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
2 PEQUAN - Performance et Qualité des Algorithmes Numériques

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.

Mots clés: Gauss-Legendre quadrature; interval arithmetic AMS subject classifications 65Y99; arbitrary-precision arithmetic; interval arithmetic; Legendre polynomials; 65G99; 33C45; Gauss–Legendre quadrature

Domaines: Analyse numérique [cs.NA]; Logiciel mathématique [cs.MS]

Fredrik Johansson, Marc Mezzarobba. Fast and rigorous arbitrary-precision computation of Gauss-Legendre quadrature nodes and weights. SIAM Journal on Scientific Computing, 2018, 40 (6), pp.C726-C747. ⟨10.1137/18M1170133⟩. ⟨hal-01705612v2⟩