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 [AIS] (alternating) Iteration series: Half-iterate using the AIS? mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 12/10/2012, 11:26 AM (This post was last modified: 12/10/2012, 11:48 AM by mike3.) (12/10/2012, 10:38 AM)Gottfried Wrote: (12/10/2012, 03:33 AM)mike3 Wrote: (...) I've tried other ways of arranging the partial sums (like Cesaro-summing the positive-"degree" terms and the negative-"degree" terms separately), but it doesn't seem to help. Do I need to use Euler summation? Hi Mike, I did not check Cesaro-summability; I just used the sumalt-procedure in Pari/GP: Code:fmt(200,12)   \\ set internal float precision to 200 and display digits to 12                     \\                   (by user defined function) tb=1.3 \\ set the exponential base in global variable tl = log(tb) \\ set the log of the base \\ procedure to allow sequential access to consecutive iterates via sumalt() \\ nextx(x,0) - initializes glbx \\ nextx(x,h) - if h>0 gives the next iterate towards the attracting fixpoint \\ nextx(x,h) - if h<0 gives the next iterate towards the repelling fixpoint       glx=0   \\ global x-variable nextx(x,h=1)=if(h==0,glx=x,if(h>0,glx=exp(glx*tl)-1,glx=log(1+glx)/tl));return(glx)     \\   nextx(1.5, 0)  \\ example call \\ == procedure for doubly infinite alternating iteration series beginning at x {asum(x)=local(su0,su1,su);      su0  = sumalt(k=0,(-1)^k*nextx(x,k));      su1  = sumalt(k=0,(-1)^k*nextx(x,-k));      su   = su0+su1-x; return(su); }     \\ asum(0.6) \\ example [update]: Even more simple: the alternating sum towards the fixpoint fp0=0 converges; but also the alternating sum towards the upper fixpoint fp1 can be made by separating the convergent sum of the $\pm (x_{-h} - fp1)$ and then add the half of the fixpoint (the dirichlet's eta at 0 is 0.5): PHP Code:sum(h=0,100,(-1)^h*(nextx(1.4,-h) -fp1 ) )  + 0.5* fp1  (take a meaningful upper limit for the sum instead of 100) Ah... now I see, it seems I wasn't using enough terms in the Cesaro summation. But dang, this thing is close to 0. (EDIT: Ah! I see the scale in your graph... I didn't see that second scale on the right hand side -- guess I'm all OK here with this, now.) Using the sumalt, or even better convergent summation, seems to yield the same values, so I guess it's rescued. « Next Oldest | Next Newest »

 Messages In This Thread [AIS] (alternating) Iteration series: Half-iterate using the AIS? - by Gottfried - 12/06/2012, 12:10 AM RE: Half-iterate using the infinite iteration-series? - by tommy1729 - 12/08/2012, 06:27 PM RE: Half-iterate using the infinite iteration-series? - by Gottfried - 12/09/2012, 03:13 PM RE: Half-iterate using the infinite iteration-series? - by Gottfried - 12/10/2012, 02:23 AM RE: Half-iterate using the infinite iteration-series? - by mike3 - 12/10/2012, 03:33 AM RE: Half-iterate using the infinite iteration-series? - by Gottfried - 12/10/2012, 10:38 AM RE: Half-iterate using the infinite iteration-series? - by mike3 - 12/10/2012, 11:26 AM RE: Iteration series: Half-iterate using the infinite iteration-series? - by sheldonison - 12/11/2012, 12:44 AM RE: Iteration series: Half-iterate using the infinite iteration-series? - by Gottfried - 12/11/2012, 01:16 AM RE: Iteration series: Half-iterate using the infinite iteration-series? - by sheldonison - 12/13/2012, 02:49 AM RE: Iteration series: Half-iterate using the infinite iteration-series? - by sheldonison - 12/15/2012, 04:47 AM RE: Iteration series: Half-iterate using the infinite iteration-series? - by Gottfried - 12/15/2012, 06:49 AM RE: Iteration series: Half-iterate using the infinite iteration-series? - by sheldonison - 12/15/2012, 01:43 PM RE: Iteration series: Half-iterate using the infinite iteration-series? - by Gottfried - 12/15/2012, 11:27 PM RE: Iteration series: Half-iterate using the infinite iteration-series? - by Gottfried - 12/16/2012, 12:05 PM RE: Iteration series: Half-iterate using the infinite iteration-series? - by sheldonison - 12/17/2012, 10:19 PM RE: Iteration series: Half-iterate using the infinite iteration-series? - by Gottfried - 12/18/2012, 11:17 AM RE: Iteration series: Half-iterate using the infinite iteration-series? - by sheldonison - 12/22/2012, 04:12 PM RE: Iteration series: Half-iterate using the infinite iteration-series? - by Gottfried - 12/23/2012, 09:01 AM RE: Iteration series: Half-iterate using the infinite iteration-series? - by sheldonison - 12/23/2012, 03:51 PM RE: Iteration series: Half-iterate using the infinite iteration-series? - by Nasser - 12/14/2012, 05:06 PM RE: Iteration series: Half-iterate using the infinite iteration-series? - by Gottfried - 12/14/2012, 07:16 PM RE: Iteration series: Half-iterate using the infinite iteration-series? - by Gottfried - 12/15/2012, 06:37 AM RE: Iteration series: Half-iterate using the infinite iteration-series? - by andydude - 12/18/2012, 06:05 AM RE: Iteration series: Half-iterate using the infinite iteration-series? - by tommy1729 - 12/15/2012, 10:06 PM RE: Iteration series: Half-iterate using the infinite iteration-series? - by Gottfried - 12/23/2012, 04:43 PM RE: Iteration series: Half-iterate using the infinite iteration-series? - by Gottfried - 12/23/2012, 05:59 PM RE: Iteration series: Half-iterate using the infinite iteration-series? - by Gottfried - 01/02/2013, 03:28 PM RE: Iteration series: Half-iterate using the infinite iteration-series? - by Gottfried - 01/02/2013, 04:05 PM RE: Iteration series: Half-iterate using the infinite iteration-series? - by Gottfried - 01/03/2013, 07:07 AM RE: Iteration series: Half-iterate using the infinite iteration-series? - by sheldonison - 01/03/2013, 09:58 PM RE: [AIS] (alternating) Iteration series: Half-iterate using the AIS? - by Gottfried - 03/26/2015, 01:01 PM RE: [AIS] (alternating) Iteration series: Half-iterate using the AIS? - by tommy1729 - 03/27/2015, 11:28 PM

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