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 Comparing the Known Tetration Solutions bo198214 Administrator Posts: 1,386 Threads: 90 Joined: Aug 2007 08/20/2007, 01:36 PM (This post was last modified: 08/20/2007, 01:46 PM by bo198214.) Now I compared Daniels method with Andrews method for the base $b=\sqrt{2}<\eta$ (hyperbolic case). Daniel's approach is to consider the fixed point of $f(x)=b^x$, and to determine the unique hyperbolic iterate $f^{\circ t}$ there and then set ${}^tb=f^{\circ t}(1)$. The lower fixed point of $f$ for $b=\sqrt{2}$ is 2 (and the upper fixed point is not reachable from 1 by iterations, so it is not of importance for our method ${}^xb=\exp_b^{\circ t}(1)$). Because I have no actual formula to compute the series expansion for the hyperbolic iterate, I used an iterative formula (which can be found in [1] and has quite some similarity with Jay's approach): $f^{\circ t}(x)=\lim_{x\to 0} f^{\circ -n}(c^t f^{\circ n}(x))$ where f is assumed to have its fixed point at 0 and $c$ is the derivative at the fixed point, which is in our case $c=f'(2)=\log(b) b^2 = \log(2^{1/2})2=\log(2)$. Of course there are some demands on the function f for the formula to be valid, but they are satisfied by our f, particularely $0. In the usual way we can move the fixed point to 0 by conjugating and after iteration move it back to its original place by inverse conjugating. Resulting in this case in the formula $f_2(x)=b^{x+2}-2, g_2(x)=\log_b(x+2)-2$ and $f^{\circ t}(x)= \lim_{n\to\infty} g_2^{\circ n}(c^t f_2(x-2))+2$ i.e. $\phantom{sqrt{2}}^t \left(sqrt{2}\right) = \lim_{n\to\infty} g_2^{\circ n}(\log(2)^t f_2(-1))+2$ And now guess how Andrew's and Daniel's slog compare! (At least on the picture, I didnt start exacter numerical computations)     [1] M. C. Zdun, Regular fractional iterations, Aequationes Mathematicae 28 (1985), 73-79 « Next Oldest | Next Newest »

 Messages In This Thread Comparing the Known Tetration Solutions - by bo198214 - 08/17/2007, 09:13 PM RE: Comparing the Known Tetration Solutions - by jaydfox - 08/20/2007, 09:59 AM RE: Comparing the Known Tetration Solutions - by bo198214 - 08/20/2007, 11:09 AM New pictures from the hyperbolic slog! - by bo198214 - 08/20/2007, 01:36 PM RE: Comparing the Known Tetration Solutions - by jaydfox - 08/20/2007, 05:35 PM RE: Comparing the Known Tetration Solutions - by jaydfox - 08/21/2007, 05:06 AM RE: Comparing the Known Tetration Solutions - by andydude - 08/22/2007, 11:28 PM RE: Comparing the Known Tetration Solutions - by bo198214 - 08/23/2007, 12:14 AM RE: Comparing the Known Tetration Solutions - by Gottfried - 08/29/2007, 06:19 AM RE: Comparing the Known Tetration Solutions - by jaydfox - 08/29/2007, 06:15 PM RE: Comparing the Known Tetration Solutions - by jaydfox - 08/29/2007, 12:06 AM RE: Comparing the Known Tetration Solutions - by jaydfox - 08/29/2007, 01:56 AM RE: Comparing the Known Tetration Solutions - by bo198214 - 08/29/2007, 07:48 AM RE: Comparing the Known Tetration Solutions - by andydude - 08/29/2007, 08:19 AM RE: computing the iterated exp(x)-1 - by andydude - 08/17/2007, 11:20 PM RE: computing the iterated exp(x)-1 - by jaydfox - 08/17/2007, 11:38 PM RE: computing the iterated exp(x)-1 - by bo198214 - 08/17/2007, 11:45 PM RE: computing the iterated exp(x)-1 - by jaydfox - 08/18/2007, 12:19 AM RE: computing the iterated exp(x)-1 - by bo198214 - 08/18/2007, 08:19 AM RE: computing the iterated exp(x)-1 - by andydude - 08/18/2007, 09:35 AM RE: computing the iterated exp(x)-1 - by bo198214 - 08/18/2007, 11:59 AM RE: computing the iterated exp(x)-1 - by jaydfox - 08/18/2007, 03:49 PM RE: computing the iterated exp(x)-1 - by jaydfox - 08/19/2007, 12:50 AM RE: Comparing the Known Tetration Solutions - by bo198214 - 08/19/2007, 10:55 AM

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