## martedì 20 giugno 2017

### Timetable: modification

Timetable: modification

# Timetable: modification

According to my timetable (that you can find here), during this last week of June, I should've work on the input validation of betainc. Since a new bug related to this function has been found and, moreover, the actual implementation doesn't accept the "lower" or "upper" tail (as MATLAB do), me and my mentor decided to use this week to start studying how to rewrite betainc (main references will be [1] and [2]) and to use the last part fo the GSoC to actually implement it. In this way, my timetable remain almost identical (I will use July to work on Bessel functions) and I will be able to fix also this problem.

[1] Abramowitz, Stegun "Handbook of Mathematical Functions"
[2] Cuyt, Brevik Petersen, Vendonk, Waadeland "Handbook of Continued Fractions for Special Functions"

## venerdì 16 giugno 2017

### gammaincinv

The gammaincinv function

# 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".

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.

## martedì 6 giugno 2017

### gammainc

The gammainc function

# The incomplete gamma function

## Definition

There are actually four types of incomplete gamma functions:
• lower (which is the default one) $$P(a,x) = \frac{1}{\Gamma(a)}\int_{0}^{x}e^{-t}t^{a-1}$$
• upper $$Q(a,x) = \frac{1}{\Gamma(a)}\int_{x}^{+\infty}t^{a-1}e^{-t}dt$$
• scaled lower and scaled upper which return the corresponding incomplete gamma function scaled by $$\Gamma(a+1)\frac{e^x}{x^a}$$

## Computation

To compute the incomplete gamma function, there is a total of seven strategies depending on the position of the arguments in the $x-a$ plane.
1. $x=0,a=0$. In this case, the output is given as follows:
• for "lower" and "scaledlower" is $1$
• for "upper" and "scaledupper" is $0$
2. $x=0,a\neq0$. In this case:
• for "lower" is $0$
• for "upper" and "scalelower" is $1$
• for "scaledupper" is $+\infty$
3. $x\neq0,a=0$. In this case:
• for "lower" is $1$
• for "scalelower" is $e^x$
• for "upper" and "scaledupper" is $0$
4. $x\neq0,a=1$. In this case:
• for "lower" is $1-e^x$
• for "upper" is $e^{-x}$
• for "scaledlower" is $\frac{e^x-1}{x}$
• for "scaledupper" is $\frac{1}{x}$
5. $0< x\leq36,a\in\{2,3,4,\ldots,17,18\}$. In this case we used the following expansion (which holds true only for $n\in\mathbb{N}$): denote $\gamma(a,x):=\Gamma(a)P(a,x)$, then $$\gamma(n,x) = (n-1)!\left( 1-e^{-x}\sum_{k=0}^{n-1} \frac{x^k}{k!}\right).$$
6. $x+0.25< a$ or $x<0$, or $x$ not real or $|x|<1$ or $a<5$. Here we used the expansion $$\gamma(a,x) = e^{-x} x^a \sum_{k=0}^{+\infty} \frac{\Gamma(a)}{\Gamma(a+1+k)}x^k.$$
7. for all the remaining cases was used a continued fraction expansion. Call $\Gamma(a,x) = \Gamma(a) Q(a,x)$, the expansion is $$\Gamma(a,x) = e^{-x}x^a\left( \frac{1}{x+1-a-}\frac{1(1-a)}{x+3-a-}\frac{2(2-a)}{x+5-a-}\cdots \right).$$ This case is handled by an external .cc function.

## My contribution

Actually I dind't participate from the start in the (re)making of this function: most of the work done on it is due to Nir and Marco (see the discussion on the bug tracker).
My contribution was, before the GSoC, to adapt the codes in such a way to make the type of the output (and the tolerances inside the algorithms) coherent on the type of the input (single, int or double). Then I corrected few small bugs and made a patch helped by Carne during the OctConf in Geneve.
During this first week of GSoC I fixed the input validation and added tests to this function, finding some problem in the implementation. Most of them depended on the fact that the continued fractions were used in cases in which they are not optimal. To solve these problems I changed the conditions to use the series expansion instead of the continued fractions.
Now gammainc works properly also with complex argument (for the $x$ variable), while Matlab doesn't accept non-real input.
You can find my work on my repository on bitbucket here, bookmark "gammainc". I will work on this same bookmark the next two weeks while I will work on gammaincinv.