Anti-Gaussian quadrature formulas

Laurie, Dirk

doi:10.1090/S0025-5718-96-00713-2

Anti-Gaussian quadrature formulas
HTML articles powered by AMS MathViewer

by Dirk P. Laurie PDF

Math. Comp. 65 (1996), 739-747 Request permission

Abstract:

An anti-Gaussian quadrature formula is an $(n+1)$-point formula of degree $2n-1$ which integrates polynomials of degree up to $2n+1$ with an error equal in magnitude but of opposite sign to that of the $n$-point Gaussian formula. Its intended application is to estimate the error incurred in Gaussian integration by halving the difference between the results obtained from the two formulas. We show that an anti-Gaussian formula has positive weights, and that its nodes are in the integration interval and are interlaced by those of the corresponding Gaussian formula. Similar results for Gaussian formulas with respect to a positive weight are given, except that for some weight functions, at most two of the nodes may be outside the integration interval. The anti-Gaussian formula has only interior nodes in many cases when the Kronrod extension does not, and is as easy to compute as the $(n+1)$-point Gaussian formula.

References

Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. MR 0167642
Annuntiato Begumisa and Ian Robinson, Suboptimal Kronrod extension formulae for numerical quadrature, Numer. Math. 58 (1991), no. 8, 807–818. MR 1098866, DOI 10.1007/BF01385655
C. W. Clenshaw and A. R. Curtis, A method for numerical integration on an automatic computer, Numer. Math. 2 (1960), 197–205. MR 117885, DOI 10.1007/BF01386223
V. I. Devjatko, On two-sided approximation in the numerical integration of ordinary differential equations, Ž. Vyčisl. Mat i Mat. Fiz. 3 (1963), 254–265 (Russian). MR 162356
Terje Espelid, Integration rules, null rules and error estimation, Technical report, Department of Informatics, University of Bergen, 1988.
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, Gauss-Kronrod quadrature—a survey, Numerical methods and approximation theory, III (Niš, 1987) Univ. Niš, Niš, 1988, pp. 39–66. MR 960329
Walter Gautschi, Algorithm 726: ORTHPOL – a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules, ACM Trans. Math. Software, 20:21–62, 1994.
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 and G. Monegato, Nonexistence of extended Gauss-Laguerre and Gauss-Hermite quadrature rules with positive weights, Z. Angew. Math. Phys. 29 (1978), no. 6, 983–986 (English, with German summary). MR 523866, DOI 10.1007/BF01590820
Aleksandr Semenovich Kronrod, Nodes and weights of quadrature formulas. Sixteen-place tables, Consultants Bureau, New York, 1965. Authorized translation from the Russian. MR 0183116
D. P. Laurie, Sharper error estimates in adaptive quadrature, BIT 23 (1983), no. 2, 258–261. MR 697788, DOI 10.1007/BF02218446
D. P. Laurie, Practical error estimation in numerical integration, J. Computat. Appl. Math, 12&13:258–261, 1985.
Dirk P. Laurie, Stratified sequences of nested quadrature formulas, Quaestiones Math. 15 (1992), no. 3, 365–384. 17th South African Symposium on Numerical Mathematics (Umhlanga, 1991). MR 1192847, DOI 10.1080/16073606.1992.9631697
Beresford N. Parlett, The symmetric eigenvalue problem, Prentice-Hall Series in Computational Mathematics, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1980. MR 570116
T. N. L. Patterson, An algorithm for generating interpolatory quadrature rules of the highest degree of precision with preassigned nodes for general weight functions, ACM Trans. Math. Software 15 (1989), no. 2, 123–136. MR 1062489, DOI 10.1145/63522.63523
T. N. L. Patterson, Modified optimal quadrature extensions, Numer. Math., 60:511–520, 1993.
Robert Piessens, Elise de Doncker-Kapenga, Christoph W. Überhuber, and David K. Kahaner, QUADPACK, Springer Series in Computational Mathematics, vol. 1, Springer-Verlag, Berlin, 1983. A subroutine package for automatic integration. MR 712135, DOI 10.1007/978-3-642-61786-7
Ju. V. Rakitskiĭ, Some properties of solutions of systems of ordinary differential equations by one-step methods of numerical integration, Ž. Vyčisl. Mat i Mat. Fiz. 1 (1961), 947–962 (Russian). MR 142199
N. P. Salihov, Polar difference methods for solving the Cauchy problem for a system of ordinary differential equations, Ž. Vyčisl. Mat i Mat. Fiz. 2 (1962), 515–528 (Russian). MR 148229