Computation of the incomplete gamma function for negative values of the argument

TL;DR

This paper presents a comprehensive algorithm for accurately computing the incomplete gamma function for negative argument values using various mathematical techniques, implemented in a Fortran 90 module.

Contribution

It introduces a new algorithm combining multiple expansion methods for reliable computation of the incomplete gamma function with negative arguments.

Findings

Achieves relative accuracy of ~10^{-13} in specified parameter region.

Provides a Fortran 90 module implementing the algorithm.

Covers a broad parameter range for real values of a and negative z.

Abstract

An algorithm for computing the incomplete gamma function $\gamma^*(a,z)$ for real values of the parameter $a$ and negative real values of the argument $z$ is presented. The algorithm combines the use of series expansions, Poincar\'e-type expansions, uniform asymptotic expansions and recurrence relations, depending on the parameter region. A relative accuracy $\sim 10^{-13}$ in the parameter region $(a,z) \in [-500,\,500] \times [-500,\,0)$ can be obtained when computing the function $\gamma^*(a,z)$ with the Fortran 90 module IncgamNEG implementing the algorithm.