  Based on a novel point of view on 1-dimensional Gaussian quadrature, we
present a new approach to the computation of d-dimensional cubature formulae.
It is well known that the nodes of 1-dimensional Gaussian quadrature can be
computed as eigenvalues of the so-called Jacobi matrix. The d-dimensional
analog is that cubature nodes can be obtained from the eigenvalues of certain
mutually commuting matrices. These are obtained by extending (adding rows and
columns to) certain noncommuting matrices A_1,...,A_d, related to the
coordinate operators x_1,...,x_d, in R^d. We prove a correspondence between
cubature formulae and "commuting extensions" of A_1,...,A_d, satisfying a
compatibility condition which, in appropriate coordinates, constrains certain
blocks in the extended matrices to be zero. Thus the problem of finding
cubature formulae can be transformed to the problem of computing (and then
simultaneously diagonalizing) commuting extensions. We give a general
discussion of existence and of the expected size of commuting extensions and
describe our attempts at computing them, as well as examples of cubature
formulae obtained using the new approach.
