van Wijngaarden transformation resources Wikipedia's treatment is superficial, with only the original Dutch paper as a reference. I'm interested in resources with more detailed discussion of the transformation. In particular I hope to learn how rapidly the transformation converges (the answer will presumably depend on how rapidly the original series converges), or why the stop-two-thirds-of-the-way trick works.
 A: A bit too large for a comment. This paper by Cohen, Villegas, and Zagier, is a nice reference. Although they introduce their own variation on the method, the exposition also covers convergence for the Euler and Euler-Van Wijngaarden methods, which they show are all part of a family of methods. See in particular Proposition 1 and the remarks that follow it.
For the Euler method (and a particular class of sequences) you get an error of $2^{-d}$ with $d$ summands, while for the Euler-Van Wijngaarden method this is improved to $3^{-d}$.
The paper also, implicitly, explains where the “stop at two thirds” comes from. All these methods are based on a polynomial approximation of $(1+x)^{-1}$ on the interval $[0,1]$. The Taylor expansion at $x=0$ results simply in the partial sums as approximation. The Taylor expansion at $x=1$ leads to the Euler method. Note that in this case the maximum error of the polynomial approximation is $2^{-d}$ at $x=0$.
The Van Wijngaarden trick is to use a polynomial that agrees to order $d_1$ at $x=0$ and to degree $d_2$ at $x=1$, where $d = d_1 + d_2$ is the total degree. It turns out that the maximal error over $[0,1]$ is minimized for $d_1$ and $d_2$ close to $d/3$ and $2d/3$ respectively.
Finally, the method proposed in the linked paper proposes an approximation based on Chebyshev polynomials to achieve a near optimal maximal error.
