Van Wijngaarden transformation

In mathematics and numerical analysis, in order to accelerate convergence, Euler's transform can be implemented as follows: compute the partial sums of an alternating series::$s\_\{k,0\}\; =\; sum\_\{n=0\}^k(-1)^n\; a\_n.$Form a new sequence by simply taking the average of two consecutive terms, i.e.:$s\_\{k,j+1\}\; =\; (s\_\{k,j\}+s\_\{k+1,j\})/2,\; quad\; k=0,1,2,ldots,$where $,\; j=0.$Keep on doing this for $j=1,2,ldots.$ The partial sums of Euler's transform are $s\_\{0,0\},\; s\_\{0,1\},\; s\_\{0,2\},dots$

Van Wijngaarden's contribution was to point out that it is better not to carry this procedurethrough to the very end, but to stop two-thirds of the way [A. van Wijngaarden, in: Cursus: Wetenschappelijk Rekenen B, Process Analyse, Stichting Mathematisch Centrum, (Amsterdam, 1965) pp 51-60] . E.g. if you have $a\_0,a\_1,ldots,a\_\{12\}$ available, then $,\; s\_\{4,8\}$ is almost always a better approximationto the sum than $s,\; \_\{0,12\}.$

For example, Leibniz's series:$4\; (1\; -\; 1/3\; +\; 1/5\; -\; 1/7\; +\; cdots)\; =\; pi\; =\; 3.14159265ldots$gives $,s\_\{0,12\}\; =\; 3.1416008,$ whereas$,s\_\{4,8\}\; =\; 3.1415931.$

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