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Algorithms for the numerical evaluation of the incomplete gamma functionratios $P(a,x)=\gamma(a,x)/\Gamma(a)$ and $Q(a,x)=\Gamma(a,x)/\Gamma(a)$ aredescribed for positive values of $a$ and $x$. Also, inversion methods are givenfor solving the equations $P(a,x)=p$, $Q(a,x)=q$, with $0<p,q<1$. Both thedirect computation and the inversion of the incomplete gamma function ratiosare used in many problems in statistics and applied probability. The analyticalapproach from earlier literature is summarized and new initial estimates arederived for starting the inversion algorithms. The performance of theassociated software to our algorithms (the Fortran 90 module {\bf IncgamFI}) isanalyzed and compared with earlier published algorithms.

Efficient and accurate algorithms for the computation and inversion of the incomplete gamma function ratios