Rational Gauss Quadrature

The existence of (standard) Gauss quadrature rules with respect to a nonnegative measure dμ with support on the real axis easily can be shown with the aid of orthogonal polynomials with respect to this measure. Efficient algorithms for computing the nodes and weights of an n-point Gauss rule use the n × n symmetric tridiagonal matrix determined by the recursion coefficients for the first n orthonormal polynomials. Many rational functions that are orthogonal with respect to the measure dμ and have real or complex conjugate poles also satisfy a short recursion relations. This paper describes how banded matrices determined by the recursion coefficients for these orthonormal rational functions can be used to efficiently compute the nodes and weights of rational Gauss quadrature rules.