Re: [Help-gsl] Question about error margins on Bessel Functions

[Brian Gough]

joe bradley writes:
 > Ok, on to my question:  I'd like to know where I can find documentation on
 > how the error margins are computed for the Bessel functions.  (Specifically,
 > I'm working with the function gsl_sf_bessel_I0_e; one of the so-called
 > regular modified cylindrical bessel functions.)
 > 
 > I've looked a little at the code for the function, but everything is
 > computed using Chebyshev polynomials, which I don't know much about. Is
 > there any kind of documentation _between_ the standard documentation (which
 > doesn't seem to address this kind of question much) and the source code?

Hello,

Thanks for your email.

Beyond the source code and the GSL reference manual the only other
documentation is the references to the original papers, code or books.

In this case the comments in the code point to SLATEC, and the error
bounds are quoted in the original source code there. The SLATEC
routines are generally described in various ACM TOMS papers, e.g. 

Cody, W. J. 1993. Algorithm 715; SPECFUN: a portable FORTRAN package
of special function routines and test drivers. ACM
Trans. Math. Softw. 19, 1 (Mar. 1993), 22-30.

For a Chebyshev approximation the error is bounded by the last term.
The original error bounds would be computed by brute force, comparing
the approximation with high-precision values across the range.  In
this case we are just using the values given by SLATEC since they
should be reliable.

-- 
Brian Gough

Network Theory Ltd,
Publishing the GSL Manual - http://www.network-theory.co.uk/gsl/manual/