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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 Steffensen's method ： ウィキペディア英語版 Steffensen's method In numerical analysis, Steffensen's method is a root-finding method, similar to Newton's method, named after Johan Frederik Steffensen. Steffensen's method also achieves quadratic convergence, but without using derivatives as Newton's method does. ==Simple description== The simplest form of the formula for Steffensen's method occurs when it is used to find the zeros, or roots, of a function $f$ ; that is: to find the value $x\_\backslash star$ that satisfies $f(x\_\backslash star)=0$ . Near the solution $x\_\backslash star$ , the function $f$ is supposed to approximately satisfy  ; this condition makes the function $f$ adequate as a correction for finding its ''own'' solution, although it is not required to work efficiently. For some functions, Steffensen's method can work even if this condition is not met, but in such a case, the starting value $x\_0\backslash$ must be ''very'' close to the actual solution $x\_\backslash star$ , and convergence to the solution may be slow. Given an adequate starting value $x\_0\backslash$ , a sequence of values $x\_0,\backslash \; x\_1,\backslash \; x\_2,\backslash dots,\backslash \; x\_n,\backslash dots$ can be generated using the formula below. When it works, each value in the sequence is much closer to the solution $x\_\backslash star$ than the prior value. The value $x\_n\backslash$ from the current step generates the value $x\_\backslash$ for the next step, via this formula:〔Germund Dahlquist, Åke Björck, tr. Ned Anderson. (1974). ''Numerical Methods'', pp. 230–231. Englewood Cliffs, NJ: Prentice Hall.〕 :$x\_\; =\; x\_n\; -\; \backslash frac$ for ''n'' = 0, 1, 2, 3, ... , where the slope function $g(x\_n)$ is a composite of the original function $f$ given by the following formula: :$g(x\_n)\; =\; \backslash frac.$ The function $g$ is the average value for the slope of the function $f$ between the last sequence point $(x,y)\; =\; (\; x\_n,\backslash \; f(x\_n)\; )$ and the auxiliary point $(x,y)=(\; x\_n\; +\; h,\backslash \; f(x\_n\; +\; h)\; )$ , with the step $h=f(x\_n)\backslash$ . It is also called the first-order divided difference of $f$ between those two points. It is only for the purpose of finding $h$ for this auxiliary point that the value of the function $f$ must be an adequate correction to get closer to its own solution, and for that reason fulfill the requirement that  . For all other parts of the calculation, Steffensen's method only requires the function $f$ to be continuous and to actually have a nearby solution. Several modest modifications of the step $h$ in the slope calculation $g$ exist to accommodate functions $f$ that do not quite meet the requirement. 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「Steffensen's method」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.