Re: [boost] [math] Contribution and TODO-list implementation.

Subject: Re: [boost] [math] Contribution and TODO-list implementation.
From: Bikineev (ant.bikineev_at_[hidden])
Date: 2013-11-06 09:12:46

Christopher Kormanyos <e_float <at> yahoo.com> writes:
 
> 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.