Error bounds for approximation in Chebyshev points
type: Item
instance of: scholarly article
title: Error bounds for approximation in Chebyshev points
zbMATH Open document ID: 1201.65040
author: Shuhuang Xiang
author: Xiaojun Chen
author: Haiyong Wang
published in: Numerische Mathematik
publication date: 2010-10-08T00:00:00Z
review text: \textit{Lloyd N. Trefethen} [SIAM Rev. 50, No.~1, 67--87 (2008; Zbl 1141.65018)] has compared the convergence behavior of the Gauss quadrature with the Clenshaw-Curtis quadrature [cf. \textit{C. W. Clenshaw} and \textit{A. R. Curtis}, Numer. Math. 2, 197--205 (1960; Zbl 0093.14006)] and the experiments show that the supposed factor-of-2 advantage of the Gauss quadrature is rarely realized and backed by the corresponding theorems explaining this effect. These results are employed in this paper to consider new error estimates for approximations of \(f\) in the Chebyshev points. It is demonstrated that polynomial interpolation in the Chebyshev points of the 1st and 2nd kind should be regarded as equally valuable and fundamental. Error bounds for Gauss, Clenshaw-Curtis and Fejér's first quadratures are improved by using new error estimates for polynomial interpolation in the Chebyshev points. Numerical results (for highly oscillatory integrals) demonstrate that the improved error bounds are reasonably sharp. These results can be employed for approximate solutions of integral equations appearing in identification and control theory of nonlinear dynamical systems [\textit{D. N. Sidorov}, Sib. Zh. Ind. Mat. 3, No.~1, 182--194 (2000; Zbl 0951.93021)].
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 65D30
Mathematics Subject Classification ID: 41A50
zbMATH DE Number: 5797349
zbMATH Keywords: numerical results
zbMATH Keywords: Chebyshev points
zbMATH Keywords: Gauss quadrature
zbMATH Keywords: oscillatory integrals
zbMATH Keywords: integral equations
zbMATH Keywords: interpolation
zbMATH Keywords: Clenshaw-Curtis quadrature
zbMATH Keywords: Fejér quadrature
zbMATH Keywords: convergence
zbMATH Keywords: error estimates
reviewed by: Denis Sidorov
describes a project that uses: Matlab
MaRDI profile type: Publication
full work available at URL: s00211-010-0309-4
OpenAlex ID: W1977463378
cites work: Product Integration Rules at Clenshaw-Curtis and Related Points: A Robust Implementation
cites work: Q5558293
cites work: Barycentric Lagrange Interpolation
cites work: A method for numerical integration on an automatic computer
cites work: Q3323133
cites work: Complex Gaussian quadrature of oscillatory integrals
cites work: A comparison of some methods for the evaluation of highly oscillatory integrals
cites work: A Fast Algorithm for the Calculation of the Roots of Special Functions
cites work: Q4515165
cites work: Error Estimation in the Clenshaw-Curtis Quadrature Formula
cites work: On numerical computation of integrals with integrands of the form \(f(x)\sin(w/x^r)\) on [0,1]
cites work: Efficient quadrature of highly oscillatory integrals using derivatives
cites work: Clenshaw-Curtis quadrature with a weighting function
cites work: Numerical Evaluation of Fourier Integrals
cites work: On high precision methods for the evaluation of Fourier integrals with finite and infinite limits
cites work: Modified Clenshaw-Curtis method for the computation of Bessel function integrals
cites work: A numerical method for the integration of oscillatory functions
cites work: Q3900444
cites work: On the numerical evaluation of singular integrals
cites work: Product-integration with the Clenshaw-Curtis and related points: Convergence properties
cites work: Product integration with the Clenshaw-Curtis points: Implementation and error estimates
cites work: Q3142876
cites work: Spectral Methods in MATLAB
cites work: Is Gauss Quadrature Better than Clenshaw–Curtis?
cites work: Fast construction of the Fejér and Clenshaw-Curtis quadrature rules
cites work: Efficient Filon-type methods for \(\int_a^b f(x)\,e^{i\omega g(x)}\, dx\)
DOI: 10.1007/S00211-010-0309-4
description: scientific article; zbMATH DE number 5797349