fast accurate Kneser sexp algorithm

[sheldonison]

I can calculate good for base 1.1.
I tried bases 1.01 and 1.001 but I get not big precision and too small precision, respectively.

but good -> http://math.eretrandre.org/tetrationforu...hp?tid=272 - other 1 code for mathematica.

Hey Nuinho,

I'm aware it doesn't converge for bases<1.02 (I had updated the post with the latest code). The code update supporting a kneser mapping to generate sexp for bases<eta is still somewhat of an experiment, since it is inherently less efficient than just using regular iteration.

There are much better methods for calculating sexp(z) for bases less than eta, and I'm sure you know much more about that than I do. When I have time, I may try to figure out if there is a way to get the code converging for smaller bases. Also on my todo list, is supporting complex bases (which I'm about 1/3rd of the way through). And I still really need to try to rigorously defend why the algorithm converges at all, which will be a difficult task.

New version of kneser.gp dowload attached below. For bases<eta, this supports generating the kneser mapping Laurent series to the get the regular iteration sexp solution via a mapping from the upper sexp. I got this working yesterday. It should be almost exactly equal to the new superf2(z) function, which is only valid for bases<eta, and is the sexp(z) from the lower attracting fixed point. If you want the kneser mapping for the newsexp function I posted earlier, type "init(sqrt(2));gennewsexp=1;loop;". Then you can compare sexp(z) to superf2(z), but you'll need to set "\p 134;" or larger, to calculate the differences.
- Sheldon
For the most recent code version: go to the Nov 21st, 2011 thread.