Christopher Kormanyos
Please be advised that writing any research thesis and contributing to Boost is a very individual effort.
John mentioned that we can help in some way, but quite honestly, you would be researching and writing independently for long stretches because we simply do not have the time for full-time research advisers.
Yes, of course. I didn't mean that research and writing should be collective work. It would be impudent on my part.
It is also always a good idea to round out any thesis or research work with practical examples. So in your thesis, you should also include some applications.
One application might be, for example, using Bessel function derivatives to assist in the computation of certain zeros of Bessel functions. When we added the zeros of Bessel functions last year, we used a "poor man's" derivative calculation in some expansion regions via a trivial recursion relation --- which is slow because it requires the calculation of multiple Bessel functions. Having "native" derivatives of Bessel functions could improve these calculations. And this would be one of your examples, and a further contribution to the code.
You should also seek out one or two other examples such as special functions expanded in Bessel derivatives, and also extend these to multiprecision.
You might also consider including in some way an investigation of the basic tenets of generic numeric programming with Boost.Math and Boost.Multiprecision. Although this is a research topic that can stand alone, you might consider include it in something like a final chapter or an appendix. This would pave the way for another research paper wholly dedicated to generic numeric programing in C++ with Boost.
Thank you so much for ideas! I will follow them.
Having "native" derivatives of Bessel functions could improve these calculations.
Does it mean that I should calculate Bessel functions derivatives with no using Bessel functions? I mean formulas like in http://functions.wolfram.com/Bessel-TypeFunctions/BesselK/20/01/02/ If no, what is the most suitable formula for calculating derivatives? For example, it seems we should always check x value for zero using first or second formula. And one more question. How to generate test input values (e.g. bessel_j_data.ipp) and their results calculated on functions.wolfram.com? Or should they be written manually? Should these values (first two) be the same for functions and their derivatives? Thanks.