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

  • Amparo Gil
  • Javier SeguraEmail author
  • Nico M. Temme


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).

Mathematics Subject Classification

65D32 65H05 33C45 34C10 



The authors thank the anonymous referees for their constructive comments and suggestions. NMT thanks CWI for scientific support.


  1. 1.
    Bogaert, I.: Iteration-free computation of Gauss–Legendre quadrature nodes and weights. SIAM J. Sci. Comput. 36(3), A1008–A1026 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bogaert, I., Michiels, B., Fostier, J.: O(1) computation of Legendre polynomials and Gauss–Legendre nodes and weights for parallel computing. SIAM J. Sci. Comput. 34(3), C83–C101 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bremer, J.: On the numerical calculation of the roots of special functions satisfying second order ordinary differential equations. SIAM J. Sci. Comput. 39(1), A55–A82 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cash, J.: A note on the numerical solution of linear recurrence relations. Numer. Math. 34, 371–386 (1980). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Davis, P., Rabinowitz, P.: Abscissas and weights for Gaussian quadratures of high order. J. Res. Nat. Bur. Stand. 56, 35–37 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Deaño, A., Huybrechs, D., Opsomer, P.: Construction and implementation of asymptotic expansions for Jacobi-type orthogonal polynomials. Adv. Comput. Math. 42(4), 791–822 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Deaño, A., Gil, A., Segura, J.: New inequalities from classical Sturm theorems. J. Approx. Theory 131(2), 208–230 (2004). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Deaño, A., Segura, J.: Global Sturm inequalities for the real zeros of the solutions of the Gauss hypergeometric differential equation. J. Approx. Theory 148(1), 92–110 (2007). MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dimitrov, D.K., Nikolov, G.P.: Sharp bounds for the extreme zeros of classical orthogonal polynomials. J. Approx. Theory 162(10), 1793–1804 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gautschi, W.: Gauss–Radau formulae for Jacobi and Laguerre weight functions. Math. Comput. Simul. 54(4–5), 403–412 (2000). 1999 International Symposium on Computational Sciences, to honor John R. Rice (West Lafayette, IN)
  11. 11.
    Gautschi, W.: Generalized Gauss–Radau and Gauss–Lobatto formulae. BIT 44(4), 711–720 (2004). MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gil, A., Segura, J., Temme, N.M.: Numerical Methods for Special Functions. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2007).
  13. 13.
    Gil, A., Segura, J., Temme, N.M.: Recent software developments for special functions in the Santander–Amsterdam project. Sci. Comput. Program. 90A, 42–54 (2014)CrossRefGoogle Scholar
  14. 14.
    Gil, A., Segura, J., Temme, N.M.: Asymptotic approximations to the nodes and weights of Gauss–Hermite and Gauss–Laguerre quadratures. Stud. Appl. Math. 140(3), 298–332 (2018).
  15. 15.
    Gil, A., Segura, J., Temme, N.M.: Noniterative computation of Gauss–Jacobi quadrature. SIAM J. Sci. Comput. 41(1), A668–A693 (2019). MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Glaser, A., Liu, X., Rokhlin, V.: A fast algorithm for the calculation of the roots of special functions. SIAM J. Sci. Comput. 29(4), 1420–1438 (2007). MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Golub, G.H., Welsch, J.H.: Calculation of Gauss quadrature rules. Math. Comput. 23(1969), 221–230. addendum, ibid. 23(106, loose microfiche suppl), A1–A10 (1969)Google Scholar
  18. 18.
    Hale, N., Townsend, A.: Fast and accurate computation of Gauss–Legendre and Gauss–Jacobi quadrature nodes and weights. SIAM J. Sci. Comput. 35(2), A652–A674 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Huybrechs, D., Opsomer, P.: Construction and implementation of asymptotic expansions for Laguerre-type orthogonal polynomials. IMA J. Numer. Anal. 38(3), 1085–1118 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Johansson, F., Mezzarobba, M.: Fast and rigorous arbitrary-precision computation of Gauss–Legendre quadrature nodes and weights. arXiv preprint arXiv:1802.03948 (2018)
  21. 21.
    Kreuser, P.: Über das Verhalten der Integrale homogener linearer Differenzengleichungen im Unendlichen. Dissertation. Tübingen, 48 S (1914)Google Scholar
  22. 22.
    Lowan, A.N., Davids, N., Levenson, A.: Table of the zeros of the Legendre polynomials of order 1–16 and the weight coefficients for Gauss’ mechanical quadrature formula. Bull. Am. Math. Soc. 48, 739–743 (1942)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Olver, F.W.J.: Asymptotics and Special Functions. AKP Classics. A K Peters, Ltd., Wellesley (1997). (Reprint of the 1974 original [Academic Press, New York; MR0435697 (55 #8655)])CrossRefGoogle Scholar
  24. 24.
    Segura, J.: Reliable computation of the zeros of solutions of second order linear ODEs using a fourth order method. SIAM J. Numer. Anal. 48(2), 452–469 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Segura, J., Temme, N.M.: Numerically satisfactory solutions of Kummer recurrence relations. Numer. Math. 111(1), 109–119 (2008). MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Swarztrauber, P.N.: On computing the points and weights for Gauss–Legendre quadrature. SIAM J. Sci. Comput. 24(3), 945–954 (2002). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Townsend, A., Trogdon, T., Olver, S.: Fast computation of Gauss quadrature nodes and weights on the whole real line. IMA J. Numer. Anal. 36(1), 337–358 (2016)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Wilf, H.S.: Mathematics for the Physical Sciences. Dover Publications, Inc., New York (1978). (Reprinting of the 1962 original, Dover Books in Advanced Mathematics)zbMATHGoogle Scholar
  29. 29.
    Yakimiw, E.: Accurate computation of weights in classical Gauss–Christoffel quadrature rules. J. Comput. Phys. 129(2), 406–430 (1996). MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada y CC. de la Computación. ETSI CaminosUniversidad de CantabriaSantanderSpain
  2. 2.Departamento de Matemáticas, Estadística y Computación. Facultad de CienciasUniversidad de CantabriaSantanderSpain
  3. 3.IAAAlkmaarThe Netherlands
  4. 4.Centrum Wiskunde & Informatica (CWI)AmsterdamThe Netherlands

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