Algorithms for the numerical evaluation of the incomplete gamma function ratios $P(a,x)=\gamma(a,x)/\Gamma(a)$ and $Q(a,x)=\Gamma(a,x)/\Gamma(a)$ are described for positive values of $a$ and $x$. Also, inversion methods are given for solving the equations $P(a,x)=p$, $Q(a,x)=q$, with $0<p,q<1$. Both the direct computation and the inversion of the incomplete gamma function ratios are used in many problems in statistics and applied probability. The analytical approach from earlier literature is summarized and new initial estimates are derived for starting the inversion algorithms. The performance of the associated software to our algorithms (the Fortran 90 module {\bf IncgamFI}) is analyzed and compared with earlier published algorithms.

Incomplete gamma function ratios, chi-squared distribution function, inversion of incomplete gamma functions, numerical evaluation of special functions, asymptotic analysis.
Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals) (msc 33B20)
SIAM Journal on Scientific Computing
Computational Dynamics

Gil, A, Segura, J, & Temme, N.M. (2012). Efficient and accurate algorithms for the computation and inversion of the incomplete gamma function ratios. SIAM Journal on Scientific Computing, 0, 1–17. doi:10.1137/120872553