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Computing the Incomplete Gamma Function to Arbitrary Precision

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Computational Science and Its Applications — ICCSA 2003 (ICCSA 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2667))

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Abstract

I consider an arbitrary-precision computation of the incomplete Gamma function from the Legendre continued fraction. Using the method of generating functions, I compute the convergence rate of the continued fraction and find a direct estimate of the necessary number of terms. This allows to compare the performance of the continued fraction and of the power series methods. As an application, I show that the incomplete Gamma function Γ (a, z) can be computed to P digits in at most O(P) long multiplications uniformly in z for Re z > 0. The error function of the real argument, erf x, requires at most O(P 2/3) long multiplications.

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Author information

Authors and Affiliations

  1. Department of Physics, Ludwig-Maximilians University, Theresienstr. 37, 80333, Munich, Germany

    Serge Winitzki

Authors
  1. Serge Winitzki
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Editor information

Editors and Affiliations

  1. Army High Performance Computing Research Center, USA

    Vipin Kumar

  2. Department of Computer Science, University of Calgary, Calgary, AB, T2N1N4, Canada

    Marina L. Gavrilova

  3. Heuchera Technologies Inc., 122 9251-8 Yonge Street, Richmond Hill, ON, Canada, L4C 9T3

    Chih Jeng Kenneth Tan

  4. Département d’informatique et de recherche opérationelle, Université de Montréal, Montréal, Québec, H3C 3J7, Canada

    Pierre L’Ecuyer

  5. Department of Computer Science and Engineering, University of Minessota, MN, 55455, USA

    Vipin Kumar

  6. The Queen’s University of Belfast, School of Computer Science, Belfast BT7 1NN, Northern Ireland, UK

    Chih Jeng Kenneth Tan

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© 2003 Springer-Verlag Berlin Heidelberg

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Winitzki, S. (2003). Computing the Incomplete Gamma Function to Arbitrary Precision. In: Kumar, V., Gavrilova, M.L., Tan, C.J.K., L’Ecuyer, P. (eds) Computational Science and Its Applications — ICCSA 2003. ICCSA 2003. Lecture Notes in Computer Science, vol 2667. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44839-X_83

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  • DOI: https://doi.org/10.1007/3-540-44839-X_83

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40155-1

  • Online ISBN: 978-3-540-44839-6

  • eBook Packages: Springer Book Archive

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