A Cardinal Function Algorithm for Computing Multivariate Quadrature Points

We present a new algorithm for numerically computing quadrature formulas for arbi-trary domains which exactly integrate a given polynomial space. An effective method for constructingquadrature formulas has been to numerically solve a nonlinear set of equations for the quadraturepoints and their associated weights. Symmetry conditions are often used to reduce the number ofequations and unknowns. Our algorithm instead relies on the construction of cardinal functions andthus requires that the number of quadrature points N be equal to the dimension of a prescribedlower dimensional polynomial space. The cardinal functions allow us to treat the quadrature weightsas dependent variables and remove them, as well as an equivalent number of equations, from thenumerical optimization procedure. We give results for the triangle, where for all degree d ≤ 25, wefind quadrature formulas of this form which have positive weights and contain no points outside thetriangle. Seven of these quadrature formulas improve on previously known results.