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 Fractional iteration of x^2+1 at infinity and fractional iteration of exp bo198214 Administrator Posts: 1,395 Threads: 91 Joined: Aug 2007 06/08/2011, 09:55 AM (This post was last modified: 06/08/2011, 06:57 PM by bo198214.) Hey guys, a polynomial pendant of moving from base e over eta to sqrt(2) may be moving from x^2+1 to x^2 to x^2-1, i.e. from no fixpoint to one fixpoint to two fixpoints on the real axis. Now for polynomials there is a technique to iterate them, that is not applicable to exp(x), which I will describe here applied to the example f(x)=x^2+1. It is the iteration at infinity. To see whats happening there one moves the fixpoint at infinity to 0, by conjugating with 1/x: $g(x)=\frac{1}{(\left(\frac{1}{x}\right)^2+1}=\frac{x^2}{x^2+1}$ $g(x)$ can be developed into a powerseries at 0, knowing that: $\frac{1}{x+1}=1-x+x^2-x^3\pm\dots$ $g(x)=x^2-x^4+x^6\mp \dots$ $g(x)$ has a so called super-attracting fixpoint at 0, which means that $g'(x)=0$. In this case one can solve the Böttcher equation: $\beta(g(x))=\beta(x)^2$ (where 2 is the first power with non-zero coefficient in the powerseries of $g(x)$). Iterating $g(x)$ can then be done similarly to the Schröder iteration: $g^t(x)=\beta^{-1}(\beta(x)^{2^t})$. Again similar to the Schröder case, we have an alternative expression: $g^t(x)=\lim_{n\to\infty} g^{-n}\left(g^n(x)^{2^t}\right)$ If we even roll back our conjugation with $1/x$ we get: $f^t(x)=\lim_{n\to\infty} f^{-n}\left(f^n(x)^{2^t}\right)$. Numerically this also looks very convincing: The following is the half-iterate h of x^2+1 accompanied by the identity function and x^2+1 itself. Computed with n=9.     And the verification h(h(x))-(x^2+1)     This may lead into a new way of computing fractional iterates of exp, because we just approximate exp(x) with polynomials $\sum_{n=0}^N \frac{x^n}{n!}$ and approximate the half-iterate of exp with the half-iterate of these polynomials. « Next Oldest | Next Newest »

 Messages In This Thread Fractional iteration of x^2+1 at infinity and fractional iteration of exp - by bo198214 - 06/08/2011, 09:55 AM RE: Fractional iteration of x^2+1 at infinity and fractional iteration of exp - by Gottfried - 06/08/2011, 01:09 PM RE: Fractional iteration of x^2+1 at infinity and fractional iteration of exp - by tommy1729 - 06/08/2011, 01:18 PM RE: Fractional iteration of x^2+1 at infinity and fractional iteration of exp - by bo198214 - 06/08/2011, 07:59 PM RE: Fractional iteration of x^2+1 at infinity and fractional iteration of exp - by JmsNxn - 06/08/2011, 07:26 PM RE: Fractional iteration of x^2+1 at infinity and fractional iteration of exp - by mike3 - 06/08/2011, 09:31 PM RE: Fractional iteration of x^2+1 at infinity and fractional iteration of exp - by bo198214 - 06/08/2011, 09:48 PM RE: Fractional iteration of x^2+1 at infinity and fractional iteration of exp - by mike3 - 06/08/2011, 10:08 PM RE: Fractional iteration of x^2+1 at infinity and fractional iteration of exp - by bo198214 - 06/08/2011, 10:12 PM RE: Fractional iteration of x^2+1 at infinity and fractional iteration of exp - by mike3 - 06/08/2011, 10:58 PM RE: Fractional iteration of x^2+1 at infinity and fractional iteration of exp - by bo198214 - 06/09/2011, 05:56 AM

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