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The numerical evaluation of the error term in a quadrature formula of Clenshaw-Curtis type for the Gegenbauer weight function

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Abstract

A method is derived for the numerical evaluation of the error term arising in 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]. The method is illustrated by an example.

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References

  1. Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions. Dover, New York (1966)

    Google Scholar 

  2. Clenshaw, C.W., Curtis, A.R.: A method for numerical integration on an automatic computer. Numer. Math. 2, 197–205 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  3. Evans, G.A.: Practical Numerical Integration. Wiley, Chichester (1993)

    MATH  Google Scholar 

  4. Golub, G.H., Welsh, J.H.: Calculation of Gauss quadrature rules. Math. Comput. 23, 221–230 (1969)

    Article  MATH  Google Scholar 

  5. Hunter, D.B., Nikolov, G.: Gaussian quadrature of Chebyshev polynomials. J. Comput. Appl. Math. 94, 123–131 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kambo, N.S.: Error bounds for Clenshaw-Curtis quadrature formulas. BIT Numer. Math. 11, 299–309 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  8. O’Hara, H., Smith, F.J.: Error estimation in the Clenshaw-Curtis quadrature formula. Comput. J. 11, 213–219 (1968)

    MathSciNet  MATH  Google Scholar 

  9. Smith, H.V.: A correction term for Gauss-Gegenbauer quadrature. Int. J. Math. Educ. Sci. Technol. 35, 363–367 (2004)

    Article  Google Scholar 

  10. 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)

    Article  Google Scholar 

  11. Smith, H.V.: The convergence of the error term in a quadrature formula of Clenshaw-Curtis type for the Gegenbauer weight function. Technical Report Number 01/07, Quadrature Research Centre, 2 Valleyfield Park, Terregles, Dumfries, Scotland DG2 9RA

  12. Trefethen, L.N.: Is Gauss quadrature better than Clenshaw-Curtis? SIAM Rev. 50, 67–87 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Quadrature Research Centre, 2 Valleyfield Park, Terregles, Dumfries, Scotland, DG2 9RA, UK

    H. V. Smith

  2. Mathematics Unit, Department of Computing, Bradford University, Bradford 7, West Yorkshire, England, BD7 1DP, UK

    D. B. Hunter

Authors
  1. H. V. Smith
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  2. D. B. Hunter
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Correspondence to H. V. Smith.

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Communicated by Lothar Reichel.

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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 Numer Math 51, 1031–1038 (2011). https://doi.org/10.1007/s10543-011-0330-8

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  • DOI: https://doi.org/10.1007/s10543-011-0330-8

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