(02/14/2016, 08:29 PM)sheldonison Wrote: The process is fairly simple. Pick n bases, equally spaced around a circle in the complex plane around the base in question. Since complex base tetration is an analytic function, you get really good convergence, out to the radius of the sample points that were chosen, using Fourier/Cauchy to generate the Taylor series coefficients.
I had a course in complex analysis, and error propagation, but I'm still chewing it.
Aside of being a bit rusted, error control is critical here. The smaller the radius of the circle, the smaller theoretical error, but fatou only gives a limited number of decimals, no matter how much I crank up precision in pari/gp, so if I reduce the radius too much, all decimals get truncated, and I lose all the information.
If I rise the radius, then the error comes from error propagation and singularites. No clue about what is the optimal radius.
also, I need a base on the real line, as close as possible to 1, but fatou doesn't calc on bases with negative imaginary parts, so I need to integrate over a semicircle; cutting the circle with the real line, and that complicates tracing the error.
Despite Sheldonison kind attempts to explain fatou.gp algorithm, it goes over my head, so I cannot tweak fatou for more decimals, and must treat it like a black box.
I have the result, but I do not yet know how to get it.
