Computing the incomplete Gamma function to arbitrary precision

Authors:
Serge Winitzki
Affiliations:
Department of Physics, Ludwig-Maximilians University, Munich, Germany
Venue:
ICCSA'03 Proceedings of the 2003 international conference on Computational science and its applications: PartI
Year:
2003

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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(P2/3) long multiplications.