Section: New Results

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

Participant : Fredrik Johansson.

In [26], F. Johansson and M. Mezzarobba 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. The 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. The 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. Rigorous enclosures of the Gauss-Legendre nodes and weights are then computed using the interval Newton method. The work provides 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.