BIT Numerical Mathematics

, Volume 35, Issue 1, pp 64–82

Some error expansions for Gaussian quadrature

  • D. B. Hunter


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.

Key words

Gaussian quadrature error expansion 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    K. J. Achieser,Theory of Approximation (tr by C. J. Hyman) Frederick Ungar Publishing Co., New York, 1956.Google Scholar
  2. 2.
    W. Barrett,Convergence properties of Gaussian quadrature formulae, Computer J., 3 (1961), pp. 272–277.Google Scholar
  3. 3.
    M. M. Chawla,On the Chebyshev polynomials of the second kind, SIAM Rev., 9 (1967), pp. 729–733.Google Scholar
  4. 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.Google Scholar
  5. 5.
    M. M. Chawla and M. K. Jain,Error estimates for Gauss quadrature formulas for analytic functions, Math. Comp., 22 (1968), pp. 82–90.Google Scholar
  6. 6.
    M. M. Chawla and M. K. Jain,Asymptotic error estimates for the Gauss quadrature formula, Math. Comp., 22 (1968), pp. 91–97.Google Scholar
  7. 7.
    A. R. Curtis and P. Rabinowitz,On the Gaussian integration of Chebyshev polynomials, Math. Comp., 26 (1972), pp. 207–211.Google Scholar
  8. 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.Google Scholar
  9. 9.
    H. E. Fettis,Numerical calculation of certain defininte integrals by Poisson's summation formula, MTAC, 9 (1955), pp. 85–92.Google Scholar
  10. 10.
    W. Gautschi,On Padé approximants associated with Hamburger series, Calcolo, 20 (1983), pp. 111–127.Google Scholar
  11. 11.
    W. Gautschi and R. S. Varga,Error bounds for Gaussian quadrature of analytic functions, SIAM J. Numer. Anal., 20 (1983), pp. 1170–1186.Google Scholar
  12. 12.
    D. B. Hunter,Some properties of orthogonal polynomials, Math. Comp., 29 (1975), pp. 559–565.Google Scholar
  13. 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.Google Scholar
  14. 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.Google Scholar
  15. 15.
    N. S. Kambo,Error of the Newton-Cotes and Gauss-Legendre quadrature formulas, Math. Comp., 24 (1970), pp. 261–269.Google Scholar
  16. 16.
    F. G. Lether,Error bounds for fully symmetric quadrature rules, SIAM J. Numer. Anal., 11 (1974), pp. 1–9.Google Scholar
  17. 17.
    F. G. Lether,Error estimates for Gaussian quadrature, Appl. Math. and Comp., 7 (1980), pp. 237–246.Google Scholar
  18. 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.Google Scholar
  19. 19.
    H. V. Smith,Global error bounds for Gauss-Christoffel quadrature, BIT, 21 (1981), pp. 481–499.Google Scholar
  20. 20.
    F. Stenger,Bounds on the error of Gauss-type quadratures, Numer. Math., 8 (1966), pp. 150–160.Google Scholar

Copyright information

© BIT Foundation 1995

Authors and Affiliations

  • D. B. Hunter
    • 1
  1. 1.Department of MathematicsUniverstiy of BradfordBradfordEngland

Personalised recommendations