Integral formulas for Chebyshev polynomials  and  the error term  of  interpolatory quadrature formulae  for analytic functions

The Fejér and Clenshaw-Curtis rules for numerical integration exhibit a curious phenomenon when applied to certain analytic functions. When N (the number of points in the integration rule) increases, the error does not decay to zero evenly but does so in two distinct stages. For N less than a critical value, the error behaves like O( −2N ), where is a constant greater than 1. For these values of N the accuracy of both the Fejér and ClenshawCurtis rules is almost indistinguishable from that of the more celebrated Gauss-Legendre quadrature rule. For larger N, however, the error decreases at the rate O( −N ), i.e., only half as fast as before. Convergence curves typically display a kink where the convergence rate cuts in half. In this paper we derive explicit as well as asymptotic error formulas that provide a complete description of this phenomenon.

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“…these estimates are poor if is near 1." Notaris makes a similar comment in the last paragraph of [12].…”

Section: Formulas For S N (Z)mentioning

confidence: 64%

“…Chelo Ferreira and José Lopez, authors of [8], generously shared their knowledge of the Lerch Φ function. Milton Maritz helped with Mathematica and Walter Gautschi brought reference [12] to our attention.…”

mentioning

confidence: 99%

The kink phenomenon in Fejér and Clenshaw–Curtis quadrature

Weideman

Trefethen

2007

Numer. Math.

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“…Here, we consider error estimation of quadrature formula (1) (special cases ( 2) and ( 3)) with multiple end nodes for analytic function. The error bounds, for integrands f having an analytic extension in a domain containing [−1, 1], can be obtained by a contour integration method, well established and described in ( [3], [4], [5], [6], [8], [9], [10], [12], [13], [14]). These papers considered integrands f having an analytic extension into a domain Γ, which encompasses the interval [−1, 1], where Γ is a simple closed curve in the complex plain.…”

Section: The Error Bounds Of Gauss-kronrod Quadrature Formulaementioning

confidence: 99%

The error estimates of Kronrod extension for Gauss-Radau and Gauss-Lobatto quadrature with the four Chebyshev weights

Jandrlić

Pejčev

Spalević

2022

Filomat

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In this paper, we consider the Kronrod extension for the Gauss-Radau and Gauss-Lobatto quadrature consisting of any one of the four Chebyshev weights. The main purpose is to effectively estimate the error of these quadrature formulas. This estimate needs a calculation of the maximum of the modulus of the kernel. We compute explicitly the kernel function and determine the locations on the ellipses where a maximum modulus of the kernel is attained. Based on this, we derive effective error bounds of the Kronrod extensions if the integrand is an analytic function inside of a region bounded by a confocal ellipse that contains the interval of integration.

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“…Finally, and for the sake of completeness, similar computations for the kernels corresponding to the other modified Chebyshev measures dσ [i] n , i = 2, 3, 4, are gathered in the final appendix. To end this introduction, let us say that the problem of estimating the quadrature error for Gauss-type rules has been thoroughly studied in the literature; see the references [12,18,19,22,23], and [26][27][28][29][30], to only cite a few. See also [5] for a very recent survey of the error estimates of Gaussian type quadrature formulae for analytic functions on ellipses.…”

Section: Secondary 41a55 1 Introductionmentioning

confidence: 99%

On the Gauss-Kronrod quadrature formula for a modified weight function of Chebyshev type

et al. 2022

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In this paper, we consider the Gauss-Kronrod quadrature formulas for a modified Chebyshev weight. Efficient estimates of the error of these Gauss–Kronrod formulae for analytic functions are obtained, using techniques of contour integration that were introduced by Gautschi and Varga (cf. Gautschi and Varga SIAM J. Numer. Anal. 20, 1170–1186 1983). Some illustrative numerical examples which show both the accuracy of the Gauss–Kronrod formulas and the sharpness of our estimations are displayed. Though for the sake of brevity we restrict ourselves to the first kind Chebyshev weight, a similar analysis may be carried out for the other three Chebyshev type weights; part of the corresponding computations are included in a final appendix.

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Integral formulas for Chebyshev polynomials and the error term of interpolatory quadrature formulae for analytic functions

Cited by 15 publications

References 5 publications

The kink phenomenon in Fejér and Clenshaw–Curtis quadrature

The kink phenomenon in Fejér and Clenshaw–Curtis quadrature

The error estimates of Kronrod extension for Gauss-Radau and Gauss-Lobatto quadrature with the four Chebyshev weights

On the Gauss-Kronrod quadrature formula for a modified weight function of Chebyshev type

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