Clenshaw-Curtis

Implementing Clenshaw-Curtis quadrature, I methodology and experience. Clenshaw-Curtis quadrature is a particularly important automatic quadrature scheme for a variety of reasons, especially the high accuracy obtained from relatively few integrand values. However, it has received little use because it requires the computation of a cosine transformation, and the arithmetic cost of this has been prohibitive. This paper is in two parts; a companion paper, “II Computing the Cosine Transformation,” shows that this objection can be overcome by computing the cosine transformation by a modification of the fast Fourier transform algorithm. This first part discusses the strategy and various error estimates, and summarizes experience with a particular implementation of the scheme.

This software is also peer reviewed by journal TOMS.


References in zbMATH (referenced in 28 articles )

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  1. Lewanowicz, Stanisław; Keller, Paweł; Woźny, Paweł: Constrained approximation of rational triangular Bézier surfaces by polynomial triangular Bézier surfaces (2017)
  2. Domínguez, V.; Graham, I.G.; Smyshlyaev, V.P.: Stability and error estimates for Filon-Clenshaw-Curtis rules for highly oscillatory integrals (2011)
  3. Hasegawa, Takemitsu; Sugiura, Hiroshi: Algorithms for approximating finite Hilbert transform with end-point singularities and its derivatives (2011)
  4. Sugiura, Hiroshi; Hasegawa, Takemitsu: A polynomial interpolation process at quasi-Chebyshev nodes with the FFT (2011)
  5. Keller, Paweł; Woźny, Paweł: On the convergence of the method for indefinite integration of oscillatory and singular functions (2010)
  6. Hasegawa, Takemitsu; Sugiura, Hiroshi: Uniform approximation to fractional derivatives of functions of algebraic singularity (2009)
  7. Wang, Haiyong; Xiang, Shuhuang: Uniform approximations to Cauchy principal value integrals of oscillatory functions (2009)
  8. Hale, Nicholas; Trefethen, Lloyd N.: New quadrature formulas from conformal maps (2008)
  9. Trefethen, Lloyd N.: Is Gauss quadrature better than Clenshaw-Curtis? (2008)
  10. Hasegawa, Takemitsu; Sugiura, Hiroshi: Quadrature rule for indefinite integral of algebraic-logarithmic singular integrands (2007)
  11. Keller, Paweł: A method for indefinite integration of oscillatory and singular functions (2007)
  12. Adam, Gh.; Adam, Sanda: Discrete group symmetry in the fast Chebyshev transform (2003)
  13. Hasegawa, Takemitsu: Numerical integration of functions with poles near the interval of integration (1997)
  14. Hasegawa, Takemitsu; Sidi, Avram: An automatic integration procedure for infinite range integrals involving oscillatory kernels (1996)
  15. Hasegawa, Takemitsu; Torii, Tatsuo: An algorithm for nondominant solutions of linear second-order inhomogeneous difference equations (1995)
  16. Favati, Paola; Lotti, Grazia; Romani, Francesco: Algorithm 691: Improving QUADPACK automatic integration routines (1991)
  17. Hasegawa, Takemitsu; Torii, Tatsuo: An automatic quadrature for Cauchy principal value integrals (1991)
  18. Favati, P.; Lotti, G.; Romani, F.: Testing automatic quadrature programs (1990)
  19. Hasegawa, Takemitsu; Torii, Tatsuo; Sugiura, Hiroshi: An algorithm based on the FFT for a generalized Chebyshev interpolation (1990)
  20. Geppini, M.; Romani, F.: Automatic quadrature on Chebyshev points (1988)

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