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

Abstract

An algorithm for computing the incomplete gamma function ${\gamma }^{\ast }(a,z)$ 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, Poincaré-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 ${\gamma }^{\ast }(a,z)$ with the Fortran 90 module IncgamNEG implementing the algorithm.

To appear in ACM Trans. Math. Softw