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DE-Sinc methods have almost the same convergence property as SE-Sinc methods even for a family of functions fitting the SE-Sinc methods

Part II: indefinite integration

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Abstract

In this paper, the theoretical convergence rate of the Sinc indefinite integration combined with the double-exponential (DE) transformation is given for a class of functions for which the single-exponential (SE) transformation is suitable. Although the DE transformation is considered as an enhanced version of the SE transformation for Sinc-related methods, the function space for which the DE transformation is suitable is smaller than that for SE, and therefore, there exist some examples such that the DE transformation is not better than the SE transformation. Even in such cases, however, some numerical observations in the literature suggest that there is almost no difference in the convergence rates of SE and DE. In fact, recently, the observations have been theoretically explained for two explicit approximation formulas: the Sinc quadrature and the Sinc approximation. The conclusion is that in such cases, the DE’s rate is slightly lower, but almost the same as that of the SE. The contribution of this study is the derivation of the same conclusion for the Sinc indefinite integration. Numerical examples that support the theoretical result are also provided.

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Notes

  1. This is taken from Okayama et al. [5, Example 9.6].

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Authors and Affiliations

  1. School of Systems Information Science, Future University Hakodate, Hakodate, Japan

    Ken’ichiro Tanaka

  2. Graduate School of Economics, Hitotsubashi University, 2-1 Naka, Kunitachi-shi, Tokyo, 186-8601, Japan

    Tomoaki Okayama

  3. Graduate School of Information Science and Technology, The University of Tokyo, Tokyo, Japan

    Takayasu Matsuo & Masaaki Sugihara

Authors
  1. Ken’ichiro Tanaka
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  2. Tomoaki Okayama
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  3. Takayasu Matsuo
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  4. Masaaki Sugihara
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Corresponding author

Correspondence to Tomoaki Okayama.

Additional information

This work was supported by JSPS Grants-in-Aid for Scientific Research. Part of this work was done while the second author visited Future University Hakodate in the summer of 2011 using its summer stay program.

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Tanaka, K., Okayama, T., Matsuo, T. et al. DE-Sinc methods have almost the same convergence property as SE-Sinc methods even for a family of functions fitting the SE-Sinc methods. Numer. Math. 125, 545–568 (2013). https://doi.org/10.1007/s00211-013-0541-9

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  • DOI: https://doi.org/10.1007/s00211-013-0541-9

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