[Maxima] Numerical issues with jacobi_sn
raymond.toy at ericsson.com
Wed Feb 11 11:14:34 CST 2009
Raymond Toy wrote:
> At this point, I'm stumped. Neither AGM nor Landen/Gauss
> transformations produce the correct answers.
I think I've found a solution to this. Use the descending Landen
transformation to jacobi_sn. This appears to work well for all real and
complex values of the argument and the parameter.
Then, for jacobi_dn, we use the Gauss transformation
(<http://functions.wolfram.com/09.29.16.0013.01>) to express jacobi_dn
in terms of jacobi_sn.
Finally, jacobi_cn can be computed using the ascending Landen
transformation to express jacobi_cn in terms of jacobi_dn.
Simple tests of selected interesting points and random values indicates
that the algorithms work without undue loss of precision and without
returning bogus results like jacobi_dn or jacobi_cn greater then 1.
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