Fast and rigorous arbitrary-precision computation of Gauss-Legendre quadrature nodes and weights
This work provides a practical and rigorous method for generating Gauss-Legendre quadrature rules, benefiting numerical analysts and scientific computing practitioners who require high-precision integration.
The authors present a rigorous arbitrary-precision algorithm for computing Gauss-Legendre quadrature nodes and weights, achieving order-of-magnitude speedups over previous implementations for high degree and precision (100 to 100,000 bits).
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.