Re: Re: Re: Limit of list

*To*: mathgroup at smc.vnet.net*Subject*: [mg57663] Re: [mg57638] Re: [mg57527] Re: Limit of list*From*: DrBob <drbob at bigfoot.com>*Date*: Fri, 3 Jun 2005 05:34:03 -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> <200506020918.FAA12003@smc.vnet.net>*Reply-to*: drbob at bigfoot.com*Sender*: owner-wri-mathgroup at wolfram.com

> 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. What "extra information" do SAT test-takers have? None, except that the pattern should be relatively "simple", or test-writers wouldn't expect them to get it. Whatever "simple" means. (Different things to different people, I fear.) And what was the OP's "_real_ problem"? What extra information does he have? We have no clue. His sequence isn't in my copy of Sloane and PLouffe's Encyclopedia of Integer Sequences, so that may rule out "relatively simple". (Or it may do just the opposite.) And how would we pass extra information to SequenceLimit? We can't, can we? SequenceLimit selects a "best" match from a small subset of all sequences that could extend the list. If it uses the wrong subset or the wrong notion of "best", it will return a wrong answer. ("Wrong" is just as unknowable as "right", of course, but it's much more likely.) Back to the OP's general question, which was: > Guy Israeli <guyi1 at netvision.net.il> wrote: > Is there a way to find out the convergence point of a list of numbers? No. Bobby On Thu, 2 Jun 2005 05:18:08 -0400 (EDT), Paul Abbott <paul at physics.uwa.edu.au> wrote: > 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 > > -- DrBob at bigfoot.com

**References**:**Re: Re: Limit of list***From:*Paul Abbott <paul@physics.uwa.edu.au>