Discussing the (extended) Kneser-method the question of fixpoints is relevant. Here I have produced a picture of the fixpoints of tetration to base(î) , I found 2 simple fixpoint (attracting for exp, attracting for log, both used for the Kneser-method), and three periodic points - making things a bit more complicated. The fixpoints were sought using the Newton-algorithm for the joint threefold exponentiation \( f(z)= \log_i(\log_i(\log_i(z))) \) and the iteration \( z_{k+1} = f(z_k) \) .
This means for example, that for the point in the top-left edge with the blue color having the z-value z_0=-5 - 5i the Newton-algorithm using iteration \( z_{k+1} = f(z_k) \) arrives at the fixpoint 3.0 (having the complex value of about -1.14+0.71I in a moderate number of iterations. The blue point at coordinate z_0=-2.5+1I needs less iterations and the a bit lighter blue points near the periodic point 3.0 need even less iterations.
Perhaps this post should be moved into a discussion of the Kneser-method or of the general problem of fixpoints.
Gottfried
Here is the picture:
This means for example, that for the point in the top-left edge with the blue color having the z-value z_0=-5 - 5i the Newton-algorithm using iteration \( z_{k+1} = f(z_k) \) arrives at the fixpoint 3.0 (having the complex value of about -1.14+0.71I in a moderate number of iterations. The blue point at coordinate z_0=-2.5+1I needs less iterations and the a bit lighter blue points near the periodic point 3.0 need even less iterations.
Perhaps this post should be moved into a discussion of the Kneser-method or of the general problem of fixpoints.
Gottfried
Here is the picture:
Gottfried Helms, Kassel
