x↑↑x = -1

[jaydfox]

I have not had time to analyze complex bases. I have my doubts that a unique solution exists, since the periods of the fixed points are different, but I'm waiting to make an analysis before I make any claims one way or the other.

Hey Jay,

...Jay, you should see if you can generate the accelerated version of the intuitive method for base 3+i.

Okay, I played around with my library this morning, to try to get it working with complex bases. For real bases greater than eta, my library was coded to calculate the taylor series for one singularity, then take twice the real part (based on the conjugated pair). I modified it to calculate each fixed point separately and add the results. I tested with bases e and 10 to make sure I didn't break anything.

I then tried base 3+i. The solutions were initially convergent (intuitive and accelerated versions). However, as early as about size 30-40 for the matrix system, the convergence became rapid divergence. I ran into the same issue several years ago, when trying to recenter the matrix system for a real base to a complex development point. The closer the development point to the real line, the longer the initial convergence would last, before diverging rapidly. I never had time to properly analyze the problem. It wasn't an issue with numerical precision (adding precision didn't affect the divergence at all).

I'll play around with it when I get time, but right now I'm still working on other tetration-related projects. For example, I'm building out my intuitive/accelerated slog library and getting the bugs out, so I can eventually upload it here. It's in SAGE/python/cython code for now, but once it's stable, I'll write a pari/gp version for those who prefer that environment. I'm also analyzing convergence (with pretty pictures!) for the intuitive and accelerated intuitive methods, in quite a bit of detail, so that takes a fair amount of time as well.