02/20/2016, 07:45 PM
(This post was last modified: 02/21/2016, 01:16 PM by sheldonison.)
(02/20/2016, 03:03 PM)marraco Wrote: I sampled 1000 points around the point r=1.1, with a radius R=.01.
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I will attempt to implement a conjugate gradient on the red blue equation, use knesser.gp to get a very good first guess, and increase precision.
That also may lead nowhere, since the red blue eq is mined with local minima.
For real bases greater than eta=exp(1/e), the fixed points are complex conjugates of each other. For real bases between 1 and eta, you have an attracting fixed point, and a repelling fixed point. The branch at b=1 is pretty bad, worst than the branch at b=eta. For real bases less than b=1, you still have a real attracting fixed point, but the sexp(z) function is again not real valued. This is true when using either the Schroeder function or the more complicated Kneser solution.
So your basic problem is the half iterate isn't analyatic at b=1, no matter which approach you use, Kneser or the simpler Schroeder function solution. For simplicity, I'm graphing the Schroeder function half iterate, sexp(-0.5), as you loop around from 1.1 to -0.9 and then back to 1.1, but the Kneser solution you get using both fixed points (which is what fatou.gp gives you) isn't visually different for this graph. If it isn't analytic, increasing the precision of the number of sample points or decreasing the radius doesn't help.
Red is imaginary; magenta is real
- Sheldon
