• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 the inconsistency depending on fixpoint-selection bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 03/06/2008, 04:58 PM Gottfried Wrote:What I've got so far is a good result for the half-iterate, which seems to be exact to more than 20 digits. I've also used a shifting of x->x' to get a triangular matrix with exact terms, where I denote $\hspace{24} \begin{matrix} x' &=& x/(1/4) - 1 \\ x'' &=& (x+1)*1/4 \\ g(z)&=& 1/4 z^2 + 3/2 z \end{matrix}$ and with the above then $\hspace{24} f(x) = g(x')''$ Wah thats strong tobacco, I just had used a *translation*, for example $x' = x-\frac{1}{4}$ $x'' = x+\frac{1}{4}$ this $g_1(x)=f(x')''$ is $f$ with the fixed point $\frac{1}{4}$ moved to 0. And $g_2(x)=f(x'')'$ corresponds to moving the fixed point $-\frac{1}{4}$ to 0. Computing the half iterate with your triangular matrices corresponds to computing the regular fractional iterate at the corresponding fixed point (to get $f^{\circ 1/2}$ you of course have to move back the fixed point 0 of $g_1$ or $g_2$ to its original place. The half iterate at both fixed points is different. And from here you already see variants: 1. You dont need to use a translation. You can use any bijective (analytic) function $\tau$ which moves the fixed point to 0 (you used a *linear* transformation). You then take the fractional iterate of $g=\tau^{-1}\circ f\circ \tau$ and get the fractional iteration of $f$ as $f^{\circ t}=\tau\circ g^{\circ t}\circ \tau^{-1}$. 2. The matrix operator method is more general. It does not only work at fixed points but also in between where you use the triangulated finite matrices as approximating sequence. So you can use *any* translation and then apply the matrix operator method. And it is quite sure that all those results differ (by a super small amount). 3. And then you can go even a step further and use any (analytic) bijective $\tau$ before applying the matrix operator method. « Next Oldest | Next Newest »

 Messages In This Thread the inconsistency depending on fixpoint-selection - by Gottfried - 02/07/2008, 03:06 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 02/07/2008, 05:20 PM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 03/04/2008, 10:14 AM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 03/04/2008, 11:17 AM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/04/2008, 11:25 AM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/04/2008, 11:32 AM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 03/04/2008, 11:43 AM RE: the inconsistency depending on fixpoint-selection - by GFR - 02/07/2008, 09:02 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 02/07/2008, 09:49 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/03/2008, 11:20 AM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/06/2008, 04:06 PM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 03/06/2008, 04:58 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/06/2008, 05:25 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/07/2008, 03:16 AM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 03/07/2008, 10:56 AM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/07/2008, 07:17 PM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 03/07/2008, 08:30 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/07/2008, 09:38 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/08/2008, 01:22 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/08/2008, 01:52 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/08/2008, 04:21 PM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 03/09/2008, 09:30 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/14/2008, 02:10 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/14/2008, 04:41 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/14/2008, 07:19 PM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 04/29/2008, 08:15 AM

 Possibly Related Threads... Thread Author Replies Views Last Post Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... Gottfried 23 33,637 10/20/2017, 08:32 PM Last Post: Gottfried (Again) fixpoint outside Period tommy1729 2 2,879 02/05/2017, 09:42 AM Last Post: tommy1729 Polygon cyclic fixpoint conjecture tommy1729 1 2,521 05/18/2016, 12:26 PM Last Post: tommy1729 The " outside " fixpoint ? tommy1729 0 1,677 03/18/2016, 01:16 PM Last Post: tommy1729 2 fixpoint pairs [2015] tommy1729 0 1,982 02/18/2015, 11:29 PM Last Post: tommy1729 [2014] The secondary fixpoint issue. tommy1729 2 4,041 06/15/2014, 08:17 PM Last Post: tommy1729 Simple method for half iterate NOT based on a fixpoint. tommy1729 2 3,764 04/30/2013, 09:33 PM Last Post: tommy1729 2 fixpoint failure tommy1729 1 3,091 11/13/2010, 12:25 AM Last Post: tommy1729 abs f ' (fixpoint) = 0 tommy1729 2 4,953 09/09/2010, 10:13 PM Last Post: tommy1729 [Regular tetration] [Iteration series] norming fixpoint-dependencies Gottfried 11 14,869 08/31/2010, 11:55 PM Last Post: tommy1729

Users browsing this thread: 1 Guest(s)