Generalized Gaussian quadrature rules for systems of arbitrary functions

A numerical algorithm is presented for the construction of generalized Gaussian quadrature rules, originally introduced by S. Karlin and W. Studden over three decades ago. The quadrature rules to be discussed possess most of the desirable properties of the classical Gaussian integration formulae, such as positivity of the weights, rapid convergence, mathematical elegance, etc. The algorithm is applicable to a wide class of functions, including smooth functions (not necessarily polynomials), as well as functions with end-point singularities, such as those encountered in the solution, of integral equations, complex analysis, potential theory, and several other areas. The performance of the algorithm is illustrated with several numerical examples.