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 (relative errors are in round brackets).
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. This will be needed in an geometric series with ratio -4. This process of successive averaging of the average of partial sum can be replaced by using formula to calculate the diagonal term.
Famous quotes containing the word van:
“It is not a certain conformity of manners that the painting of Van Gogh attacks, but rather the conformity of institutions themselves. And even external nature, with her climates, her tides, and her equinoctial storms, cannot, after van Gogh’s stay upon earth, maintain the same gravitation.”
—Antonin Artaud (1896–1948)