[Bug-gsl] [bug #37209] gsl_sf_bessel_jl_e returns NaN for large inputs without setting error code

[Patrick Alken]

Follow-up Comment #2, bug #37209 (project gsl):

The two issues below are in fact separate bugs in the spherical bessel code.

The bug report from 28 Aug 2012 involves an instability condition in the
downward recurrence relation after computing the CF1 continued fraction. This
instability appears when x is small relative to l. There is a short discussion
of this in Numerical Recipes, but I have been unable to find a fully
acceptable solution in the literature. Many other bessel codes are susceptable
to this same bug (including NR and Barnett's SBESJY.F routine).

I was able to fix this particular test case by starting with a smaller initial
value to the recurrence relation, which delays (but does not completely fix)
the overflow condition. The code now works for l = 364, x = 36.62, however
there are probably other combinations of large l, small x which will cause an
overflow in the downward recurrence.

This is partially fixed in rev 4850.

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The bug report from 23 Sep 2013 involves a different branch of the code and
the gsl_sf_bessel_IJ_taylor_e function - this is still open.

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