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Dear GSL developers,
I found a bug in the function gsl_sf_coulomb_wave_F_array which uses the recurrence relation to obtain F for different l's.
To be short, when eta is large, and l is at least moderately large, inaccuracy occurs in the implementation of the recurrence such that F has unphysical spikes at certain x for smaller l's.
My observation is that F is determined directly for the largest l, and F for smaller l's are determined from the recurrence relation.
When l is large, the number of recurrence is large to obtain F for small l's.
Therefore, errors in F's get accumulated as we proceed to get F for small l's.
Attached please find a test program.
The program uses two methods to obtain F for different l's and x's.
The first method obtains F directly, and the second method obtains F from recurrence relation.
Please compile it using "g++ testCoulomb.cpp lgsl lgslcblas".
Now run the program using "./a.out > resultCoulomb".
We can now take a look at the output of the program using gnuplot.
After launching gnuplot, just type plot "resultCoulomb" using 1:2 every ::2::2000 to see the output for l=0 at different x's using the first method.
And type plot "resultCoulomb" using 1:2 every ::2002::4000 to see the output for l=0 at different x's using the second method.
Now you can find, F is fine using the first method, but F has unphysical spikes.
You can see similar behaviors for other l's by "using 1:3", "using 1:4" etc in gnuplot.
I can of course always stick to the first method, but the second method is much faster.
So it would be great if you could fix this problem.
