Computing the Incomplete Gamma Function to Arbitrary Precision

  • Serge Winitzki
Conference paper

DOI: 10.1007/3-540-44839-X_83

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

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(P2/3) long multiplications.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Serge Winitzki
    • 1
  1. 1.Department of PhysicsLudwig-Maximilians UniversityMunichGermany

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