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The superconvergence of Newton–Cotes rules for the Hadamard finite-part integral on an interval

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Abstract

We study the general (composite) Newton–Cotes rules for the computation of Hadamard finite-part integral with the second-order singularity and focus on their pointwise superconvergence phenomenon, i.e., when the singular point coincides with some a priori known point, the convergence rate is higher than what is globally possible. We show that the superconvergence rate of the (composite) Newton–Cotes rules occurs at the zeros of a special function and prove the existence of the superconvergence points. Several numerical examples are provided to validate the theoretical analysis.

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Authors and Affiliations

  1. Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing, 100088, People’s Republic of China

    Jiming Wu

  2. Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

    Weiwei Sun

Authors
  1. Jiming Wu
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  2. Weiwei Sun
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Correspondence to Weiwei Sun.

Additional information

The work of J. Wu was partially supported by the National Natural Science Foundation of China (No. 10671025) and a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (No. CityU 102507).

The work of W. Sun was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (No. City U 102507) and the National Natural Science Foundation of China (No. 10671077).

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Wu, J., Sun, W. The superconvergence of Newton–Cotes rules for the Hadamard finite-part integral on an interval. Numer. Math. 109, 143–165 (2008). https://doi.org/10.1007/s00211-007-0125-7

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  • DOI: https://doi.org/10.1007/s00211-007-0125-7

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