Abstract
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.
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References
K. J. Achieser,Theory of Approximation (tr by C. J. Hyman) Frederick Ungar Publishing Co., New York, 1956.
W. Barrett,Convergence properties of Gaussian quadrature formulae, Computer J., 3 (1961), pp. 272–277.
M. M. Chawla,On the Chebyshev polynomials of the second kind, SIAM Rev., 9 (1967), pp. 729–733.
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.
M. M. Chawla and M. K. Jain,Error estimates for Gauss quadrature formulas for analytic functions, Math. Comp., 22 (1968), pp. 82–90.
M. M. Chawla and M. K. Jain,Asymptotic error estimates for the Gauss quadrature formula, Math. Comp., 22 (1968), pp. 91–97.
A. R. Curtis and P. Rabinowitz,On the Gaussian integration of Chebyshev polynomials, Math. Comp., 26 (1972), pp. 207–211.
D. Elliott,The evaluation and estimation of the coefficients in the Chebyshev series expansion of a function, Math. Comp., 18 (1964), pp. 274–284.
H. E. Fettis,Numerical calculation of certain defininte integrals by Poisson's summation formula, MTAC, 9 (1955), pp. 85–92.
W. Gautschi,On Padé approximants associated with Hamburger series, Calcolo, 20 (1983), pp. 111–127.
W. Gautschi and R. S. Varga,Error bounds for Gaussian quadrature of analytic functions, SIAM J. Numer. Anal., 20 (1983), pp. 1170–1186.
D. B. Hunter,Some properties of orthogonal polynomials, Math. Comp., 29 (1975), pp. 559–565.
D. B. Hunter,The positive-definiteness of the complete symmetric functions of even order, Math. Proc. Camb. Phil. Soc., 82 (1977), pp. 255–258.
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.
N. S. Kambo,Error of the Newton-Cotes and Gauss-Legendre quadrature formulas, Math. Comp., 24 (1970), pp. 261–269.
F. G. Lether,Error bounds for fully symmetric quadrature rules, SIAM J. Numer. Anal., 11 (1974), pp. 1–9.
F. G. Lether,Error estimates for Gaussian quadrature, Appl. Math. and Comp., 7 (1980), pp. 237–246.
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.
H. V. Smith,Global error bounds for Gauss-Christoffel quadrature, BIT, 21 (1981), pp. 481–499.
F. Stenger,Bounds on the error of Gauss-type quadratures, Numer. Math., 8 (1966), pp. 150–160.
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Hunter, D.B. Some error expansions for Gaussian quadrature. Bit Numer Math 35, 64–82 (1995). https://doi.org/10.1007/BF01732979
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DOI: https://doi.org/10.1007/BF01732979