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 Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation... sheldonison Long Time Fellow Posts: 640 Threads: 22 Joined: Oct 2008 10/20/2017, 08:19 PM (This post was last modified: 10/20/2017, 08:38 PM by sheldonison.) (10/20/2017, 07:55 PM)Gottf ried Wrote: (10/20/2017, 06:00 PM)sheldonison Wrote: For example, here is the pari-gp program for the formal inverse schroeder function.  I don't know how to turn this into a matrix function, but not many programming languages support the powerful polyonomial functions that pari-gp has. Code:```formalischroder(fx,n) = {   local(lambda,i,j,z,f1t,f2t,ns,f1s);   lambda = polcoeff(fx,1);   f1t=x;   i=2;   while (i<=n,     f1s=f1t;     f1t=f1t+acoeff*x^i+O(x^(i+1));     f2t=subst(f1t,x,lambda*x)-subst(fx+O(x^(i+1)),x,f1t);     z = polcoeff(f2t, i);     z = subst(z,acoeff,x);     ns=-polcoeff(z,0)/polcoeff(z,1);     f1t=f1s+ns*x^i;     i++;   );   return(Pol(f1t)); } fz1=x^2+(1-sqrt(3))*x; [size=small][font=Monaco, Consolas, Courier, monospace]lambda1=polcoeff(fz1,1);[/font][/size] [size=small][font=Monaco, Consolas, Courier, monospace]fs1=formalischroder(fz1,20);[/font][/size] superfunction1(z)=subst(fs2,x,lambda2^z);``` Sheldon - I find some unexplained terms: what is "acoeff" / how is this defined before it is queried for "f1t"? the same with "fz2" in "lambda1=pol..." and "fs2" in "superfuncion...subst(fs2..."acoeff is just like an unknown; like "x" in the equations "x^3+2*x^2+3*x+1".  In the routine, Pari-gp will treat acoeff as an unknown unassigned variable just like "x", that happens to be the multiplier for the "i'th" coefficient we are trying to determine.   The others are typos:  I was originally going to post both superfunctions, from both fixed points....  The other fixed point equation would be fz2=x^2+(1+sqrt(3))*x.  Depending on whether |lambda|>1 or <1, determines where the superfunction equation converges, either negative or positive values of . Actually, the other fixed point also converges better, so that's what I meant to post.  Here superfunction1 converges pretty good for $\Re(z)<=2$ Code:```fz1=x^2+(1+sqrt(3))*x; lambda1=polcoeff(fz1,1); fs1=formalischroder(fz1,20); superfunction1(z)=subst(fs1,x,lambda1^z);``` - Sheldon « 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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