Fast and rigorous arbitrary-precision computation of Gauss-Legendre quadrature nodes and weights
Language
en
Article de revue
This item was published in
SIAM Journal on Scientific Computing. 2018, vol. 40, n° 6, p. C726-C747
Society for Industrial and Applied Mathematics
English 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 ...Read more >
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.Read less <
English Keywords
Gauss-Legendre quadrature
interval arithmetic AMS subject classifications 65Y99
arbitrary-precision arithmetic
interval arithmetic
Legendre polynomials
65G99
33C45
Gauss–Legendre quadrature
ANR Project
Approximation rapide et fiable - ANR-14-CE25-0018
Origin
Hal imported