12/05/2011, 11:22 AM
(11/21/2011, 11:19 PM)sheldonison Wrote: I generated a taylor series and theta mapping, from the secondary fixed point....Below, there are graphs of the sexp(z) from the secondary fixed point, at the real axis, from sexp(-1.5) to sexp(1.5). I also graphed the first and second derivatives, and the equivalent functions from the primary fixed point. Notice how the derivative goes to zero at integer values of z.
I was able to get fairly clean convergence using two different algorithms, both with identical results. The simplest algorithm, with the quickest convergence required an initialization, very similar to the initialization used in the my kneser.gp program, followed by an initial approximation
\( \text{sexp_{l2}}(z)=\text{sexp}(z-sin(2z\pi)/(2\pi)) \). The initial sexp(z) need only have three terms in its Taylor series. Then this initial approximation required an additional 42 iterations, generating a theta(z) approximation from the secondary fixed point, followed by an sexp(z) approximation, from both the theta(z) and the sexp(z) approximation around z=-1. This gave results accurate to ~32 decimal digits. At each iteration, I forced the first three terms in the Taylor series to zero.
- Sheldon
