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 38 articles )

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  1. Gilles, Marc Aurèle; Townsend, Alex: Continuous analogues of Krylov subspace methods for differential operators (2019)
  2. Hasegawa, Takemitsu; Sugiura, Hiroshi: Uniform approximation to finite Hilbert transform of oscillatory functions and its algorithm (2019)
  3. Liu, Guidong; Xiang, Shuhuang: Clenshaw-Curtis-type quadrature rule for hypersingular integrals with highly oscillatory kernels (2019)
  4. Hasegawa, Takemitsu; Sugiura, Hiroshi: Uniform approximation to Cauchy principal value integrals with logarithmic singularity (2018)
  5. Kang, Myeongmin; Jung, Miyoun; Kang, Myungjoo: Higher-order regularization based image restoration with automatic regularization parameter selection (2018)
  6. Townsend, Alex; Webb, Marcus; Olver, Sheehan: Fast polynomial transforms based on Toeplitz and Hankel matrices (2018)
  7. Wang, Haiyong: On the convergence rate of Clenshaw-Curtis quadrature for integrals with algebraic endpoint singularities (2018)
  8. Hasegawa, Takemitsu; Sugiura, Hiroshi: A user-friendly method for computing indefinite integrals of oscillatory functions (2017)
  9. Lewanowicz, Stanisław; Keller, Paweł; Woźny, Paweł: Constrained approximation of rational triangular Bézier surfaces by polynomial triangular Bézier surfaces (2017)
  10. Motygin, Oleg V.: Numerical approximation of oscillatory integrals of the linear ship wave theory (2017)
  11. Ruijter, M. J.; Oosterlee, C. W.: A Fourier cosine method for an efficient computation of solutions to BSDEs (2015)
  12. Domínguez, V.; Graham, I. G.; Smyshlyaev, V. P.: Stability and error estimates for Filon-Clenshaw-Curtis rules for highly oscillatory integrals (2011)
  13. Hasegawa, Takemitsu; Sugiura, Hiroshi: Algorithms for approximating finite Hilbert transform with end-point singularities and its derivatives (2011)
  14. Sugiura, Hiroshi; Hasegawa, Takemitsu: A polynomial interpolation process at quasi-Chebyshev nodes with the FFT (2011)
  15. Keller, Paweł; Woźny, Paweł: On the convergence of the method for indefinite integration of oscillatory and singular functions (2010)
  16. Hasegawa, Takemitsu; Sugiura, Hiroshi: Uniform approximation to fractional derivatives of functions of algebraic singularity (2009)
  17. Wang, Haiyong; Xiang, Shuhuang: Uniform approximations to Cauchy principal value integrals of oscillatory functions (2009)
  18. Hale, Nicholas; Trefethen, Lloyd N.: New quadrature formulas from conformal maps (2008)
  19. Trefethen, Lloyd N.: Is Gauss quadrature better than Clenshaw-Curtis? (2008)
  20. Hasegawa, Takemitsu; Sugiura, Hiroshi: Quadrature rule for indefinite integral of algebraic-logarithmic singular integrands (2007)

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