Re: Re: Limit of list
- To: mathgroup at smc.vnet.net
- Subject: [mg57638] Re: [mg57527] Re: Limit of list
- From: Paul Abbott <paul at physics.uwa.edu.au>
- Date: Thu, 2 Jun 2005 05:18:08 -0400 (EDT)
- References: <d79ejm$lb1$1@smc.vnet.net> <200505310859.EAA03396@smc.vnet.net> <opsrnpltapiz9bcq@monster.ma.dl.cox.net> <p06210287bec30fedd446@[130.95.156.21]> <opsro8xdopiz9bcq@monster.ma.dl.cox.net>
- Sender: owner-wri-mathgroup at wolfram.com
On 1/6/05, DrBob wrote: >>>Just as well, I guess, since it can't possibly work. >>Why not? > >Because a finite sequence has an uncountable number of extensions, >most of which don't converge, others of which converge to anything >one cares to arbitrarily choose. Of course. But _real_ questions do not emerge from a vacuum. Their context can provide a definite answer to such a question. >SequenceLimit simply gives an answer to the question, "What is the >result of Wynn's epsilon algorithm for this list of numbers?" SequenceLimit gives the limit of a sequence (computed using Wynn's epsilon algorithm). Since the poster was asking a question about the limit of a sequence, surely SequenceLimit is an appropriately named function? >Since I've never heard of the algorithm until now, it's not likely I >would ask that question. But surely the question you would ask (not necessarily to Mathematica) involves the keyword "convergence"? If you search for this keyword at MathWorld then the second match is to http://mathworld.wolfram.com/ConvergenceImprovement.html and there is a link from there to http://mathworld.wolfram.com/WynnsEpsilonMethod.html >SAT questions that ask, "What's the next term in this sequence?" are >written by mathematical morons who (apparently) don't realize all >the choices are equally valid. I agree that SAT questions of this type are stupid. However, in _real_ problems all the choices are _not_ equally valid -- there is some extra information to help guide us. Also, it is often that case that when one has n terms of a sequence, one can produce additional terms if required, to do a sanity check. There is also another sense in which this type of approach can be optimal. In the paper "Maximum entropy summation of divergent perturbation series" by Carl M. Bender, Lawrence R. Mead, and N. Papanicolaou (Journal of Mathematical Physics (1987) 28(5): 1016-1018) one can extract sense from a divergent series using maximum entropy as guiding principle. The analogy presented there is nice: If you heat an object and let it cool down and then measure its heat distribution, because the diffusion of heat is a smoothing process, there is no unique initial state leading to the observed final stated. However, you can use maximum entropy to find the most likely initial state. I am arguing that there is an analogy to the problem of determining the limit of a sequence (especially one that emerges from a "real" problem). A similar situation is de-blurring a photograph. Cheers, Paul >On Wed, 1 Jun 2005 15:13:49 +0800, Paul Abbott ><paul at physics.uwa.edu.au> wrote: > >>>SequenceLimit. Another COMPLETELY undocumented feature. >> >>Not COMPLETELY undocumented. Try >> >> ?SequenceLimit >> >>>Just as well, I guess, since it can't possibly work. >> >>Why not? SequenceLimit returns the approximation given by Wynn's >>epsilon algorithm to the limit of a sequence whose first few terms >>are given by list. This algorithm can give finite results for >>divergent sequences. As I understand it, SequenceLimit is used by >>NIntegrate when Method->Oscillatory. >> >>Cheers, >>Paul >> >>> >>>Bobby >>> >>>On Tue, 31 May 2005 04:59:40 -0400 (EDT), Paul Abbott >>><paul at physics.uwa.edu.au> wrote: >>> >>>>In article <d79ejm$lb1$1 at smc.vnet.net>, >>>> Guy Israeli <guyi1 at netvision.net.il> wrote: >>>> >>>>>Is there a way to find out the convergence point of a list of numbers? >>>>> >>>>>for example if I have >>>>> >>>>>{1,2,5,6,8,9,10,11,10,11,12,11,12.. and so on} >>>>> >>>>>it will give me something around 10-12 >>>> >>>>Try SequenceLimit: >>>> >>>> SequenceLimit[{1,2,5,6,8,9,10,11,10,11,12,11,12}] >>>> >>>>Also, if your list is entering a cycle there have been previous >>>>MathGroup postings on methods for detecting cycles. >>>> >>>>Cheers, >>>>Paul >>>> >>> >>> >>> >>>-- >>>DrBob at bigfoot.com >> >> >> >> >> > > > >-- >DrBob at bigfoot.com -- ____________________________________________________________________ Paul Abbott Phone: +61 8 6488 2734 School of Physics, M013 Fax: +61 8 6488 1014 The University of Western Australia (CRICOS Provider No 00126G) 35 Stirling Highway Crawley WA 6009 mailto:paul at physics.uwa.edu.au AUSTRALIA http://physics.uwa.edu.au/~paul Conference Chair for International Mathematica Symposium 2005 http://InternationalMathematicaSymposium.org/IMS2005/ ____________________________________________________________________
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