Which method is currently "the best"?

[sheldonison]

Hi there, I'm a reader of this forum.
I've also tried to make some tetration code but currently failed.

Can some of You please point me to currently "the best" algorithm for tetration available?

This algorithm handles a large range of real bases, greater than \( \exp(1/e) \) to around b=100,000. For each base, it can generate sexp(z) for z anywhere in the complex plane. http://math.eretrandre.org/tetrationforu...hp?tid=486

By "the best" I mean algorithm which can handle most bases (both base and rank should be complex numbers)

I have a version of the program for complex bases, but it is still experimental, and handles a fairly small range of bases. I would recommend the real base version linked to above, instead, but here is the link to the tetcomplex program. http://math.eretrandre.org/tetrationforu...hp?tid=729 One consequence of studying how tetration works for complex bases, is that there is no one correct routine for real bases<\( \exp(1/e) \), and that regular iteration from the fixed point is probably more meaningful than a Kneser style Riemann mapping for real bases<\( \exp(1/e) \).

Also if possible point me to some underlying math explaination....?

Here are two links to look at: http://math.eretrandre.org/tetrationforu...hp?tid=487, especially this picture: http://math.eretrandre.org/tetrationforu...e=threaded and also take a look at this post for the connection between Kneser's solution and my algorithm.
http://math.eretrandre.org/tetrationforu...hp?tid=700

I also want to rewrite it in some wide available language like C or C++ etc.
I will of course publish C/C++ code if I manage to translate it.

Go for it. Please give me credit if you do so, and put a link on this forum. There are some other analytic solution methods that work quite well. Mike reports good results with Kouznetsov's method for both real and complex bases. Additionally, Mike is working on some new methods right now. There is Andrew Robbins's slog method, which Jay Daniels was able to get working reasonably well, with some acceleration techniques. The nice thing about my method is it has theoretical connections with Kneser's proven Riemann mapping solution, and it is more computationally efficient than Kouznetsov's method or Andrew's method.
- Sheldon Levenstein