Authors: Andrej Liptaj
If several independent algorithms for a computer-calculated quantity exist, then one can expect their results (which differ because of numerical errors) to follow approximately Gaussian distribution. The mean of this distribution, interpreted as the value of the quantity of interest, can be determined with much better precision than what is the precision provided by a single algorithm. Many practical algorithms introduce a bias using a parameter, e.g. a small but finite number to compute a limit or a large but finite number (cutoff) to approximate infinity. One may vary such parameter of a single algorithm, interpret the resulting numbers as generated by several algorithms and compute the average. A numerical evidence for the validity of this approach is, in the context of a fixed machine epsilon, shown for differentiation: the method greatly improves the precision and leads, presumably, to the most precise numerical differentiation nowadays known.
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