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07/15/2008, 12:52 PM
(This post was last modified: 07/15/2008, 12:57 PM by Gottfried.)
Fiddling with alternate interpolation-approaches I came to an interpolation-technique, which is not "alternate" in the sense as I was searching, but has some interesting aspect on its own.

It also seems to back the diagonalization-method from another point of view.

I've not seen this before (nor in a more common serial representation) - may be someone recognizes it though I used the matrix-notation.

It is at

http://go.helms-net.de/math/tetdocs/Expo...lation.pdf
and -if of interest here- I'd upload it to the forum-resources.

Gottfried

Gottfried Helms, Kassel

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There is a recurrence equation for hyperbolic iteration that Henryk describes

here, and I later noticed

here and since this is a "finite" recurrence equation, everything is eventually defined in terms of

, which is equivalent to your

in the case

. I have also noticed that you can better describe hyperbolic iteration as a polynomial in

, but I have yet to find any patterns of note. Henryk's recurrence equation for hyperbolic iteration seems to be a great resource for explaining how and why it works the way it does.

Andrew Robbins

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07/16/2008, 11:21 AM
(This post was last modified: 02/09/2022, 12:58 PM by Gottfried.
*Edit Reason: mathjax*
)
andydude Wrote:There is a recurrence equation for hyperbolic iteration that Henryk describes here, and I later noticed here and since this is a "finite" recurrence equation, everything is eventually defined in terms of , which is equivalent to your in the case .

Hi Andrew -

thanks for the hint; I just reread that. I'll try to translate this into my matrix-lingo and see, how it is related. Though my Eigensytem-solver does not require the iterates \(g = f°^t\) I think the interpolation-approach may be related to it this way.

Quote: I have also noticed that you can better describe hyperbolic iteration as a polynomial in , but I have yet to find any patterns of note.

Do you remember my diagonalization formula?

(pg 21 in "ContinuousFunctionalIteration" )
Gottfried

Gottfried Helms, Kassel

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First, I think you meant

ContinuousfunctionalIteration (your URL was broken). Second, I think that actually

. Yes, I remember your U-tetration formula... I think you've found the most patterns in that so far...

Andrew Robbins