Efficient and Accurate Algorithms for the Computation and Inversion of the Incomplete Gamma Function Ratios

@article{Gil2012EfficientAA,
  title={Efficient and Accurate Algorithms for the Computation and Inversion of the Incomplete Gamma Function Ratios},
  author={Amparo Gil and J. Segura and N. Temme},
  journal={SIAM J. Sci. Comput.},
  year={2012},
  volume={34}
}
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… Expand
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References

SHOWING 1-10 OF 17 REFERENCES
Asymptotic inversion of incomplete gamma functions
The normalized incomplete gamma functions P(a, x) and Q(a, x) are inverted for large values of the parameter a. That is, x-solutions of the equations P(a, x) = p, Q(a, x) = q, p E [O, 1], q = I -p,Expand
On the calculation of the inverse of the error function
Formulas are given for computing the inverse of the error function to at least 18 significant decimal digits for all possible arguments up to 1-10-300 in magnitude. A formula which yields erf (x) toExpand
A Computational Procedure for Incomplete Gamma Functions
We develop a computational procedure, based on Taylor's series and continued fractions, for evaluating Tncomi's incomplete gamma functmn 7*(a, x) = (x-"/F(a))S~ e-~t'-ldt and the complementaryExpand
Computation of the incomplete gamma function ratios and their inverse
TLDR
An algorithm, employing third-order Schröder iteration supported by Newton-Raphson iteration, is provided for computing the incomplete gamma function ratios, using Temme's uniform asymptotic expansions. Expand
Numerical methods for special functions
TLDR
This book provides an up-to-date overview of methods for computing special functions and discusses when to use them in standard parameter domains, as well as in large and complex domains. Expand
Numerical Aspects of Special Functions
This paper describes methods that are important for the numerical evaluation of certain functions that frequently occur in applied mathematics, physics and mathematical statistics. This includes whatExpand
The asymptotic expansion of the incomplete gamma functions : (preprint)
Earlier investigations on uniform asymptotic expansions of the incomplete gamma functions are reconsidered. The new results include estimations for the remainder and the extension of the results toExpand
Rational Chebyshev approximations for the inverse of the error function
This report presents near-minimax rational approximations for the inverse of the error function inverf x, for 0 S x S 1-1
Capacity of Multi-antenna Gaussian Channels
  • E. Telatar
  • Computer Science
  • Eur. Trans. Telecommun.
  • 1999
TLDR
The use of multiple transmitting and/or receiving antennas for single user communications over the additive Gaussian channel with and without fading is investigated, and formulas for the capacities and error exponents are derived. Expand
Error functions
  • Dawson’s and Fresnel integrals. In NIST handbook of mathematical functions, pages 159–171. U.S. Dept. Commerce, Washington, DC
  • 2010
...
1
2
...