Revisting my accelerated slog solution using Abel matrix inversion

[jaydfox]

The Jay's slog Taylor series is centered at zero!  The region of "good convergence" extends all the way out to nearly real(2)!  The plot goes from real(-1) to real(2).    On the sides, the error is dominated by approximately the 100th term of the JaySlog series, and in the vertical, it is limited by both Jay's slog and the corresponding 8 term theta mapping.  For the most part, these two functions are consistent to 10^-32 or so.  Without the theta mapping, the error is much larger; around 10^-17 or so. 

I remember Andrew Robbins noticed something similar with his (unaccelerated) slog.  It appears to be centered at 0, with a radius of convergence (which I determined was limited by the logarithmic singularities), but the series is also convergent in a second approximately circular region centered at 1, and also limited in radius by the singularities.  To me, the region looks like a peanut shell.

This seemed odd to me at first, because this seems to run counter to the concept of a radius of convergence.  But the key to understanding this (and understanding how the solution converges as the matrix size is increased) is to recognize that the inverse Abel matrix solution (accelerated or not) is a polynomial.  It is not an infinite Taylor series.  It's not even a finite truncation of an infinite Taylor series.  It is a polynomial.  It will eventually converge on a Taylor series (hopefully), but any particular solution is a polynomial.

For example, for my 4096-term solution, I estimate that the first 600-700 terms are accurate to within 1% of the true value.  So what are the remaining 3300 terms doing?

They encode the information (in higher order derivatives) necessary so that you can recenter the polynomial from 0 to 1, and still have 600-700 terms accurate to within 1%.  And you might be asking yourself, why does it have the information encoded to have 600-700 accurate terms for polynomials centered at 0 and at 1?

Because that's how the Abel matrix was defined!!!

By definition, the solution to an N-term finite truncation of the Abel matrix will have the first N derivatives equal at x=0 and at exp(x) = 1.  That's also why the internal consistency is so high, even though the accuracy is rather poor.  The 4096-term solution has internal consistency with an error term that is O(x^4097).  The difference between Andrew's slog and my slog is that my acceleration technique exponentially reduces the error constant in the big-Oh error term.