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 polynomial interpolation to fractional iteration Gottfried Ultimate Fellow Posts: 787 Threads: 121 Joined: Aug 2007 12/22/2007, 05:28 PM (This post was last modified: 12/23/2007, 01:13 PM by Gottfried.) Hi - triggered by a discussion in sci.math I tried to explain to someone, how one could naively use interpolation to obtain a version of continuous tetration. For simplicitiness I used U-tetration (x -> exp(x)-1) In a second shot I made this a bit more general and - whoops - it comes out to be the matrix-method in disguise (but now with a bit more general approach). Nothing new to the experienced tetration-diggers here, but maybe still a nice exercise. Happy christmas to all - Gottfried Interpolation [update 4 23.12.2007] Attached Files   polynomial_interpolation.pdf (Size: 64.05 KB / Downloads: 946) Gottfried Helms, Kassel andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 12/23/2007, 05:27 AM Very nice discussion! I like the colors of the coefficients. I also briefly discuss this in this thread, and Jay discusses this in this thread, just to let you know, if you forgot. Also why do you call it U-tetration? I call it iterated decremented exponentials, since: iterated = repeating the same function over and over decremented = subtracting one from something exponential = a function from x to $b^x$ so an expression like $f(x) = b^x-1$ would be a decremented exponential, and an expression like $f^{\circ n}(x)$ would be an iterated decremented exponential. Andrew Robbins Gottfried Ultimate Fellow Posts: 787 Threads: 121 Joined: Aug 2007 12/23/2007, 10:04 AM andydude Wrote:Very nice discussion! I like the colors of the coefficients.Nice! Thanks Quote: I also briefly discuss this in this thread, and Jay discusses this in this thread, just to let you know, if you forgot. Yepp, thanks. Our forum is a rich resource - sometimes I just browse through older threads and understand today, what I didn't understand before... I'll have a look at it. Quote: Also why do you call it U-tetration? I call it iterated decremented exponentials, Yes, I know. But just count the number of letters... In informal exchange I tend to use the name of the matrices, which I use in Pari/Gp. And I don't know why, but U-tetration as some low-level association for me. If my other tetration-article is finished, I'll replace some of the nicks by the more expressive denotations. Thanks again for your comment - Gottfried Gottfried Helms, Kassel Gottfried Ultimate Fellow Posts: 787 Threads: 121 Joined: Aug 2007 12/23/2007, 03:40 PM andydude Wrote:thread, and Jay discusses this in this thread, just to let you know, if you forgot. :-) I was even involved in that thread ... For whatever reason I did not catch its contents then... So it goes - Gottfried Gottfried Helms, Kassel « Next Oldest | Next Newest »

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