Final Resume

During the GSoC I worked on different special functions that needed to be improved or implemented from scratch. Discussing with my mentors and the community, we decided that my work should be pushed on a copy of the scource code of Octave on my repository [1] and then I should have work with different bookmarks for each function I had to work on. When different functions happened to be related (e.g. gammainc and gammaincinv), I worked on these on the same bookmark. I present now a summary and the bookmarks related to the functions.

Incomplete gamma function

bookmark: gammainc
first commit: d1e03faf080b
last commit: 107dc1d24c1b
added files: /libinterp/corefcn/__gammainc_lentz__.cc, /scripts/specfun/gammainc.m, /scripts/specfun/gammaincinv.m
removed files:/libinterp/corefcn/gammainc.cc, /liboctave/external/slatec-fn/dgami.f, /liboctave/external/slatec-fn/dgamit.f, /liboctave/external/slatec-fn/gami.f, /liboctave/external/slatec-fn/gamit.f, /liboctave/external/slatec-fn/xdgami.f, /liboctave/external/slatec-fn/xdgamit.f, /liboctave/external/slatec-fn/xgmainc.f, /liboctave/external/slatec-fn/xsgmainc.f
modified files: NEWS, /doc/interpreter/arith.txi, /libinterp/corefcn/module.mk, /liboctave/external/slatec-fn/module.mk, /liboctave/numeric/lo-specfun.cc, /scripts/specfun/module.mk

Summary of the work

On this bookmark I worked on the incomplete gamma function and its inverse.
The incomplete gamma function gammainc had both missing features (it were missed the "scaled" options) and some problem of inaccurate result type (see bug # 47800). Part of the work was already been done by Marco and Nir, I had to finish it. We decided to implement it as a single .m file (gammainc.m) which call (for some inputs) a subfunction written in C++ (__gammainc_lentz__.cc).
The inverse of the incomplete gamma function was missing in Octave (see bug # 48036). I implemented it as a single .m file (gammaincinv.m) which uses a Newton method.

Bessel functions

bookmark: bessel
first commit: aef0656026cc
last commit: e9468092daf9
modified files: /liboctave/external/amos/README, /liboctave/external/amos/cbesh.f, /liboctave/external/amos/cbesi.f, /liboctave/external/amos/cbesj.f, /liboctave/external/amos/cbesk.f, /liboctave/external/amos/zbesh.f, /liboctave/external/amos/zbesi.f, /liboctave/external/amos/zbesj.f, /liboctave/external/amos/zbesk.f, /liboctave/numeric/lo-specfun.cc, /scripts/specfun/bessel.m

Summary of the work

On this bookmark I worked on Bessel functions.
There was a bug reporting NaN as output when the argument $x$ was too large in magnitude (see bug # 48316). The problem was given by Amos library, which refuses to compute the output in such cases. I started "unlocking" this library, in such a way to compute the output even when the argument was over the limit setted by the library. Then I compared the results with other libraries (e.g. Cephes [2], Gnu Scientific library [3] and C++ special function library [4]) and some implementations I made. In the end, I discovered that the "unlocked" Amos were the best one to use, so we decided to maintain them (in the "unlocked" form), modifying the error variable to explain the loss of accuracy.

Incomplete beta function

bookmark: betainc
first commit: 712a069d2860
last commit: e0c0dd40f096
added files: /libinterp/corefcn/__betainc_lentz__.cc, /scripts/specfun/betainc.m, /scripts/specfun/betaincinv.m
removed files: /libinterp/corefcn/betainc.cc, /liboctave/external/slatec-fn/betai.f, /liboctave/external/slatec-fn/dbetai.f, /liboctave/external/slatec-fn/xbetai.f, /liboctave/external/slatec-fn/xdbetai.f
modified files: /libinterp/corefcn/module.mk, /liboctave/external/slatec-fn/module.mk, /liboctave/numeric/lo-specfun.cc, /liboctave/numeric/lo-specfun.h, /scripts/specfun/module.mk, /scripts/statistics/distributions/betainv.m, /scripts/statistics/distributions/binocdf.m

Summary of the work

On this bookmark I worked on the incomplete beta function and its inverse.
The incomplete beta function missed the "upper" version and had reported bugs on input validation (see bug # 34405) and inaccurate result (see bug # 51157). We decided to rewrite it from scratch. It is now implemented ad a single .m file (betainc.m) which make the input validation part, then the output is computed using a continued fraction evaluation, done by a C++ function (__betainc_lentz__.cc).
The inverse was present in Octave but missed the "upper" version (since it was missing also in betainc itself). The function is now written as a single .m file (betaincinv.m) which implement a Newton method where the initial guess is computed by few steps of bisection method.

Integral functions

bookmark: expint
first commit: 61d533c7d2d8
last commit: d5222cffb1a5
added files:/libinterp/corefcn/__expint_lentz__.cc, /scripts/specfun/cosint.m, /scripts/specfun/sinint.m
modified files: /doc/interpreter/arith.txi, /libinterp/corefcn/module.mk, /scripts/specfun/expint.m, /scripts/specfun/module.mk

Summary of the work

On this bookmark I worked on exponential integral, sine integral and cosine integral. I already rewrote the exponential integral before the GSoC. Here I just moved the Lentz algorithm to an external C++ function (__expint_lentz__.cc), accordingly to gammainc and betainc. I've also modified the exit criterion for the asymptotic expansion using [5] (pages 1 -- 4) as reference.
The functions sinint and cosint were present only in the symbolic package of Octave but was missing a numerical implementation in the core. I wrote them as .m files (sinint.m and cosint.m). Both codes use the series expansion near the origin and relations with expint for the other values.

To do

There is still room for improvement for some of the functions I wrote. In particular, gammainc can be improved in accuracy for certain couple of values, and I would like to make a template version for the various Lentz algorithms in C++ so to avoid code duplication in the functions.
In October I will start a PhD in Computer Science, still here in Verona. This will permit me to remain in contact with my mentor Marco Caliari, so that we will work on these aspects.

[1] https://bitbucket.org/M_Ginesi/octave
[2] http://www.netlib.org/cephes/
[3] https://www.gnu.org/software/gsl/
[4] http://en.cppreference.com/w/cpp/numeric/special_math
[5] N. Bleistein and R.A. Handelsman, "Asymptotic Expansions of Integrals", Dover Publications, 1986.

Integral functions

During the last week I made few modifications to expint.m and wrote sinint.m and cosint.m from scratch. All the work done can be found on the bookmark expint of my repository.

expint

As I mentioned here I rewrote expint.m from scratch before the GSoC. During the last week I moved the Lentz algorithm to a .cc function (in order to remain coherent with the implementations of gammainc and betainc) and added few tests.

sinint

The sinint function is present in the symbolic package, but is not present a numerical implementation in the core.
The sine integral is defined as $$ \text{Si} (z) = \int_0^z \frac{\sin(t)}{t}\,dt. $$ To compute it we use the series expansion $$ \text{Si}(z) = \sum_{n=0}^\infty \frac{(-1)^n z^{2n+1}}{(2n+1)(2n+1)!} $$ when the module of the argument is smaller than 2. For bigger values we use the following relation with the exponential integral $$ \text{Si} = \frac{1}{2i} (E_1(iz)-E_1(-iz)) + \frac{\pi}{2},\quad |\text{arg}(z)| < \frac{\pi}{2}$$ and the following simmetry relations $$ \text{Si}(-z) = -\text{Si}(z), $$ $$ \text{Si}(\bar{z}) = \overline {\text{Si}(z)}. $$ The function is write as a single .m file.

cosint

As the sinint function, also cosint is present in the symbolic package, but there is not a numerical implementation in the core.
The cosine integral is defined as $$ \text{Ci} (z) = -\int_z^\infty \frac{\cos(t)}{t}\,dt. $$ An equivalent definition is $$ \text{Ci} (z) = \gamma + \log z + \int_0^z \frac{\cos t - 1}{t}\,dt. $$ To compute it we use the series expansion $$ \text{Ci}(z) = \gamma + \log z + \sum_{n=1}^\infty \frac{(-1)^n z^{2n}}{(2n)(2n)!} $$ when the module of the argument is smaller than 2. For bigger values we use the following relation with the exponential integral $$ \text{Ci} = -\frac{1}{2} (E_1(iz)+E_1(-iz)),\quad |\text{arg}(z)| < \frac{\pi}{2}$$ and the following simmetry relations $$ \text{Ci}(-z) = \text{Ci}(z) -i\pi,\quad 0<\text{arg}(z)<\pi, $$ $$ \text{Ci}(\bar{z}) = \overline{\text{Ci}(z)} .$$ As for sinint, also cosint is written as a single .m file.

betaincinv

The inverse of the incomplete beta function was present in Octave, but without the "upper" option (since it was missing in betainc itself). We decided to rewrite it from scratch using Newton method, as for gammaincinv (see my post on it if you are interested).
To make the code numerically more accurate, we decide which version ("lower" or "upper") invert depending on the inputs.
At first we compute the trivial values (0 and 1). Then the remaining terms are divided in two sets: those that will be inverted with the "lower" version, and those that will be inverted with the "upper" one. For both cases, we perform 10 iterations of bisection method and then we perform a Newton method.
The implementation (together with the new implementation of betainc) can be found on my repository, bookmark "betainc".

betainc

The betainc function has two bugs reported: #34405 on the input validation and #51157 on inaccurate result. Moreover, it is missing the "upper" version, which is present in MATLAB.

The function

The incomplete beta function ratio is defined as $$I_x(a,b) = \dfrac{B_x(a,b)}{B(a,b)},\quad 0\le x \le 1,\,a>0,\,b>0,$$ where $B(a,b)$ is the classical beta function and $$B_x(a,b)=\int_0^x t^{a-1}(1-t)^{b-1}\,dt.$$ In the "upper" version the integral goes from $x$ to $1$. To compute this we will use the fact that $$\begin{array}{rcl} I_x(a,b) + I_x^U(a,b) &=& \dfrac{1}{B(a,b)}\left( \int_0^x t^{a-1}(1-t)^{b-1}\,dt + \int_x^1 t^{a-1}(1-t)^{b-1}\,dt\right)\\ &=&\dfrac{1}{B(a,b)}\int_0^1 t^{a-1}(1-t)^{b-1}\,dt\\ &=&\dfrac{B(a,b)}{B(a,b)}\\ &=&1 \end{array}$$ and the relation $$I_x(a,b) + I_{1-x}(b,a) = 1$$ so that $$I_x^U(a,b) = I_{1-x}(b,a).$$

The implementation

Even if it is possible to obtain a Taylor series representation of the incomplete beta function, it seems to not be used. Indeed the MATLAB help cite only the continuous fraction representation present in "Handbook of Mathematical Functions" by Abramowitz and Stegun: $$I_x(a,b) = \dfrac{x^a(1-x)^b}{aB(a,b)}\left(\dfrac{1}{1+} \dfrac{d_1}{1+} \dfrac{d_2}{1+}\ldots\right)$$ with $$d_{2m+1} = -\dfrac{(a+m)(a+b+m)}{(a+2m)(a+2m+1)}x$$ and $$d_{2m} = \dfrac{m(b-m)}{(a+2m-1)(a+2m)}x$$ which seems to be the same strategy used by GSL. To be more precise, this continued fraction is computed directly when $$x<\dfrac{a-1}{a+b-2}$$ otherwise, the computed fraction is used to compute $I_{1-x}(b,a)$ and then it is used the fact that $$I_x(a,b) = 1-I_{1-x}(b,a).$$ In my implementation I use a continued fraction present in "Handboob of Continued Fractions for Special Functions" by Cuyt, Petersen, Verdonk, Waadeland and Jones, which is more complicated but converges in fewer steps: $$\dfrac{B(a,b)I_x(a,b)}{x^a(1-x)^b} = \mathop{\huge{\text{K}}}_{m=1}^\infty \left(\dfrac{\alpha_m(x)}{\beta_m(x)}\right),$$ where $$\begin{array}{rcl} \alpha_1(x) &=&1,\\ \alpha_{m+1}(x) &=&\dfrac{(a+m-1)(a+b+m-1)(b-m)m}{(a+2m-1)^2}x^2,\quad m\geq 1,\\ \beta_{m+1}(x) &=&a + 2m + \left( \dfrac{m(b-m)}{a+2m-1} - \dfrac{(a+m)(a+b+m)}{a+2m+1} \right)x,\quad m\geq 0. \end{array}$$ This is most useful when $$x\leq\dfrac{a}{a+b},$$ thus, the continued fraction is computed directly when this condition is satisfied, while it is used to evaluate $I_{1-x}(b,a)$ otherwise.
The function is now written as a .m file, which check the validity of the inputs and divide the same in the values which need to be rescaled and in those wo doesn't need it. Then the continued fraction is computed by an external .c function. Finally, the .m file explicit $I_x(a,b)$.

betaincinv

Next step will be to write the inverse. It was already present in Octave, but is missing the upper version, so it has to be rewritten.