polynomial interpolation to fractional iteration

polynomial interpolation to fractional iteration - Printable Version

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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