A Cardinal Function Algorithm for Computing Multivariate Quadrature Points

We present a new algorithm for numerically computing quadrature formulas for arbitrary domains which exactly integrate a given polynomial space. An effective method for constructing quadrature formulas has been to numerically solve a nonlinear set of equations for the quadrature points and their associated weights. Symmetry conditions are often used to reduce the number of equations and unknowns. Our algorithm instead relies on the construction of cardinal functions and thus requires that the number of quadrature points $N$ be equal to the dimension of a prescribed lower dimensional polynomial space. The cardinal functions allow us to treat the quadrature weights as dependent variables and remove them, as well as an equivalent number of equations, from the numerical optimization procedure. We give results for the triangle, where for all degrees $d \le 25$, we find quadrature formulas of this form which have positive weights and contain no points outside the triangle. Seven of these quadrature formulas improve on previously known results.