Van Wijngaarden transformation

In mathematics and numerical analysis, the van Wijngaarden transformation is a variant on the Euler transform used to accelerate the convergence of an alternating series.

One algorithm to compute Euler's transform runs as follows: "Compute a row of partial sums $s_{0,k} = \sum_{n=0}^k(-1)^n a_n$ and form rows of averages between neighbors $s_{j+1,k} = \frac{s_{j,k}+s_{j,k+1}}2$ The first column $s_{j,0}$ then contains the partial sums of the Euler transform.|undefined"

Adriaan van Wijngaarden's contribution was to point out that it is better not to carry this procedure through to the very end, but to stop two-thirds of the way. If $$a_0,a_1,\ldots,a_{12}$$ are available, then $$s_{8,4}$$ is almost always a better approximation to the sum than $$s_{12,0}$$. In many cases the diagonal terms do not converge in one cycle so process of averaging is to be repeated with diagonal terms by bringing them in a row. (For example, this will be needed in a geometric series with ratio $$-4$$.) This process of successive averaging of the average of partial sum can be replaced by using the formula to calculate the diagonal term.

For a simple-but-concrete example, recall the Leibniz formula for pi  The algorithm described above produces the following table:

These correspond to the following algorithmic outputs: