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 Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... tommy1729 Ultimate Fellow Posts: 1,358 Threads: 330 Joined: Feb 2009 06/04/2011, 09:43 PM (This post was last modified: 06/04/2011, 10:26 PM by tommy1729.) (06/04/2011, 01:13 PM)Gottfried Wrote: Sometimes we find easter-eggs even after easter... For the alternating iteration-series $\hspace{48} sn(x,p)=\sum_{h=0}^{\infty} (-1)^h g(x,h) ^p \\ \hspace{48} \text{where } g(x)=\sqrt{0.5+x} \hspace{48} g(x,h)=g(g(x,h-1)) \hspace{48} g(x,1)=g(x)$ (definitions as copied and extended from previous post, see below) we find a rational polynomial for p=4. That means $\hspace{48} x^4 - g(x)^4 + g(x,2)^4-g(x,3)^4+ ... - ... \\ \hspace{48}= sn(x,4) = 1/8 - x^2 + x^4$ (maybe this is trivial and a telescoping sum only, didn't check this thorough)
Another one: $\hspace{48} sn(x,1)+sn(x,2) = -0.25 + x^2 $
Code:\\ define function f(x) for forward iteration and g(x) for backward iteration (=negative height) \\(additional parameter h for positive integer heights is possible) f(x,h=1) = for(k=1,h,x = x^2 - 0.5 ); return (x) ; g(x,h=1) = for(k=1,h,x = sqrt(0.5 + x) ); return (x) ; \\ do analysis at central value for alternating sums x0=1 x = 1.0 sp(x) = sumalt(h=0,(-1)^h * f(x , h)) sn(x) = sumalt(h=0,(-1)^h * g(x , h)) y(x) = sp(x) + sn(x) - x this is not my expertise ... yet. but i think i have seen those before in some far past. for starters , i related your sums to equations of type f(x) = f(g(x)). also , ergodic theory studies averages of type F(x) = lim n-> oo 1/n (f^[0](x) + f^[1](x) + ... f^[n](x).) hidden telescoping can indeed occur. and sometimes we can rewrite to an integral. but again , this is not my expertise yet. you gave me extra question instead of an answer :p in particular i do not understand your matrix idea in this thread. my guess is that when you start at 1.0 , you use carleman matrices to compute the sum and one carleman matrix will not converge ( lies outside the radius ) for 1.0 ; so one is wrong and the other is not. talking about alternating series 1/2 -1/3 + 1/5 -1/7 + 1/11 - ... i believe this has a closed form/name and if i recall correctly its called the first mertens constant ... there was something else i wanted to say ... forgot :s edit : i do not know how to rewrite an average as a sum or superfunction ( do know integral and perhaps infinite product )... i say that because it might be usefull to see the link with the " ergodic average " ( or whatever its called ). it bothers me , i wanna get rid of this " lim **/n " term for averages. ( might also be of benefit for number theory and statistics ) « 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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