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+--- Thread: exponential polynomial interpolation (/showthread.php?tid=190)
exponential polynomial interpolation - Gottfried - 07/15/2008
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/ExponentialPolynomialInterpolation.pdf
and -if of interest here- I'd upload it to the forum-resources.
Gottfried
RE: exponential polynomial interpolation - andydude - 07/15/2008
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
RE: exponential polynomial interpolation - Gottfried - 07/16/2008
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
RE: exponential polynomial interpolation - andydude - 07/16/2008
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