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 Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... Gottfried Ultimate Fellow Posts: 764 Threads: 118 Joined: Aug 2007 03/02/2009, 02:50 PM (This post was last modified: 03/02/2009, 02:55 PM by Gottfried.) Hi - I'm considering the properties of the regular iteration after fixpoint-shift. To have easier conditions I look at the function f(x) = x^2 - 0.5 and its iterates. It has the fixpoints $x_a = \frac{1+\sqrt{3}}{2}$ and $x_b = \frac{1-\sqrt{3}}{2}$. I compute the half-iterate f°0.5(x) by translating f(x) -> g(x - x_a) + x_a and determine the powerseries for g°0.5(x) The result is interesting; the curve for the half-iterate meets the first fixpoint, but misses the second. Instead, the series diverges in that region. However, using the more reliable results of f°0.5(x) at other values of x, I can construct a bit of continuation, which gives one winding around (xb,xb). Naively, I'd expected that the function crossed both fixpoint but it seems, that this needed a completely different powerseries, and thus a modified procedure. I'm without an idea currently, how to proceed here. Does anyone have a comment? Here are two plots: a) an overview. Integer iterates f°-1(x), f°0, f°1,f°2, f°3,f°4 and the regular f°0.5 iteration based on the powerseries representation     b) a detail, f°0.5 and f°1 in the vicinity of (xb,xb) and a continuation based on more reliable results of f°0.5 at other values of x     Gottfried Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Messages In This Thread Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... - by Gottfried - 03/02/2009, 02:50 PM RE: Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... - by Gottfried - 03/02/2009, 04:48 PM RE: Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... - by bo198214 - 03/02/2009, 04:50 PM RE: Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... - by tommy1729 - 03/02/2009, 08:48 PM RE: Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... - by Gottfried - 03/03/2009, 12:52 AM RE: Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... - by Gottfried - 03/03/2009, 07:45 AM RE: Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... - by Gottfried - 03/03/2009, 12:15 PM RE: Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... - by tommy1729 - 06/05/2011, 01:45 PM RE: Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... - by Gottfried - 06/05/2011, 05:17 PM RE: Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... - by tommy1729 - 06/02/2011, 08:36 PM RE: Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... - by Gottfried - 06/04/2011, 10:01 AM RE: Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... - by Gottfried - 06/04/2011, 01:13 PM RE: Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... - by tommy1729 - 06/04/2011, 09:43 PM RE: Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... - by Gottfried - 06/05/2011, 10:50 AM RE: Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... - by Gottfried - 06/05/2011, 11:40 AM RE: Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... - by tommy1729 - 06/06/2011, 11:01 AM RE: Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... - by Gottfried - 06/06/2011, 12:47 PM RE: Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... - by Gottfried - 10/19/2017, 10:38 AM RE: Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... - by sheldonison - 10/19/2017, 04:50 PM RE: Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... - by Gottfried - 10/19/2017, 09:33 PM RE: Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... - by sheldonison - 10/20/2017, 06:00 PM RE: Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... - by Gottfried - 10/20/2017, 07:55 PM RE: Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... - by sheldonison - 10/20/2017, 08:19 PM RE: Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... - by Gottfried - 10/20/2017, 08:32 PM

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