new fatou.gp program

[sheldonison]

I posted a new version of the pari-gp fatou.gp program; the pari-gp code is in the original post. There's a minor bug fix along with some updated functions. The bug was that sometimes near the fixed points, the precision of the sexp(z)/invabel(z) function was lower than it was supposed to be. The closest points to the real axis with errant behavior was around sexp(0.03+3.6*I), which is only accurate to 13 decimal digits accurate instead of the expected 24 decimal digits with the default \p 28 precision. The sexp(z) routine used in fatou actually has to find the inverse of the Riemann surface of the slog, and the slog routine itself uses a different algorithm near the fixed point. The support for a wide range of complex bases led to some complicated logic in the "betterest" subroutine called in the main invabel(z)/slog(z) routine. A more minor bug fix was that the slog branch picked for points near the fixed point was not always consistent with drawing the cut-line vertically, which was the intended slog(z) cutpoint branch behavior. With this update, its easy to accurately see the difference between the two branches of \( \exp^{0.5}(z) \) near the fixed point, which I needed for http://math.eretrandre.org/tetrationforu...p?tid=1043 this post

Some of the other updated functions are actually kind of cool, and they come for free. For example, there's an implementation of Andrew's slog, and Jay's accelerated slog; some details are in the "andrewjay();" help menu. Someday, I may post my comments on the convergence of these simultaneous equation matrix solutions and the "(andrewslog(z)-slog(z))" error term. The error term turns out to be modeled surprisingly well as 1-cyclic fourier series, with respect to Kneser's solution. The provides a window into why the fatou.gp solution is unique, and why Andrew's slog converges so slowly, and might allow for fascinating future posts on the forum.

Other minor functions in this update include a generic formalhalf(fx,n) routine, to go along with the formalschroder routines.