gammaincinv

The inverse of the incomplete gamma function

Definition

Given two real values $y\in[0,1]$ and $a>0$, gammaincinv(y,a,tail) gives the value $x$ such that \[P(x,a) = y,\quad\text{ or }\quad Q(x,a) = y\] depending on the value of tail ("lower" or "upper").

Computation

The computation is carried out via standard Newton method, solving, for $x$ \[ y - P(x,a) = 0 \quad\text{ or }\quad y - Q(x,a) = 0. \] The Newton method uses as exit criterion a control on the residual and on the maximum number of iterations.
For numerical stability, the initial guess and which gammainc function is inverted depends on the values of $y$ and $a$. We use the following property: \[\text{gammaincinv}(y,a,\text{`upper'})=\text{gammaincinv}(1-y,a,\text{`lower'})\] which follows immediately by the property \[P(x,a) + Q(x,a) = 1.\] We start computing the trivial values, i.e. $a=1,y=0$ and $y=1$. Then we rename the variables as follows (in order to obtain a more classical notation): if tail = "lower", then $p=y,q=1-y$; while if tail = "upper", $q=y,p=1-y$. We will invert with $p$ as $y$ in the cases in which we invert the lower incomplete gamma function, and with $q$ as $y$ in the cases in which we invert the upper one.
Now, the initial guesses (and the relative version of the incomplete gamma function to invert) are the following:

1. $p<\dfrac{(0.2(1+a))^a}{\Gamma(1+a)}$. We define \[r = (p\,\Gamma(1+a))^{\frac{1}{a}}\] and the parameters \begin{align*} c_2 &= \dfrac{1}{a+1}\\ c_3 &= \dfrac{3a+5}{2(a+1)^2(a+2)}\\ c_4 &= \dfrac{8a^2+33a+31}{3(a+1)^3(a+2)(a+3)}\\ c_5 &= \dfrac{125a^4+1179a^3+3971a^2+5661a+2888}{24(1+a)^4(a+2)^2(a+3)(a+4)}\\ \end{align*} Then the initial guess is \[x_0 = r + c_2\,r^2+c_3\,r^3+c_4\,r^4+c_5\,r^5\] and we invert the lower incomplete gamma function.
2. $q<\dfrac{e^{-a/2}}{\Gamma(a+1)}, a\in(0,10)$. The initial guess is \[x_0 = - \log (q) - \log(\Gamma(a))\] and we invert the upper incomplete gamma function.
3. $a\in(0,1)$. The initial guess is \[x_0 = (p\Gamma(a+1))^{\frac{1}{a}}\] and we invert the lower incomplete gamma function.
4. Other cases. The initial guess is \[x_0 = a\,t^3\] where \[t = 1-d-\sqrt{d}\,\text{norminv}(q),\quad d=\dfrac{1}{9a}\] (where norminv is the inverse of the normal distribution function) and we invert the upper incomplete gamma function.

The first three initial guesses can be found in the article "Efficient and accurate algorithms for the computation and inversion of the incomplete gamma function rations" by Gil, Segura ad Temme; while the last one can be found in the documentation of the function igami.c of the cephes library.
You can find my work on my repository on bitbucket here, bookmark "gammainc".

Comments

Working on this function I found some problems in gammainc. To solve them, I just changed the flag for the condition 5 and added separate functions to treat Inf cases.