polynomial interpolation to fractional iteration - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: polynomial interpolation to fractional iteration (/showthread.php?tid=104) polynomial interpolation to fractional iteration - Gottfried - 12/22/2007 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] RE: polynomial interpolation to fractional iteration - andydude - 12/23/2007 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 RE: polynomial interpolation to fractional iteration - Gottfried - 12/23/2007 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 RE: polynomial interpolation to fractional iteration - Gottfried - 12/23/2007 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