Van Wijngaarden transformation
In mathematics and numerical analysis, in order to accelerate convergence of an alternating series, Euler's transform can be computed as follows.
Compute a row of partial sums :
and form rows of averages between neighbors,
The first column then contains the partial sums of the Euler transform.
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 are available, then is almost always a better approximation to the sum than
Leibniz formula for pi,, gives the partial sum, the Euler transform partial sum and the van Wijngaarden result .
This table results from the J formula 'b11.8'8!:2-:&(
