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fast accurate Kneser sexp algorithm
#20
(01/06/2011, 02:48 AM)JmsNxn Wrote: Thank you for this. This is exactly what I need.

I was wondering though, since sexp(5*i) = 0.999 + 4.999i is it safe for me to write sexp(fi) = 1 + fi and assume it's an error in approximation?
Perhaps you didn't execute the "loop". Also, the current version gives a warning if you evaluate sexp(z) for imag(z)>I. If you download the kneser.gp program (I would suggest the most recent version, go to this thread), and type in:
Code:
init(exp(1));loop;
.... goes through 13 iterations .....
(07:43) gp > sexp(5*I)
!!! WARNING, riemaprx(z) much better than sexp(z) for imag(z)>I !!!
%2 = 1.3786691576693131111676650899624 E40 - 3.2923562701722998997666622377240 E40*I
To get the correct result for imag(z)>I, use the riemaprx(z) instead, which is accurate for imag(z)>=0.12*I. Or use splicesexp(z), which is valid everywhere in the complex plane.
Code:
(07:43) gp > riemaprx(5*I)
%3 = 0.32106749434792621140043570196732 + 1.3394349973320786642136709026508*I
I plan to post again, giving a more clear overview of the sexp and riemaprx routines and how they are generated, and how the different versions of the code improved convergence. The latest version leads to a definition of an infinite sequence of both functions. In particular, the current version allows for a continuous Cauchy integral for the sexp function, as opposed to the earlier version which was required to be strictly a discreet Cauchy integral estimation, which I think had more theoretical problems. I think the current version will help prove convergence, but I'm not there yet.
- Shel
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Messages In This Thread
The pari-GP code - by sheldonison - 08/07/2010, 09:17 PM
updated kneser.gp code - by sheldonison - 08/19/2010, 02:35 AM
RE: updated kneser.gp code - by nuninho1980 - 08/19/2010, 12:08 PM
RE: updated kneser.gp code - by sheldonison - 08/20/2010, 01:05 AM
update to support B<eta - by sheldonison - 11/15/2010, 02:53 PM
RE: update to support B<eta - by nuninho1980 - 11/15/2010, 03:26 PM
another new version - by sheldonison - 11/17/2010, 06:52 PM
RE: fast accurate Kneser sexp algorithm - by sheldonison - 01/06/2011, 02:53 PM

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