Re: Bessel Functions and Thumbscrews

[PhilipNienhuis]

Robert T. Short wrote
> 
> Following up on a post and bug report several weeks ago.
> 
> I have attached a patch in which I added a bunch of tests to 
> bessel{j,y,i,k}.  The existing tests were fine, but the Amos code 
> (libcruft/amos) uses different algorithms for different orders and 
> arguments and the existing tests were for small values of the order and 
> magnitude.  I created a subset of the tables in Abramowitz and Stegun 
> and compared the octave results to the tables.  In particular for large 
> order/arguments Amos used the asymptotic expansion and so one might 
> expect some differences (and there are).
> 
> I also added tests to cover the bug that set me off on this mission - 
> checking that negative integer orders give the correct answers.
> 
> Bob
> 

(Sorry for reacting a bit late)

The A&S Bessel functions are fairly "rough" approximations. 

Some years back I used Bessel functions in an numerical inverse Laplace
transform (Stehfest) and it turned out that an accuracy of at least 10-12
significant digits was needed to get some stability. The A&S approximations
were insufficiently accurate by far.

I my programs of the time (some 15+ years back) I used (FORTRAN) versions
for Ix and Kx that I based on ALGOL functions from the NUMAL library (CWI,
Amsterdam) from decades ago. I only used real arguments, but perhaps the
principle works for complex arguments as well.

AFAICR my (those) FORTRAN versions were quite a bit more involved than
earlier ones I based on A&S and thus a bit slower (containing a series of
IFs depending on argument magnitude), but MUCH more accurate. A bit of a
trade-off.

Perhaps this NUMAL library can be used for Octave as well. It seems to live
here these days:

http://www.softwarepreservation.org/projects/ALGOL/applications

and the license seems a bit BSD like.
Any license gurus who can shed a light here?

Philip


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