Revisting my accelerated slog solution using Abel matrix inversion

[sheldonison]

Also, here is a copy of the full 4096-term accelerated solution, at 256 bits of precision.  As I noted before, the terms start to become inaccurate after about 600-700 terms, so the remaining terms are mostly there for completeness, and to validate the solution.

Hey Jay, welcome back. I've always been fascinated by your accelerated matrix solution and I 've always wondered how stable the solution is.  I use the same accelerator to cancel out a lot of the slog singularity at L and thereby speed up computation.  However, other than the speedup, I use a completely different slog algorithm in my fatou.gp program.  I also included an implementation of your accelerated Matrix solution in my program, although it is limited by pari-gp's memory limitations.  I've typically generated your matrix solution for a 100x100 matrix which is only a 16-17 decimal digit solution; its been awhile since I experimented with it.

When I used your accelerated solution to improve my fatou.gp abel function convergence, I center my solution between the two fixed points, so the results aren't immediately comparable.  To help compare the results, I re-sampled  my solution, in an attempt to match your accelerated solution, centered at x=0.   

Here are the first 40 taylor series term of the result, printed to 78 decimal digits of precision, centered at 0.  I used fatou.gp to generated a solution which should match Kneser's Riemann mapping solution to about 111 decimal digits.  Then I resampled and used 1024 equally spaced points on a unit circle centered at zero so the resample should have all of the precision of the 111 decimal digit solution.  Term x^692 was 10^-111, so I could post 692 terms if interested for comparison if interested.  The results match your post at the beginning of this thread for about 25-26 decimal digits; which is interesting and way too good to be due to chance, but the differences will require more investigation... edit: perhaps something like this explains what's going on where theta is a 1-cyclic function.
\( \text{JaySlogSeries}(z)\approx\text{KneserSlog}(z)+\theta(\text{KneserSlog}(z))\;\;\;??? \)
Such a theta mapping might explain why Jay's computation is internally consistent to 450 decimal digits, of which Jay posted ~78 decimal digits here on the forum.  Perhaps one could calculate such a theta mapping to help explain what's going on; more work is required   Monday evening I'll post some updates on how to bring Jay's slogseries into fatou.gp ...

Code:

{jaykneserslog=
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