Error estimates for Clenshaw-Curtis quadrature

Riess, R. D.; Johnson, L. Clark

doi:10.1007/bf01404685

Cited by 26 publications

(15 citation statements)

References 13 publications

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“…Estimates for these quadrature errors can be obtained as in the case of the Clenshaw-Curtis quadrature formula, for which a detailed discussion can be found in Riess and Johnson [5]. We give here a simple, though rough, estimate for the error E, (f) for analytic f similar to that for the Clenshaw-Curtis quadrature formula given in Chawla [6].…”

Section: The Errormentioning

confidence: 96%

Quadrature formulas for cauchy principal value integrals

Chawla

Jayarajan

1975

Computing

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Section: The Errormentioning

confidence: 96%

Quadrature formulas for cauchy principal value integrals

Chawla

Jayarajan

1975

Computing

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“…Table 10 illustrates the cpu time of the computation of the weights {w k } for the computation of Clenshaw-Curtis, Fejér's first and second-type rules by the algorithms given in [23], compared with the cpu time of the computation of the coefficients {a k } for the three quadrature by the FFT, DCT and IDST in section 3. 0.7152e-3s 0.4199e-3s 0.3785e-3s N = 2 10 0.2847e-3s 0.3710e-3s 0.2905e-3s N = 2 15 0.0053s 0.0071s 0.0087s N = 2 15 0.0052s 0.0061s 0.0072s N = 2 18 0.0725s 0.0871s 0.1394s N = 2 18 0.0609s 0.0604s 0.0567s N = 2 20 0.2170s 0.2821s 0.2830s N = 2 20 0.2066s 0.2477s 0.2345s…”

Section: Numerical Examplesmentioning

confidence: 99%

On Fast and Stable Implementation of Clenshaw-Curtis and Fejér-Type Quadrature Rules

Xiang

Wang

2014

Abstract and Applied Analysis

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Based upon the fast computation of the coefficients of the interpolation polynomials at Chebyshev-type points by FFT, DCT and IDST, respectively, together with the efficient evaluation of the modified moments by forwards recursions or by the Oliver's algorithm, this paper presents interpolating integration algorithms, by using the coefficients and modified moments, for Clenshaw-Curtis, Fejér's first and second-type rules for Jacobi or Jacobi weights multiplied by a logarithmic function. The corresponding Matlab codes are included. Numerical examples illustrate the stability, accuracy of the Clenshaw-Curtis, Fejér's first and second rules, and show that the three quadratures have nearly the same convergence rates as Gauss-Jacobi quadrature for functions of finite regularities for Jacobi weights, and are more efficient upon the cpu time than the Gauss evaluated by fast computation of the weights and nodes by Chebfun.

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“…In der Literatur finden sich eine gauze Reihe von Abschtttzungen for [R~ [/] [, so bei Fraser und Wilson [4], Locher [5], Riess und Johnson [6] (dort aueh weitere Literatur). Der ,,klassische" AbscNitzungstyp der numerischen Mathematik, n~imlieh Absch~itzungen mittels der Maxima der Betr~tge h6herer Ableitungen yon [, ist auf unser Problem in den genannten Arbeiten angewandt worden.…”

Section: Rn[/]:= F /(X)dx--fin[l] (X) DX --I --1unclassified

“…Locher [5] und Riess und Johnson [6] geben Formeln, in denen c, die GrSBenordhung 2-"n -x hat. In dieser Arbeit soil bewiesen werden: Satz 1.…”

Section: Rn[/]:= F /(X)dx--fin[l] (X) DX --I --1unclassified