• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 exponential polynomial interpolation Gottfried Ultimate Fellow Posts: 787 Threads: 121 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: 787 Threads: 121 Joined: Aug 2007 07/16/2008, 11:21 AM (This post was last modified: 10/11/2009, 01:18 PM by Gottfried.) 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 »

 Possibly Related Threads... Thread Author Replies Views Last Post My interpolation method [2020] tommy1729 1 2,457 02/20/2020, 08:40 PM Last Post: tommy1729 Math overflow question on fractional exponential iterations sheldonison 4 8,782 04/01/2018, 03:09 AM Last Post: JmsNxn Taylor polynomial. System of equations for the coefficients. marraco 17 28,251 08/23/2016, 11:25 AM Last Post: Gottfried Tribonacci interpolation ? tommy1729 0 3,192 09/08/2014, 10:37 AM Last Post: tommy1729 [Update] Comparision of 5 methods of interpolation to continuous tetration Gottfried 30 50,141 02/04/2014, 12:31 AM Last Post: Gottfried An exponential "times" table MikeSmith 0 3,012 01/31/2014, 08:05 PM Last Post: MikeSmith exponential baby Mandelbrots? sheldonison 0 3,255 05/08/2012, 06:59 PM Last Post: sheldonison exponential distributivity bo198214 4 9,894 09/22/2011, 03:27 PM Last Post: JmsNxn Iteration series: Different fixpoints and iteration series (of an example polynomial) Gottfried 0 4,355 09/04/2011, 05:59 AM Last Post: Gottfried Base 'Enigma' iterative exponential, tetrational and pentational Cherrina_Pixie 4 13,545 07/02/2011, 07:13 AM Last Post: bo198214

Users browsing this thread: 1 Guest(s)