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 exponential polynomial interpolation Gottfried Ultimate Fellow Posts: 852 Threads: 126 Joined: Aug 2007 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 andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 07/15/2008, 03:46 PM 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 $f_1^t$, which is equivalent to your $u^h$ in the case $f(x) = e^{ux} - 1$. I have also noticed that you can better describe hyperbolic iteration as a polynomial in $f_1^t$, 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 Gottfried Ultimate Fellow Posts: 852 Threads: 126 Joined: Aug 2007 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 $f_1^t$, which is equivalent to your $u^h$ in the case $f(x) = e^{ux} - 1$. 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 $f_1^t$, but I have yet to find any patterns of note. Do you remember my diagonalization formula? (pg 21 in "ContinuousFunctionalIteration" ) Gottfried Gottfried Helms, Kassel andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 07/16/2008, 10:32 PM First, I think you meant ContinuousfunctionalIteration (your URL was broken). Second, I think that actually $g = f^{\circ t-1}$. Yes, I remember your U-tetration formula... I think you've found the most patterns in that so far... Andrew Robbins « Next Oldest | Next Newest »

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