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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 Van Wijngaarden transformation ： ウィキペディア英語版 Van Wijngaarden transformation 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 : :$s\_\; =\; \backslash sum\_^k(-1)^n\; a\_n$ and form rows of averages between neighbors, :$\backslash ,\; s\_\; =\; \backslash frac\}2$ The first column $\backslash scriptstyle\; s\_$ 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.〔A. van Wijngaarden, in: Cursus: Wetenschappelijk Rekenen B, Proces Analyse, Stichting Mathematisch Centrum, (Amsterdam, 1965) pp. 51-60〕 If $\backslash scriptstyle\; a\_0,a\_1,\; \backslash ldots,\; a\_$ are available, then $\backslash scriptstyle\; s\_$ is almost always a better approximation to the sum than $\backslash scriptstyle\; s\backslash ,\; \_.$ Leibniz formula for pi, $\backslash scriptstyle\; 1\; -\; \backslash frac\; 1\; 3\; +\; \backslash frac\; 1\; 5\; -\; \backslash frac\; 1\; 7\; +\; \backslash cdots\; =\; \backslash frac\; \backslash pi\; 4\; =\; 0.7853981\backslash ldots$, gives the partial sum $\backslash scriptstyle\; \backslash ,s\_\; =\; 0.8046006...\; (+2.4\backslash \%)$, the Euler transform partial sum $\backslash scriptstyle\; \backslash ,s\_\; =\; 0.7854002...\; (+2.6\; \backslash times\; 10^)$ and the van Wijngaarden result $\backslash scriptstyle\; \backslash ,s\_\; =\; 0.7853982...\; (+4.7\; \backslash times\; 10^)$ (relative errors are in round brackets). 1.00000000 0.66666667 0.86666667 0.72380952 0.83492063 0.74401154 0.82093462 0.75426795 0.81309148 0.76045990 0.80807895 0.76460069 0.80460069 0.83333333 0.76666667 0.79523810 0.77936508 0.78946609 0.78247308 0.78760129 0.78367972 0.78677569 0.78426943 0.78633982 0.78460069 0.80000000 0.78095238 0.78730159 0.78441558 0.78596959 0.78503719 0.78564050 0.78522771 0.78552256 0.78530463 0.78547026 0.79047619 0.78412698 0.78585859 0.78519259 0.78550339 0.78533884 0.78543410 0.78537513 0.78541359 0.78538744 0.78730159 0.78499278 0.78552559 0.78534799 0.78542111 0.78538647 0.78540462 0.78539436 0.78540052 0.78614719 0.78525919 0.78543679 0.78538455 0.78540379 0.78539555 0.78539949 0.78539744 0.78570319 0.78534799 0.78541067 0.78539417 0.78539967 0.78539752 0.78539847 0.78552559 0.78537933 0.78540242 0.78539692 0.78539860 0.78539799 0.78545246 0.78539087 0.78539967 0.78539776 0.78539829 0.78542166 0.78539527 0.78539871 0.78539803 0.78540847 0.78539699 0.78539837 0.78540273 0.78539768 0.78540021 This table results from the J formula 'b11.8'8!:2-:&(}:+}.)^:n+/\(_1^n) *%1+2 *n=.i.13 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. == References == 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「Van Wijngaarden transformation」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.