Talk:Van Wijngaarden transformation

Is this article correct?

I did a little digging around, and found this reference to the van Wijngaarden transform (defined on page 4 of the .pdf file). The transformation described in that paper is not applied to an alternating series, but is used to transform a monotone series into an alternating series. And the transformation formulae in that paper do not correspond with the formula presented in this article. So I'm a little bit confused.

Is the transformation defined in this article also described as a "van Wijngaarden transform" in the literature? Did van Wijngaarden invent several different series transformation formulas, any one of which may be called a "van Wijngaarden transform"? This article may need some help! DavidCBryant 01:16, 8 August 2007 (UTC)

The description of this elegant transformation is quite incomplete. Numerical Recipes section 5.1 has a clear concise description of this transformation and shows how the tranformation can be extended from alternating series to monotone series. --24.22.30.78 16:56, 1 November 2007 (UTC)

No, I don't think it is correct. Numerical Recipes Webnote No 5 says:

"There is an elegant and subtle implementation of Euler’s transformation due to van Wijngaarden : It incorporates the terms of the original alternating series one at a time, in order. For each incorporation it either increases p by 1, equivalent to computing one further difference, or else retroactively increases n by 1, without having to redo all the difference calculations based on the old n value! The decision as to which to increase, n or p, is taken in such a way as to make the convergence most rapid."

Which is certainly not the procedure described on this page, which I haven't yet come across under this or any other name, although I am new to this area. Maybe the 2/3 emerges as approx. best, in practice. Likely it's "What van W's algorithm mostly amounts to in practice". I'm not sure if it's "original research" - it does seem like it - their misconception or mis-remembering what van W's method is. Although the method on this page at least worked well when I tried it. I think the page should probably be deleted though, if not corrected, and it seems no-one's ever going to do that. It's only lasted this long because no-one has a clue what to make of it, I guess.

p.s. I just checked who created the page, and it turns out the only edits the creator, Dirk.laurie, ever did on wikipedia, was to create this page in 2007, and some minor edits to it, (including the "2/3" bit) all within a 6 minute period. Hehe. Of course, maybe they changed their username. 110.20.158.134 (talk) 09:59, 20 November 2015 (UTC)

Dirk Laurie wrote the chapter on convergence acceleration in the book The SIAM 100-Digit Challenge: A Study in High-Accuracy Numerical Computing. The Van Wijngaarden transformation is not discussed there, but the recursive method for calculating the Euler transformation is. However, the Van Wijngaarden method fits into the framework of the discussion of Operator Polynomial Extrapolation on p239-243 of that book. The Euler method corresponds to the polynomial $(t+1)^{n+1}$, as the book states, whereas the Van Wyngaarden method (when $n+1=3m$) corresponds to the polynomial $t^{m}(t+1)^{2m}$. The error bound is proportional to $\max p(t)/p(1)$, with the maximum taken over $[-1,0]$. 105.228.218.79 (talk) 09:52, 8 April 2016 (UTC)

Two Thirds?

It's also not very clear what "stop two-thirds of the way" actually means. This should be clarified. --207.172.223.249 (talk) 12:58, 12 September 2012 (UTC)

Illustrative table

I included the table. Bo Jacoby (talk) 21:09, 2 June 2010 (UTC).