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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 I: definite integration and function approximation

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Abstract

In this paper, the theoretical convergence rate of the trapezoidal rule combined with the double-exponential (DE) transformation is given for a class of functions for which the single-exponential (SE) transformation is suitable. It is well known that the DE transformation enables the rule to achieve a much higher rate of convergence than the SE transformation, and the convergence rate has been analyzed and justified theoretically under a proper assumption. Here, it should be emphasized that the assumption is more severe than the one for the SE transformation, and there actually exist some examples such that the trapezoidal rule with the SE transformation achieves its usual rate, whereas the rule with DE does not. Such cases have been observed numerically, but no theoretical analysis has been given thus far. This paper reveals the theoretical rate of convergence in such cases, and it turns out that the DE’s rate is almost the same as, but slightly lower than that of the SE. By using the analysis technique developed here, the theoretical convergence rate of the Sinc approximation with the DE transformation is also given for a class of functions for which the SE transformation is suitable. The result is quite similar to above; the convergence rate in the DE case is slightly lower than in the SE case. Numerical examples which support those two theoretical results are also given.

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Notes

  1. Computation programs in this section were written in C with quadruple-precision floating-point arithmetic, which is available on a PowerPC CPU by using “long double” type. We set \(\epsilon =0.001\) for the computation.

  2. We used the value 0.71194382297059827888000405031539396435 as an answer, which was calculated using Mathematica 7 with sufficient accuracy. This is similar to the approach adopted by Tanaka et al. [18, § 4.2].

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Acknowledgments

We would like to thank Professor Kazuo Murota for his insightful comments. We would also like to express our gratitude to the reviewers for their comments and suggestions, which enabled us to improve this paper. This work was supported by JSPS Grants-in-Aid for Scientific Research. Part of this work was conducted while the first author visited Future University Hakodate in the summer of 2011 under its summer stay program.

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Authors and Affiliations

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

    Tomoaki Okayama

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

    Ken’ichiro Tanaka

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

    Takayasu Matsuo & Masaaki Sugihara

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

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Okayama, T., Tanaka, K., 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, 511–543 (2013). https://doi.org/10.1007/s00211-013-0540-x

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  • DOI: https://doi.org/10.1007/s00211-013-0540-x

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