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Some error expansions for Gaussian quadrature. (English) Zbl 0824.41032
Summary: Complex-variable methods are used to obtain some expansions in the error in Gaussian quadrature formulae over the interval \([-1,1]\). Much of the work is based on an approach due to Stenger, and both circular and elliptical contours are used. Stenger’s theorem on monotonicity of convergence of Gaussian quadrature formulae is generalized, and a number of error bounds are obtained.

MSC:
41A55 Approximate quadratures
41A80 Remainders in approximation formulas
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[1] K. J. Achieser,Theory of Approximation (tr by C. J. Hyman) Frederick Ungar Publishing Co., New York, 1956. · Zbl 0072.28403
[2] W. Barrett,Convergence properties of Gaussian quadrature formulae, Computer J., 3 (1961), pp. 272–277. · Zbl 0098.31703 · doi:10.1093/comjnl/3.4.272
[3] M. M. Chawla,On the Chebyshev polynomials of the second kind, SIAM Rev., 9 (1967), pp. 729–733. · Zbl 0176.35301 · doi:10.1137/1009115
[4] M. M. Chawla,On Davis’s method for the estimation of errors of Gauss-Chebyshev quadratures, SIAM J. Numer. Anal., 6 (1969), pp. 108–117. · Zbl 0191.44902 · doi:10.1137/0706012
[5] M. M. Chawla and M. K. Jain,Error estimates for Gauss quadrature formulas for analytic functions, Math. Comp., 22 (1968), pp. 82–90. · Zbl 0155.21602 · doi:10.1090/S0025-5718-1968-0223093-3
[6] M. M. Chawla and M. K. Jain,Asymptotic error estimates for the Gauss quadrature formula, Math. Comp., 22 (1968), pp. 91–97. · Zbl 0155.21701 · doi:10.1090/S0025-5718-1968-0223094-5
[7] A. R. Curtis and P. Rabinowitz,On the Gaussian integration of Chebyshev polynomials, Math. Comp., 26 (1972), pp. 207–211. · Zbl 0261.65019 · doi:10.1090/S0025-5718-1972-0298934-5
[8] D. Elliott,The evaluation and estimation of the coefficients in the Chebyshev series expansion of a function, Math. Comp., 18 (1964), pp. 274–284. · Zbl 0119.32904 · doi:10.1090/S0025-5718-1964-0166903-7
[9] H. E. Fettis,Numerical calculation of certain defininte integrals by Poisson’s summation formula, MTAC, 9 (1955), pp. 85–92. · Zbl 0066.10601
[10] W. Gautschi,On Padé approximants associated with Hamburger series, Calcolo, 20 (1983), pp. 111–127. · Zbl 0571.41013 · doi:10.1007/BF02575588
[11] W. Gautschi and R. S. Varga,Error bounds for Gaussian quadrature of analytic functions, SIAM J. Numer. Anal., 20 (1983), pp. 1170–1186. · Zbl 0545.41040 · doi:10.1137/0720087
[12] D. B. Hunter,Some properties of orthogonal polynomials, Math. Comp., 29 (1975), pp. 559–565. · Zbl 0304.42016 · doi:10.1090/S0025-5718-1975-0374792-8
[13] D. B. Hunter,The positive-definiteness of the complete symmetric functions of even order, Math. Proc. Camb. Phil. Soc., 82 (1977), pp. 255–258. · Zbl 0369.42015 · doi:10.1017/S030500410005386X
[14] D. B. Hunter and H. V. Smith,Some problems involving orthogonal polynomials, in International Series of Numerical Mathematics 112, H. Brass and G. Hämmerlin, eds., Birkhauser Verlag, Basel (1993), pp. 189–197. · Zbl 0792.41020
[15] N. S. Kambo,Error of the Newton-Cotes and Gauss-Legendre quadrature formulas, Math. Comp., 24 (1970), pp. 261–269. · Zbl 0264.65018 · doi:10.1090/S0025-5718-1970-0275671-2
[16] F. G. Lether,Error bounds for fully symmetric quadrature rules, SIAM J. Numer. Anal., 11 (1974), pp. 1–9. · Zbl 0273.65017 · doi:10.1137/0711001
[17] F. G. Lether,Error estimates for Gaussian quadrature, Appl. Math. and Comp., 7 (1980), pp. 237–246. · Zbl 0442.41026 · doi:10.1016/0096-3003(80)90046-6
[18] J. McNamee,Error-bounds for the evaluation of integrals by the Euler-Maclaurin formula and by Gauss-type formulae, Math. Comp., 18 (1964), pp. 368–381. · Zbl 0125.36202 · doi:10.1090/S0025-5718-1964-0185804-1
[19] H. V. Smith,Global error bounds for Gauss-Christoffel quadrature, BIT, 21 (1981), pp. 481–499. · Zbl 0475.41031 · doi:10.1007/BF01932845
[20] F. Stenger,Bounds on the error of Gauss-type quadratures, Numer. Math., 8 (1966), pp. 150–160. · Zbl 0149.12002 · doi:10.1007/BF02163184
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