Fast, reliable and unrestricted iterative computation of Gauss--Hermite and Gauss--Laguerre quadratures

TL;DR

This paper presents fast, reliable iterative algorithms for computing Gauss--Hermite and Gauss--Laguerre quadrature nodes and weights with guaranteed convergence and high precision, suitable for high-degree and high-accuracy applications.

Contribution

The authors introduce high-order, convergent iterative methods for quadrature rule computation that are effective for a wide range of parameters and degrees, enabling extremely high-precision results.

Findings

Achieved full relative precision for Hermite nodes at any degree.

Demonstrated high-accuracy computations up to 1000 digits.

Provided algorithms with almost unrestricted parameter validity.

Abstract

Methods for the computation of classical Gaussian quadrature rules are described which are effective both for small and large degree. These methods are reliable because the iterative computation of the nodes has guaranteed convergence, and they are fast due to their fourth-order convergence and its asymptotic exactness for an appropriate selection of the variables. For Gauss--Hermite and Gauss--Laguerre quadratures, local Taylor series can be used for computing efficiently the orthogonal polynomials involved, with exact initial values for the Hermite case and first values computed with a continued fraction for the Laguerre case. The resulting algorithms have almost unrestricted validity with respect to the parameters. Full relative precision is reached for the Hermite nodes, without any accuracy loss and for any degree, and a mild accuracy loss occurs for the Hermite and Laguerre weights as well as for the Laguerre nodes. These fast methods are exclusively based on convergent processes, which, together with the high order of convergence of the underlying iterative method, makes them particularly useful for high accuracy computations. We show examples of very high accuracy computations (of up to $1000$ digits of accuracy).