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[Help-gsl] Bessel functions and continued fractions


From: Jonny Taylor
Subject: [Help-gsl] Bessel functions and continued fractions
Date: Thu, 29 Jul 2010 12:42:57 +0100

Is there anybody where who knows the ins and outs of the Bessel function 
implementation in GSL? Under certain circumstances my code is spending over 90% 
of its time in gsl_sf_bessel_J_CF1. This occurs for large x<1000 and n<50. The 
time taken to evaluate the continued fraction grows with x.

This observation is supported by the discussion here on the implementation in 
the Boost library:
http://www.boost.org/doc/libs/1_38_0/libs/math/doc/sf_and_dist/html/math_toolkit/special/bessel/bessel.html
This suggests that a different continued fraction (CF2) converges much faster 
than CF1 for x>n.

Is there a reason that CF2 is not used in GSL? I don't know about the reasoning 
behind the implementation and the regimes where each of the various evaluation 
methods are used. Is there a reason for not using CF2 in the appropriate 
circumstances, or is it just that nobody has felt the need to implement it up 
to now? I'd be interested to hear peoples thoughts on this - if it's purely 
that nobody has bothered yet I might have a go at implementing it if I get a 
chance.

Thanks for any comments
Jonny


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