Quote:I'm not saying we can't, but I can't think of any method available that gives exact derivatives of tetration yet, but granted, we do have approximations.

Easy. We solve the slog with a very large matrix, the larger, the better. Then we shift it to be centered at z=1. Then we take a reversion of the series (the math took me a while to figure out how to do, and only after I figured it out did I discover that PARI has a pretty fast series reversion solver).

This gives us a power series for sexp at z=0, which effectively allows us to find derivatives by basic manipulation of the series itself. We now have the very power series that we must take powers of, as I described, in order to compose, in order to solve the Abel equation.

Of course, loss of precision is the killer here. I'm not talking precision of the individual terms, but of the series itself. For my accelerated 900x900 solution, shifting the center to z=1 already knocks about half the terms off (in other words, the root test looks relatively flat, then spikes about halfway through). I try to work with the "residues" (after subtracting the basic logarithms), since this slightly reduces the effect, but there's not really much you can do except use more terms. I.e., this is standard loss of precision when moving a power series that has a radius of convergence. The best you can do is make small shifts and truncate the series a little after each step, to reduce the effect, but even this only buys a small increase in precision.

Then, the reversion of the series loses some precision as well. When all is said and done, probably only the first 300 to 400 terms are even accurate enough to bother using. So to solve the Abel equation for the pentalog with 1000 terms, you'd probably need to solve an accelerated 2000x2000 or possibly up to 3000x3000 slog system.

~ Jay Daniel Fox