Gaussian Quadrature Formulae for Arbitrary Positive Measures

Abstract

We present computational methods and subroutines to compute Gaussian quadrature integration formulas for arbitrary positive measures. For expensive integrands that can be factored into well-known forms, Gaussian quadrature schemes allow for efficient evaluation of high-accuracy and -precision numerical integrals, especially compared to general ad hoc schemes. In addition, for certain well-known density measures (the normal, gamma, log-normal, Student’s t, inversegamma, beta, and Fisher’s F) we present exact formulae for computing the respective quadrature scheme.