On computing Gauss-Kronrod quadrature formulae
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- by Franca Caliò, Walter Gautschi and Elena Marchetti PDF
- Math. Comp. 47 (1986), 639-650 Request permission
Abstract:
We discuss the use of Newton’s method for computing Gauss-Kronrod quadrature formulae from modified moments. The underlying nonlinear maps are analyzed from the point of view of numerical condition. A method is indicated of computing the polynomial whose zeros are the Kronrod nodes. Examples include Gauss-Kronrod formulae for integrals with a logarithmic and algebraic singularity at one endpoint. Pertinent numerical results are tabulated in the supplements section at the end of this issue.References
- Paola Baratella, An optimal extension of the Radau quadrature formula, Rend. Sem. Mat. Univ. Politec. Torino 37 (1979), no. 1, 147–158 (Italian, with English summary). MR 547779
- F. Caliò, E. Marchetti, and G. Pizzi, Numerical evaluation of certain integrals with a singularity of logarithmic type, Rend. Sem. Fac. Sci. Univ. Cagliari 54 (1984), no. 2, 31–40 (Italian, with English summary). MR 821191
- Catterina Dagnino and Cinzia Fiorentino, Computation of nodes and weights of extended Gaussian rules, Computing 32 (1984), no. 3, 271–278 (English, with German summary). MR 745186, DOI 10.1007/BF02243577 J. J. Dongarra et al., LINPACK Users’ Guide, SIAM, Philadelphia, Pa., 1979. S. Elhay & J. Kautsky, "A method for computing quadratures of the Kronrod Patterson type," Proc. 7th Australian Computer Science Conf., Austral. Comput. Sci. Comm., v. 6, no. 1, February 1984, pp. 15.1-15.9. Department of Computer Science, University of Adelaide, Adelaide, South Australia. S. Elhay & J. Kautsky, IQPACK: Fortran Subroutines for the Weights of Interpolatory Quadratures, School of Mathematical Sciences, The Flinders University of South Australia, April 1985.
- James L. Blue, A Legendre polynomial integral, Math. Comp. 33 (1979), no. 146, 739–741. MR 521287, DOI 10.1090/S0025-5718-1979-0521287-8
- Walter Gautschi, On generating orthogonal polynomials, SIAM J. Sci. Statist. Comput. 3 (1982), no. 3, 289–317. MR 667829, DOI 10.1137/0903018
- Walter Gautschi, On the sensitivity of orthogonal polynomials to perturbations in the moments, Numer. Math. 48 (1986), no. 4, 369–382. MR 834326, DOI 10.1007/BF01389645
- Walter Gautschi, Questions of numerical condition related to polynomials, Studies in numerical analysis, MAA Stud. Math., vol. 24, Math. Assoc. America, Washington, DC, 1984, pp. 140–177. MR 925213
- G. H. Golub and J. Kautský, Calculation of Gauss quadratures with multiple free and fixed knots, Numer. Math. 41 (1983), no. 2, 147–163. MR 703119, DOI 10.1007/BF01390210
- Gene H. Golub and John H. Welsch, Calculation of Gauss quadrature rules, Math. Comp. 23 (1969), 221-230; addendum, ibid. 23 (1969), no. 106, loose microfiche suppl, A1–A10. MR 0245201, DOI 10.1090/S0025-5718-69-99647-1
- D. K. Kahaner, J. Waldvogel, and L. W. Fullerton, Addition of points to Gauss-Laguerre quadrature formulas, SIAM J. Sci. Statist. Comput. 5 (1984), no. 1, 42–55. MR 731880, DOI 10.1137/0905003
- J. Kautsky and S. Elhay, Calculation of the weights of interpolatory quadratures, Numer. Math. 40 (1982), no. 3, 407–422. MR 695604, DOI 10.1007/BF01396453
- J. Kautsky and S. Elhay, Gauss quadratures and Jacobi matrices for weight functions not of one sign, Math. Comp. 43 (1984), no. 168, 543–550. MR 758201, DOI 10.1090/S0025-5718-1984-0758201-8
- Aleksandr Semenovich Kronrod, Nodes and weights of quadrature formulas. Sixteen-place tables, Consultants Bureau, New York, 1965. Authorized translation from the Russian. MR 0183116
- Giovanni Monegato, A note on extended Gaussian quadrature rules, Math. Comp. 30 (1976), no. 136, 812–817. MR 440878, DOI 10.1090/S0025-5718-1976-0440878-3
- Giovanni Monegato, Positivity of the weights of extended Gauss-Legendre quadrature rules, Math. Comp. 32 (1978), no. 141, 243–245. MR 458809, DOI 10.1090/S0025-5718-1978-0458809-0
- Giovanni Monegato, Some remarks on the construction of extended Gaussian quadrature rules, Math. Comp. 32 (1978), no. 141, 247–252. MR 458810, DOI 10.1090/S0025-5718-1978-0458810-7
- Giovanni Monegato, An overview of results and questions related to Kronrod schemes, Numerische Integration (Tagung, Math. Forschungsinst., Oberwolfach, 1978) Internat. Ser. Numer. Math., vol. 45, Birkhäuser, Basel-Boston, Mass., 1979, pp. 231–240. MR 561296
- Giovanni Monegato, Stieltjes polynomials and related quadrature rules, SIAM Rev. 24 (1982), no. 2, 137–158. MR 652464, DOI 10.1137/1024039
- T. N. L. Patterson, Table errata: “The optimum addition of points to quadrature formulae” (Math. Comp. 22 (1968), 847–856; addendum, ibid. 22 (1968), no. 104, loose microfiche suppl. C1-C11), Math. Comp. 23 (1969), no. 108, 892. MR 400633, DOI 10.1090/S0025-5718-1969-0400633-3
- Robert Piessens and Maria Branders, A note on the optimal addition of abscissas to quadrature formulas of Gauss and Lobatto type, Math. Comp. 28 (1974), 135–139; supplement, ibid. 28 (1974), 344–347. MR 343552, DOI 10.1090/S0025-5718-1974-0343552-5
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Math. Comp. 47 (1986), 639-650
- MSC: Primary 65D32
- DOI: https://doi.org/10.1090/S0025-5718-1986-0856708-8
- MathSciNet review: 856708