High-order asymptotic expansions of Gaussian quadrature rules with classical and generalized weight functions

Gaussian quadrature rules are a classical tool for the numerical approximation of inte-grals with smooth integrands and positive weight functions. We derive and explicitly 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. Explicit expressions for these cases are included in the appendix. 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.& COPY; 2023 Elsevier B.V. All rights reserved.

Automated summary

Gaussian quadrature rules are effective in numerically approximating integrals with smooth integrands and positive weight functions. In this study, we derive and list asymptotic expressions for the points and weights of Gaussian quadrature rules for three general classes of positive weight functions. These rules cover various situations including classical orthogonal polynomials. Additionally, we provide experimental evidence for the precision achieved by these expressions and present an algorithm to compute the expansions for classical and generalized cases.