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 33 articles )

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  1. Hasegawa, Takemitsu; Sugiura, Hiroshi: Uniform approximation to Cauchy principal value integrals with logarithmic singularity (2018)
  2. Wang, Haiyong: On the convergence rate of Clenshaw-Curtis quadrature for integrals with algebraic endpoint singularities (2018)
  3. Hasegawa, Takemitsu; Sugiura, Hiroshi: A user-friendly method for computing indefinite integrals of oscillatory functions (2017)
  4. Lewanowicz, Stanisław; Keller, Paweł; Woźny, Paweł: Constrained approximation of rational triangular Bézier surfaces by polynomial triangular Bézier surfaces (2017)
  5. Motygin, Oleg V.: Numerical approximation of oscillatory integrals of the linear ship wave theory (2017)
  6. Ruijter, M.J.; Oosterlee, C.W.: A Fourier cosine method for an efficient computation of solutions to BSDEs (2015)
  7. Domínguez, V.; Graham, I.G.; Smyshlyaev, V.P.: Stability and error estimates for Filon-Clenshaw-Curtis rules for highly oscillatory integrals (2011)
  8. Hasegawa, Takemitsu; Sugiura, Hiroshi: Algorithms for approximating finite Hilbert transform with end-point singularities and its derivatives (2011)
  9. Sugiura, Hiroshi; Hasegawa, Takemitsu: A polynomial interpolation process at quasi-Chebyshev nodes with the FFT (2011)
  10. Keller, Paweł; Woźny, Paweł: On the convergence of the method for indefinite integration of oscillatory and singular functions (2010)
  11. Hasegawa, Takemitsu; Sugiura, Hiroshi: Uniform approximation to fractional derivatives of functions of algebraic singularity (2009)
  12. Wang, Haiyong; Xiang, Shuhuang: Uniform approximations to Cauchy principal value integrals of oscillatory functions (2009)
  13. Hale, Nicholas; Trefethen, Lloyd N.: New quadrature formulas from conformal maps (2008)
  14. Trefethen, Lloyd N.: Is Gauss quadrature better than Clenshaw-Curtis? (2008)
  15. Hasegawa, Takemitsu; Sugiura, Hiroshi: Quadrature rule for indefinite integral of algebraic-logarithmic singular integrands (2007)
  16. Keller, Paweł: A method for indefinite integration of oscillatory and singular functions (2007)
  17. Adam, Gh.; Adam, Sanda: Discrete group symmetry in the fast Chebyshev transform (2003)
  18. Hasegawa, Takemitsu: Numerical integration of functions with poles near the interval of integration (1997)
  19. Hasegawa, Takemitsu; Sidi, Avram: An automatic integration procedure for infinite range integrals involving oscillatory kernels (1996)
  20. Hasegawa, Takemitsu; Torii, Tatsuo: An algorithm for nondominant solutions of linear second-order inhomogeneous difference equations (1995)

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