Find all fixed points of exp[b]

[jaydfox]

No one can help how to find all fixed points of log(b), for all complex bases b.

I know exp() have infinite numbe rof fixed points, but two of them are "main" and are the same as log(), I want to find them using some algorithm and elementary functions. I know there is no closed for for them, but I want to find algorithm that approximates them for all bases.

Any ideas?
My program ffrom 1st post inds fixed point - but only one, how to find another one?

Well, unfortunately, I don't have c++ code for anything related to tetration. I don't know how easily you can work with PARI/gp or SAGE/python, but those are the languages I've been working with. Don't get me wrong; I like the speed of c++ (I've been working in c/c++ for over 20 years).

But when it comes to tetration and similar mathematical projects, I've been working primarily with "arbitrary precision" math lately, and that would require linking GMP or MPFR or similar libraries. It can be done, and the speed benefits would be nice, but the extra time to write the code nullifies any benefits. I'd only resort to c++ with MPFR if I needed to solve a particularly slow/complex problem. For example, some day I might port my accelerated slog solution.

Anyway, if you can use PARI/gp, the following code might be useful to you. (Or to anyone else, like Sheldon?)


  jsloglib_v0.5.gp (Size: 5.47 KB / Downloads: 1,224)

To test the fixed point stuff:

  jslog_fptest.gp (Size: 305 bytes / Downloads: 1,308)

To test the Schroeder functions (this is more for Sheldon's benefit):

  jslog_test.gp (Size: 933 bytes / Downloads: 1,274)

Here's an example of the output for the jslog_fptest.gp attachment: