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Verified Error Bounds for the Real Gamma Function Using Double Exponential Formula over Semi-infinite Interval

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Mathematical Aspects of Computer and Information Sciences (MACIS 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9582))

Abstract

An algorithm is presented for computing verified error bounds for the value of the real gamma function. It has been shown that the double exponential formula is one of the most efficient methods for calculating integrals of the form. Recently, an useful evaluation based on the double exponential formula over the semi-infinite interval has been proposed. However, the evaluation would be overflow when applied to the real gamma function directly. In this paper, we present a theorem so as to overcome the problem in such a case. Numerical results are presented for illustrating effectiveness of the proposed theorem in terms of the accuracy of the calculation.

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References

  1. Rump, S.M.: Verified sharp bounds for the real gamma function over the entire floating-point range. Nonlinear Theor. Appl. IEICE 5, 339–348 (2014)

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Author information

Authors and Affiliations

  1. Faculty of Modern Life, Teikyo Heisei University, 4-21-2, Nakano, Nakano, Tokyo, 164-8530, Japan

    Naoya Yamanaka

  2. Graduate School of Information Sciences, Hiroshima City University, 3-4-1, Ozuka-higashi, Asaminami, Hiroshima, 731-3194, Japan

    Tomoaki Okayama

  3. Faculty of Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku, Tokyo, 169-8555, Japan

    Shin’ichi Oishi

  4. CREST, Japan Science and Technology Agency, 4-1-8, Honcho, Kawaguchi, Saitama, 332-0012, Japan

    Naoya Yamanaka & Shin’ichi Oishi

Authors
  1. Naoya Yamanaka
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  2. Tomoaki Okayama
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  3. Shin’ichi Oishi
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Corresponding author

Correspondence to Naoya Yamanaka .

Editor information

Editors and Affiliations

  1. Wilfrid Laurier University, Waterloo, Ontario, Canada

    Ilias S. Kotsireas

  2. Hamburg University of Technology, Hamburg, Germany

    Siegfried M. Rump

  3. New York University, New York, New York, USA

    Chee K. Yap

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© 2016 Springer International Publishing Switzerland

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Yamanaka, N., Okayama, T., Oishi, S. (2016). Verified Error Bounds for the Real Gamma Function Using Double Exponential Formula over Semi-infinite Interval. In: Kotsireas, I., Rump, S., Yap, C. (eds) Mathematical Aspects of Computer and Information Sciences. MACIS 2015. Lecture Notes in Computer Science(), vol 9582. Springer, Cham. https://doi.org/10.1007/978-3-319-32859-1_19

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  • DOI: https://doi.org/10.1007/978-3-319-32859-1_19

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-32858-4

  • Online ISBN: 978-3-319-32859-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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