f)+(1−λ)T(f), where λ=23. We generalize Simpsonʹs rule to multiple integrals as follows. Let Dn be some polygonal region in Rn, 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 Simpsonʹs 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 Qjʹs.