sabato 19 agosto 2017

Integral functions

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.

sabato 12 agosto 2017

betaincinv

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".

mercoledì 2 agosto 2017

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.