Accurate and efficient computation of classical Gaussian quadrature rules

We are developing methods for the fast and accurate computation of classical Gaussian quadrature rules (Hermite, Laguerre and Jacobi). A recent talk on the subject can be found here

We consider two complementary approaches:

1. Asymptotic approximations for moderately large degrees (n>100): these methods produce accurate explicit expressions for the nodes and weights of the quadratures (with accuracy close to IEEE double precision).
 
         Aymptotic metods are described in:

         A. Asymptotic approximations to the nodes and weights of Gauss-Hermite and Gauss-Laguerre quadratures.
                 A. Gil, J. Segura, N. M. Temme. Stud. Appl. Math.,140(3) (2018) 298-332 arxiv  doi: 10.1111/sapm.1220 Clck here for an online only-read version
         B. Non-iterative computation of  Gauss-Jacobi  quadrature. 
              A. Gil, J. Segura, N. M. Temme. SIAM J. Sci. Comput. 41(1) (2019) A668-A693.  doi: 10.1137/18M1179006  arxiv.
              For a sneak preview of some of the coefficients in the expansions click here

2. Globally convergent iterative methods: these methods can be used for any degree no matter how small or large, and they are based on convergent processes. No aymptotics are used. The methods employ a globally convergent fourth order method for solving non-linear equations, with the associated orthogonal polynomials computed with exact or convergent methods (recurrence for small degree, continued fractions, Taylor series). We have developed algoritms for all the three classical cases: Hermite, Laguerre and Gauss-Jacobi:
   
         C. Fast, reliable and unrestricted iterative computation of Gauss--Hermite and Gauss--Laguerre quadratures
             A. Gil, J. Segura, N. M. Temme. Numer. Math. 143(3) (2019) 649-682  arxiv doi: 10.1007/s00211-019-01066-2 Click here for an online only-read version

         D. Fast and reliable high accuracy computation of Gauss-Jacobi quadrature. Numer. Algo. (accepted) arxiv

The final goal is to produce fast and accurate algorithms for the classical quadratures. The asymptotic methods produce acurate and very fast results for moderately large degrees and they are suited for fast fixed precision computations. On the other hand, the iterative methods are ideal for high precision computations; they can be used almost without restriction on the parameters and they display fast (fourth order) convergence.

Software

We provide some examples of software for the iterative computation of Gauss quadratures:

          I. A Maple worksheet for the computation of Gauss-Hermite quadratures (it can even be used for 1000 digits accuracy or better).

        II. Fortran 90 algorithms for Gauss-Hermite in double and quadruple precision.

       III. Fortran 90 algorithms for Gauss-Laguerre in double and quadruple precision.

       IV. A Maple worksheet for the computation of Gauss-Hermite quadratures (high accuracy!)  NEW

        V. MATLAB program for Gauss-Jacobi quadrature  NEW

        
DISCLAIMER: this software may still contain some bugs, and it has not been optimized. If you find any bug or you have any suggestion, please contact me: javier . segura @ unican . es

This software is distributed under the MIT license
Copyright 2019 A. Gil, J. Segura, N. M. Temme