Arbitrary-order asymptotic expansions of Gaussian quadrature rules with classical and generalised weight functions

TL;DR

This paper derives explicit asymptotic expansions for Gaussian quadrature points and weights across various weight functions, enhancing understanding and computational efficiency for numerical integration.

Contribution

It provides new asymptotic formulas for Gaussian quadrature rules with general weight functions, including classical and generalized cases, and offers algorithms for their computation.

Findings

Asymptotic expressions achieve high precision for large number of points.

Formulas include classical orthogonal polynomials as special cases.

Algorithms enable computation of many terms in the expansions.

Abstract

Gaussian quadrature rules are a classical tool for the numerical approximation of integrals with smooth integrands and positive weight functions. We derive and expicitly list asymptotic expressions for the points and weights of Gaussian quadrature rules for three general classes of positive weight functions: analytic functions on a bounded interval with algebraic singularities at the endpoints, analytic weight functions on the halfline with exponential decay at infinity and an algebraic singularity at the finite endpoint, and analytic functions on the real line with exponential decay in both directions at infinity. The results include the Gaussian rules of classical orthogonal polynomials (Legendre, Jacobi, Laguerre and Hermite) as special cases. We present experiments indicating the range of the number of points at which these expressions achieve high precision. We provide an algorithm that can compute arbitrarily many terms in these expansions for the classical cases, and many though not all terms for the generalized cases.