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# gammaincinv

Inverse incomplete gamma function

## Syntax

x = gammaincinv(y,a)
x = gammaincinv(y,a,tail)

## Description

x = gammaincinv(y,a) evaluates the inverse incomplete gamma function for corresponding elements of y and a, such that y = gammainc(x,a). The elements of y must be in the closed interval [0,1], and those of a must be nonnegative. y and a must be real and the same size (or either can be a scalar).

x = gammaincinv(y,a,tail) specifies the tail of the incomplete gamma function. Choices are 'lower' (the default) to use the integral from 0 to x, or 'upper' to use the integral from x to infinity.

These two choices are related as:

gammaincinv(y,a,'upper') = gammaincinv(1-y,a,'lower').

When y is close to 0, the 'upper' option provides a way to compute x more accurately than by subtracting y from 1.

## Definitions

The lower incomplete gamma function is defined as:

$\mathrm{gammainc}\left(x,a\right)=\frac{1}{\Gamma \left(a\right)}\underset{0}{\overset{x}{\int }}{e}^{-t}{t}^{\left(a-1\right)}dt$

The upper incomplete gamma function is defined as:

$\mathrm{gammainc}\left(x,a\right)=\frac{1}{\Gamma \left(a\right)}\underset{x}{\overset{\infty }{\int }}{e}^{-t}{t}^{\left(a-1\right)}dt$

gammaincinv computes the inverse of the incomplete gamma function with respect to the integration limit x using Newton's method.

For any a>0, as y approaches 1, gammaincinv(y,a) approaches infinity. For small x and a, gammainc(x,a)$\cong {x}^{a}$, so gammaincinv(1,0) = 0.

## References

[1] Cody, J., An Overview of Software Development for Special Functions, Lecture Notes in Mathematics, 506, Numerical Analysis Dundee, G. A. Watson (ed.), Springer Verlag, Berlin, 1976.

[2] Abramowitz, M. and I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series #55, Dover Publications, 1965, sec. 6.5.