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

TL;DR

This paper presents efficient algorithms for computing and inverting incomplete gamma function ratios, improving accuracy and performance for statistical and probabilistic applications.

Contribution

It introduces new algorithms and initial estimates for the computation and inversion of incomplete gamma function ratios, with software implementation and performance analysis.

Findings

Algorithms outperform previous methods in accuracy and speed.

New initial estimates improve convergence of inversion algorithms.

Software IncgamFI demonstrates competitive performance.

Abstract

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.