## Bessel functions

During the second part of GSoC I have to work on Bessel functions. The bug is related in particular on the Bessel function of first kind and regard the fact that the function is not computed if the $x$ argument is too big.
During this week I studied the actual implementation i.e. the Amos library, written in Fortran. The problem is related to the fact that zbesj.f simply refuse to compute the result when the input argument is too large (the same problem happens for all bessel functions, both for real and complex arguments). What I did was to "unlock" the amos in such a way to compute the value in any case (which seems to be the strategy used by MATLAB). Then I compared the relative errors with the relative errors done by the Gnu Scientific Library (GSL).
At first, I would precise some limitation present in GSL:
• The parameter $x$ must be real, while it can be complex in Amos and MATLAB. The same holds for the parameter $\alpha$.
• The class of the output does not adapt to the class of the input, returning always a double.
Doing some tests, it seems that the amos work better in terms of accuracy (even if we are talking about errors which do not differ in the order). I concentrated on values in which Amos usually refuse to compute, since in every other zone of the $x-\alpha$ plane it is known the error is in the order of 1e-15. For $\alpha\in\{-1,0,1\}$ they return the same result, while for other values of $\alpha$, Amos are in general more accurate.
Anyway, I would remark the fact that there are not, as far as I know, accurate algorithms in double precision for very large magnitude. In fact, for such arguments, both Amos and GSL make a relative error of the order of the unity. This problem is evident when using SageMath to compute accurate values, e.g.
sage: bessel_J(1,10^40).n()
7.95201138653707e-21
sage: bessel_J(1,10^40).n(digits=16)
-7.522035532561300e-21
sage: bessel_J(1,10^40).n(digits=20)
-5.7641641376196142462e-21
sage: bessel_J(1,10^40).n(digits=25)
1.620655257788838811378880e-21
sage: bessel_J(1,10^40).n(digits=30)
1.42325562284462027823158499095e-21
sage: bessel_J(1,10^40).n(digits=35)
1.4232556228446202782315849909515310e-21
sage: bessel_J(1,10^40).n(digits=40)
1.423255622844620278231584990951531026335e-21
The values "stabilize" only when the number of digits is bigger than 30, far away from double precision.

## Incomplete gamma function

I also gave a look on how to use GSL to eventually improve the incomplete gamma function. It is not possible, however, to use only GSL functions gsl_sf_gamma_inc_P and gsl_sf_gamma_inc_Q due to their limitations:
• There is no the "scaled" option.
• The value of $x$ must be real. This may be not a problem since even MATLAB does not accept complex value (while the version I worked on does).
• The parameter $x$ must be positive. This is actually a problem of MATLAB compatibility.
• The class of the output does not adapt to the class of the input, returning always a double.
I tested the accuracy of the GSL and they work better when $a<1$ and $x\ll1$ so I think I will fix gammainc.m using the algorithm of gsl_sf_gamma_inc_P and gsl_sf_gamma_inc_Q for that values of the input arguments.

## Incomplete Beta function

The actual incomplete beta function needs to be replaced. I've already studied the continued fraction exapnsion (which seems to be the best one to use). In GSL is implemented in a good way but still present two limitations:
• Does not adapt to the input class (it always work in double precision).
• There is no "upper" version. This is missing also in the current betainc function, but it is present in MATLAB. Unfortunately, it is not sufficient to compute it as $1 - I_x(a,b)$, it is necessary to find an accurate way to compute directly it.
So I will take inspiration to GSL version of the function but I think I will write beatinc as a single .m file.