Generalized Gaussian Quadrature Rules for Systems of Arbitrary Functions

In an earlier work, a far-reaching generalization of the classical Gaussian quadrature rules is introduced, replacing the polynomials with a wide class of functions. While the rules of in that report possess most of the desirable properties of the classical Gaussian integration formulae (positivity of the weights, etc.), it is not clear from siad research how much quadrature rules can obtained numerically. In this paper, we present a numerical scheme for the generation of such generalized Gaussian quadratures. The algorithm is applicable to a variety of functions, including smooth functions as well as functions with end-point singularities. The performance of the algorithm is demonstrated with several numerical examples