x↑↑x = -1

[jaydfox]

Numerically, it can't possibly give results of any reasonable precision, due to the really nasty singularity for the Abel function of 0.

Ha! Don't I know it! You can see the convergence analysis in my post. Using a 2835x2835 system, and using 11-point Newton-Cotes interpolation, I still barely managed to get convergence on the order of 10^-7, or about 20-25 bits of accuracy.

I have no idea how you got results that are that good, for a Kneser Riemann mapping. I just generated a 2000 term Taylor series for the Riemann Circle mapping, from the theta(z) mapping, using the equations I posted. If \( \Im(z)>0.002 \), then a 2000 term series is sufficient for approximately 10^-15 decimal digits accuracy for the end result sexp(z), and I verified that. But for a 2000 term series, at the real axis, I get an error term of a little bit bigger than 10^-4 when compared to sexp(z). That's nowhere near 10^-7.

Bear in mind, we're calculating the mapping in a different way. They should be equivalent in the infinite limit, but they will be different for finite truncations. I'm not sure why mine would have converged faster though, since they should be equivalent at a first order approximation.

I would estimate that each doubling of the number of terms would give you one binary bit additional accuracy.

If you look back at my original analysis, I stated "convergence is slow, yielding just under 3 bits of precision for each tripling of system size. This more or less corresponds to 2 bits per doubling".

I'll try to wrap up my slog library in the next couple weeks so I can go back to working on my Kneser code. I have a partial Kneser library that I worked on last year. I hadn't gotten as far as testing it with the Kneser construction. The last time I worked on it, I was still testing the Riemann mapping code with the iterated logarithm, which I mentioned in post #33 of this thread.