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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Aitken's delta-squared process ： ウィキペディア英語版 Aitken's delta-squared process In numerical analysis, Aitken's delta-squared process is a series acceleration method, used for accelerating the rate of convergence of a sequence. It is named after Alexander Aitken, who introduced this method in 1926.〔Alexander Aitken, "On Bernoulli's numerical solution of algebraic equations", ''Proceedings of the Royal Society of Edinburgh'' (1926) 46 pp. 289–305.〕 Its early form was known to Seki Kōwa (end of 17th century) and was found for rectification of the circle, i.e. the calculation of π. It is most useful for accelerating the convergence of a sequence that is converging linearly. ==Definition== Given a sequence $x=\_$, one associates with this sequence the new sequence :$A\; x=)^2\}+x\_n\}\backslash right)\}\_,$ which can, with improved numerical stability, also be written as :$(A\; x)\_n=x\_n-\backslash frac,$ or equivalently $=\; x\_\; -\; \backslash frac$ where :$\backslash Delta\; x\_=)\},\backslash \; \backslash Delta\; x\_=)\},$ and :$\backslash Delta^2\; x\_n=x\_n\; -2x\_\; +\; x\_=\backslash Delta\; x\_-\backslash Delta\; x\_,\backslash$ for $n\; =\; 0,\; 1,\; 2,\; 3,\; \backslash dots\; \backslash ,$ Obviously, ''A x'' is ill-defined if Δ2x contains a zero element, or equivalently, if the sequence of first differences has a repeating term. From a theoretical point of view, assuming that this occurs only for a finite number of indices, one could easily agree to consider the sequence ''A x'' restricted to indices ''n>n0'' with a sufficiently large ''n0''. From a practical point of view, one does in general rather consider only the first few terms of the sequence, which usually provide the needed precision. Moreover, when numerically computing the sequence, one has to take care to stop the computation when rounding errors become too important in the denominator, where the Δ² operation may cancel too many significant digits. (It would be better for numerical calculation to use $\backslash Delta\; x\_\; -\; \backslash Delta\; x\_\backslash \; =\; (x\_-x\_)-(x\_-x\_)\backslash$ rather than $x\_n\; -\; 2x\_\; +\; x\_\backslash$ .) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Aitken's delta-squared process」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.