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

Fredrik Johansson 1 Marc Mezzarobba 2
1 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
2 PEQUAN - Performance et Qualité des Algorithmes Numériques
LIP6 - Laboratoire d'Informatique de Paris 6
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.
Type de document :
Pré-publication, Document de travail
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Contributeur : Fredrik Johansson <>
Soumis le : mardi 16 octobre 2018 - 17:38:50
Dernière modification le : vendredi 16 novembre 2018 - 02:20:22


  • HAL Id : hal-01705612, version 2
  • ARXIV : 1802.03948


Fredrik Johansson, Marc Mezzarobba. Fast and rigorous arbitrary-precision computation of Gauss-Legendre quadrature nodes and weights. 2018. 〈hal-01705612v2〉



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