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[Regular tetration] [Iteration series] norming fixpoint-dependencies
#4
Another picture which shows the wobble of different values for the regular height-function when fp0 or fp1 is used. Example: base b=sqrt(2).

Then the value "normzero" is the map of 1 into the segment seg1 between 2 and 4 using fp0-powerseries and has reference-height 0 in that segment. It is that value of 2.46... = tet0(1,Pi*I/ln(fp0))

To have computations numerically nearer at the fixpointvalue 2, I increase its height (tetrate it using tet0) by 13.5.

Then I generate a set of x-coordinates in small steps in the height-interval hgh0(x)= 13... 17. Now I determine the heights of these x-coordinates using the hgh1()-function which employs the second fixpoint fp1.
Then the height-values using hgh0(x) and hgh1(x) differ periodically by small differences of about 1e-25.

This is the basic idea of the curves in the plot.

But we find, that the norming process has more implications. If we connect the tet0 and tet1-functions using a common x at a fractional iterate from "normzero", then the difference-curve becomes asymmetric.
Examples: if we use the connection-value at tet0(normzero,+0.25) all differences are positive, if at tet0(normzero,+0.5) we have nearly the same curve as with tet0(normzero,0) itself, and if we connect tet0 and tet1 at tet0(normzero,+0.75) all differences become negative.

So the selection of the connection-point for the norming is an important aspect. However, the matter is not yet satisfactorily solved: still we have a small (but seemingly constant ~ 2e-26) difference of the curves for connection-point tet0(normzero,+0) and tet0(normzero,+0.5). So the wobbling is not exact the same even at half-integer steps of the connection-point.


Gottfried




   
Gottfried Helms, Kassel
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RE: [Regular tetration] norming fixpoint-dependencies - by Gottfried - 07/29/2010, 11:36 AM

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