DE-Sinc methods have almost the same convergence property as SE-Sinc methods even for a family of functions fitting the SE-Sinc methods | Numerische Mathematik Skip to main content
Springer Nature Link
Log in

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

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

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].

References

  1. Bornemann, F., Laurie, D., Wagon, S., Waldvogel, J.: The SIAM 100-Digit Challenge. SIAM, Philadelphia (2004)

    Book  MATH  Google Scholar 

  2. Moran, P.A.P.: Numerical integration in the presence of singularities. Acta Math. Sci. 1, 83–85 (1981)

    MathSciNet  MATH  Google Scholar 

  3. Mori, M., Nurmuhammad, A., Murai, T.: Numerical solution of Volterra integral equations with weakly singular kernel based on the DE-sinc method. Jpn. J. Ind. Appl. Math. 25, 165–183 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Mori, M., Sugihara, M.: The double-exponential transformation in numerical analysis. J. Comput. Appl. Math. 127, 287–296 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Muhammad, M., Mori, M.: Double exponential formulas for numerical indefinite integration. J. Comput. Appl. Math. 161, 431–448 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Okayama, T., Matsuo, T., Sugihara, M.: Improvement of a Sinc-collocation method for Fredholm integral equations of the second kind. BIT Numer. Math. 51, 339–366 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Okayama, T., Matsuo, T., Sugihara, M.: Error estimates with explicit constants for Sinc approximation, Sinc quadrature and Sinc indefinite integration. Numer. Math. (2013). doi:10.1007/s00211-013-0515-y

  8. Schwartz, C.: Numerical integration of analytic functions. J. Comput. Phys. 4, 19–29 (1969)

    Article  MATH  Google Scholar 

  9. Squire, W.: Numerical evaluation of integrals using Moran transformations. Aerosp. Eng. Report No. Tr-14 (1969)

  10. Stenger, F.: Optimal convergence of minimum norm approximations in $H_p$. Numer. Math. 29, 345–362 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  11. Stenger, F.: Numerical Methods Based on Sinc and Analytic Functions. Springer, New York (1993)

    Book  MATH  Google Scholar 

  12. Stenger, F.: Summary of Sinc numerical methods. J. Comput. Appl. Math. 121, 379–420 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Sugihara, M.: Optimality of the double exponential formula—functional analysis approach. Numer. Math. 75, 379–395 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Sugihara, M.: Near optimality of the sinc approximation. Math. Comput. 72, 767–786 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Sugihara, M., Matsuo, T.: Recent developments of the Sinc numerical methods. J. Comput. Appl. Math. 164–165, 673–689 (2004)

    Article  MathSciNet  Google Scholar 

  16. Takahasi, H., Mori, M.: Quadrature formulas obtained by variable transformation. Numer. Math. 21, 206–219 (1973–1974)

    Google Scholar 

  17. Takahasi, H., Mori, M.: Double exponential formulas for numerical integration. Publ. Res. Inst. Math. Sci. Kyoto Univ. 9, 721–741 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  18. Tanaka, K., Sugihara, M., Murota, K., Mori, M.: Function classes for double exponential integration formulas. Numer. Math. 111, 631–655 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

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.

Author information

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
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Ken’ichiro Tanaka
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Takayasu Matsuo
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Masaaki Sugihara
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Tomoaki Okayama.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

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

Mathematics Subject Classification (2000)