Fast and rigorous arbitrary-precision computation of Gauss-Legendre quadrature nodes and weights

TL;DR

This paper presents a fast, rigorous method for arbitrary-precision computation of Gauss-Legendre quadrature nodes and weights, enabling high-accuracy numerical integration with verified error bounds.

Contribution

It introduces a novel algorithm combining hypergeometric series expansions, fixed-point recurrence, and interval Newton method for efficient, verified high-precision quadrature rule generation.

Findings

Achieves order-of-magnitude speedups over previous methods

Provides rigorous error bounds for all computational steps

Enables high-precision numerical integration up to 100,000 bits

Abstract

We describe a strategy for rigorous arbitrary-precision evaluation of Legendre polynomials on the unit interval and its application in the generation of Gauss-Legendre quadrature rules. Our focus is on making the evaluation practical for a wide range of realistic parameters, corresponding to the requirements of numerical integration to an accuracy of about 100 to 100 000 bits. Our algorithm combines the summation by rectangular splitting of several types of expansions in terms of hypergeometric series with a fixed-point implementation of Bonnet's three-term recurrence relation. We then compute rigorous enclosures of the Gauss-Legendre nodes and weights using the interval Newton method. We provide rigorous error bounds for all steps of the algorithm. The approach is validated by an implementation in the Arb library, which achieves order-of-magnitude speedups over previous code for computing Gauss-Legendre rules with simultaneous high degree and precision.