Talk:Sequence transformation

Just created Aitken's delta-squared process (suggested on "missing pages"), didn't notice this page before. Merging might be a good idea, but I think Aitken's method deserves an own page (even if it's a stub at present), so I suggest to move from here to there what is not yet there. &mdash; MFH:Talk 18:15, 12 October 2006 (UTC)

Merging Series Acceleration
I am merging Series Acceleration into this article. I just added MPE and realized that there were several related articles on wikipedia where this information was needed, so I am collecting it all together here.

I have not merged the following bit of text from Series Acceleration.

bad formula for aitken method
The article derives
 * $$s = s_n + \frac{(s_{n+1} - s_n)^2}{s_{n+2} - 2s_{n+1} + s_{n}}.$$

but this formula is clearly mistaken. The error is seen very easily: consider s_n to be a strictly increasing series. Then the above formula gives an estimate that is smaller than the current s_n, because the correction term is negative. Clearly that is erroneous. The correction term needs to be subtracted, not added.

The correct formula is in fact
 * $$s = s_{n+2} - \frac{(s_{n+1} - s_n)^2}{s_{n+2} - 2s_{n+1} + s_{n}}.$$

Besides the minus sign; note also the term being corrected. This can be vareified in any text that actually discusses the Aitken method; I'm looking at "numerical methods in C" right now. Can someone please fix this? linas (talk) 03:27, 2 June 2008 (UTC)


 * I think you don't give the correct formula either, which is:
 * s_n^'=s_(n+1)-((s_(n+1)-s_n)^2)/(s_(n+1)-2s_n+s_(n-1)).
 * or something equivalent to it. However, I think it is not a good idea to suff in lots of details about one particular method into one section of this article (which has lots of other problems). Most of the details (upon correction and rewriting) should be moved to the main article Aitken's method. &mdash; MFH:Talk 13:47, 2 June 2008 (UTC)


 * Yes, right, thanks; I see that there was a bit of a mess made with redirects and what-not. Things seem a bit improved now. linas (talk) 17:02, 2 June 2008 (UTC)

Those who edit formulas should be careful. Besides the (correct) formula in the original version of the page, it is possible to derive from it different expressions like


 * $$ s_n^'= s_{n+2} - \frac{(s_{n+2} - s_{n+1})^2}{s_{n+2} - 2s_{n+1} + s_{n}}$$

and


 * $$ s_n^'= s_{n+1} - \frac{(s_{n+2} - s_{n+1})(s_{n+1} - s_n)}{s_{n+2} - 2s_{n+1} + s_{n}}$$

In the context of iteration of the Delta-Squared method, these can behave differently as shown by Weniger.

DerHannes (talk) 19:18, 21 March 2010 (UTC)

change title to "series acceleration"
The most common name (that I know of) for the techniques described in this areticle is not "sequence transformations", but "series acceleration". There are many sequence transformations that do not accelerate convergence. Wikipedia has some articles for these (I'm drawing a blank for what some of these are called, otherwise I'd give an example here). The merge was incorrect, it needs to be un-merged. linas (talk) 03:32, 2 June 2008 (UTC)


 * I fully agree. The initial phrase, "In mathematics, a sequence transformation is a resummation of a sequence." is plainly wrong.&mdash; MFH:Talk 13:52, 2 June 2008 (UTC)


 * I resurrected the old series acceleration article, and moved much of the content of this article to there. linas (talk) 18:17, 2 June 2008 (UTC)

cut & paste of Aitken's chunk
I cut & paste here a part of the paragraph on Aitken's method, which I believe (a) too much detailed, and (b) simply wrong to a large extend.

The sequence can be derived by assuming that the individual $$s_n$$ converge at a constant rate towards some $$s$$ with a convergence rate $$q < 1$$. Therefore, we can write the error of $$s_{n+1}$$ as


 * $$(s_{n+1} - s) = q(s_n - s),$$

from which we can extract the convergence rate


 * $$q = \frac{(s_{n+1}-s)}{(s_n - s)}.$$

We can then use this convergence rate in the error of $$s_{n+2}$$:




 * $$(s_{n+2}-s)$$||$$=$$||$$q(s_{n+1} - s)$$
 * ||$$=$$||$$\frac{(s_{n+1}-s)}{(s_n - s)}(s_{n+1} - s)$$
 * }
 * }

from which we can then extract the only unknown, $$s$$:




 * align=right|$$(s_{n+2}-s)(s_n-s)$$||=||$$(s_{n+1}-s)^2$$
 * align=right|$$s_{n+2}s_n - s(s_n + s_{n+2}) + s^2$$||=||$$s_{n+1}^2 - 2s_{n+1}s + s^2$$
 * align=right|$$s(s_{n+2} - 2s_{n+1} + s_{n})$$||=||$$s_{n+2}s_n - s_{n+1}^2$$
 * align=right|$$s$$||=||$$s_n + \frac{(s_{n+1} - s_n)^2}{s_{n+2} - 2s_{n+1} + s_{n}}.$$
 * }
 * align=right|$$s$$||=||$$s_n + \frac{(s_{n+1} - s_n)^2}{s_{n+2} - 2s_{n+1} + s_{n}}.$$
 * }
 * }

Taking the sequence of partial sums


 * $$s_n=\sum_{k=0}^{n} q^k=\frac{1-q^{n+1}}{1-q}$$

of the geometric series as untransformed sequence, we obtain for the transformed sequence by simple algebra


 * $$s'_n=\frac{1}{1-q}$$

independent of $$n$$. This, however is the limit of the geometric series for &#124;q&#124; < 1. Thus, the Aitken method yields the exact result by extrapolation of only three consecutive partial sums. For &#124;q&#124; > 1, the geometric series diverges since it is a power series in q outside its radius of convergence. But even for this divergent series, the Aitken method sums the geometric series to its analytic continuation for all complex numbers q &ne; 1.

The Aitken method is often used iteratively by applying it again to the transformed sequence S&prime;:


 * $$\displaystyle S''=\mathbb{A}(S') $$


 * $$\displaystyle S'=\mathbb{A}(S) $$

and so on. (end of paste) Actually, I don't know of any realistic application of the last statement, and even think that it is wrong in several regards; to start with, I don't think repeated application improves anything (but rather will give worse results due to problems in the denominator.) &mdash; MFH:Talk 15:26, 2 June 2008 (UTC)

redirect Aitken's method
Hi Oleg, you changed the redirect of Aitken delta-squared process to Sequence transformations, I just undid this after re-establishing the former article. I think Sequence transformations is extremely ill written, it starts with a plainly wrong statement (in fact it seems to be about acceleration of convergence, rather). IMHO sequence transformations should be a) spelled in singular, b) contain material about generic sequence transformations (binomial, ...). Since you know the math part of WP way better than me, I invite you to participate in the discussion at Talk:Sequence transformations and give links to relevant material.&mdash; MFH:Talk 14:02, 2 June 2008 (UTC)
 * As far as i can tell, by this I just fixed a double redirect, following this. I have no comment on the actual content, feel free to write it as you feel best. Cheers, Oleg Alexandrov (talk) 16:39, 2 June 2008 (UTC)