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
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

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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