Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

Numerical evaluation of the Gauss hypergeometric function by power summationsHTML articles powered by AMS MathViewer

by Jurgen A. Doornik
Math. Comp. 84 (2015), 1813-1833 Request permission

Abstract:

Numerical evaluation of the Gauss hypergeometric function ${}_2F_1(a,b;c;z)$, with complex parameters $a,b,c$ and complex argument $z$ is notoriously difficult. Carrying out the summation that defines the function may fail, even for moderate values of $z$. Formulae are available to transform the effective argument in the series, potentially leading to a numerically successful summation. Unfortunately, these transformations have a singularity when $b-a$ or $c-a-b$ is an integer, and suffer numerical instability near that. This singularity has to be removed analytically after collecting powers in $z$.

The contributions in this paper are fourfold. First, analytical expressions are provided that remove the singularity from Bühring’s $1/(z-z_0)$ transformation. This is more difficult, because the singularity occurs twice, and it is necessary to collect powers of $z_0$, as well as $z$. The resulting expression has a three-term recursion, like the original. Next, improved expressions are derived for the cases that have been addressed before. We study a transformation that converges outside $|z-0.32| > 0.32$ for ${\mathcal {R}}z>0$, which is tighter than the $|z-0.5| > 0.5$ which is normally considered. Finally, we derive an improved algorithm for the numerical evaluation of ${}_2F_1$.

References
Similar Articles
• Retrieve articles in Mathematics of Computation with MSC (2010): 33C05, 65D20
• Retrieve articles in all journals with MSC (2010): 33C05, 65D20