fast accurate Kneser sexp algorithm

[sheldonison]

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