Inverse incomplete gamma function
x = gammaincinv(y,a)
y = gammaincinv(x,a,tail)
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).
y = gammaincinv(x,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.
The lower incomplete gamma function is defined as:
The upper incomplete gamma function is defined as:
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) , so gammaincinv(1,0) = 0.
 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.
 Abramowitz, M. and I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series #55, Dover Publications, 1965, sec. 6.5.