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: <blockquote>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.</blockquote>
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 . 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 .) 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 ÃÂ: The algorithm described above produces the following table:
These correspond to the following algorithmic outputs: