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complex base tetration program
#19
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 and the iteration .

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 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:


Attached Files Image(s)
   
Gottfried Helms, Kassel
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Messages In This Thread
complex base tetration program - by sheldonison - 02/29/2012, 10:28 PM
RE: complex base tetration program - by mike3 - 03/10/2012, 06:59 AM
RE: complex base tetration program - by mike3 - 03/11/2012, 12:27 AM
RE: complex base tetration program - by Gottfried - 02/06/2016, 01:37 AM
RE: complex base tetration program - by Gottfried - 02/06/2016, 05:36 PM
RE: complex base tetration program - by Gottfried - 02/07/2016, 05:27 AM
RE: complex base tetration program - by Gottfried - 02/07/2016, 01:25 PM
RE: complex base tetration program - by Gottfried - 10/24/2016, 11:50 PM
RE: complex base tetration program - by Gottfried - 10/26/2016, 10:02 AM

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