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BIT Numerical Mathematics

, Volume 55, Issue 3, pp 823–842 | Cite as

Convergence properties of a quadrature formula of Clenshaw–Curtis type for the Gegenbauer weight function

  • H. V. SmithEmail author
Article
  • 185 Downloads

Abstract

A quadrature formula of Clenshaw–Curtis type for functions of the form \((1-x^2)^{\lambda - \frac{1}{2}}f(x)\) over the interval [\(-\)1,1] exhibits a curious phenomenon when applied to certain analytic functions. As the number of points in the quadrature rule increases the error may sometimes decay to zero in two distinct stages rather than in one depending on the value of \(\lambda \). In this paper we shall derive explicit and asymptotic error formulae which describe this phenomenon.

Keywords

Gegenbauer weight function Clenshaw–Curtis quadrature   Convergence rate Lobatto-Chebyshev quadrature 

Mathematics Subject Classification

65D32 

Notes

Acknowledgments

The author is indebted to both referees for their very helpful comments which have led to a considerable improvement of the paper and my grateful thanks go to Nairn Kennedy whose technical expertise proved invaluable.

References

  1. 1.
    Bell, W.W.: Special Functions for Scientists and Engineers. Van Nostrand Reinhold Co., London (1968)zbMATHGoogle Scholar
  2. 2.
    Clenshaw, C.W., Curtis, A.R.: A method for numerical integration on an automatic computer. Numer. Math. 2, 197–205 (1960)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Elliott, D., Johnston, B.M., Johnston, P.R.: Clenshaw-Curtis and Gauss-Legendre quadrature for certain boundary element integrals. SIAM J Sci. Comput 31, 510–530 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Hunter, D.B., Smith, H.V.: A quadrature formula of Clenshaw-Curtis type for the Gegenbauer weight function. J. Comput. Appl. Math. 177, 389–400 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Hyslop, J.M.: Infinte Series. Oliver and Boyd Ltd., Edinburgh (1959)Google Scholar
  6. 6.
    Krylov, V.I.: Approximate Calculation of Integrals. The Macmillan Co., New York (1962)zbMATHGoogle Scholar
  7. 7.
    Smith, H.V.: A correction term for Gauss-Legendre quadrature. Int. J. Math. Educ. Sci. Technol. 34, 53–56 (2003)CrossRefGoogle Scholar
  8. 8.
    Smith, H.V.: A correction term for Gauss-Gegenbauer quadrature. Int. J. Math. Educ. Sci. Technol. 35, 363–367 (2004)CrossRefGoogle Scholar
  9. 9.
    Smith, H.V.: Numerical integration–a different approach. Math. Gaz. 90, 21–24 (2006)Google Scholar
  10. 10.
    Smith, H.V.: The numerical evaluation of the error term in Gaussian quadrature rules. Int. J. Math. Educ. Sci. Technol. 37, 201–205 (2006)CrossRefGoogle Scholar
  11. 11.
    Smith, H.V.: The evaluation of the error term in some Gauss-type formulae for the approximation of Cauchy Principal Value integrals. Int. J. Math. Educ. Sci. Technol. 39, 69–76 (2008)CrossRefGoogle Scholar
  12. 12.
    Smith, H.V., Hunter, D.B.: The numerical evaluation of the error term in a quadrature formula of Clenshaw-Curtis type for the Gegenbauer weight function. BIT 51, 1031–1038 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Szegö, G.: Orthogonal Polynomials. Amer. Math. Soc. Coll. Pub, XXIII (1975)zbMATHGoogle Scholar
  14. 14.
    Trefethen, L.N.: Is Gauss quadrature better than Clenshaw-Curtis? SIAM Rev. 50, 67–87 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Weideman, J.A.C., Trefethen, L.N.: The kink phenomenon in Fejér and Clenshaw-Curtis quadrature. Numer. Math. 107, 707–727 (2007)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Quadrature Research CentreDumfriesScotland, UK

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