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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 Sidi's generalized secant method ： ウィキペディア英語版 Sidi's generalized secant method Sidi's generalized secant method is a root-finding algorithm, that is, a numerical method for solving equations of the form $f(x)=0$ . The method was published by Avram Sidi.〔 Sidi, Avram, "Generalization Of The Secant Method For Nonlinear Equations", Applied Mathematics E-notes 8 (2008), 115–123, http://www.math.nthu.edu.tw/~amen/2008/070227-1.pdf 〕 The method is a generalization of the secant method. Like the secant method, it is an iterative method which requires one evaluation of $f$ in each iteration and no derivatives of $f$. The method can converge much faster though, with an order which approaches 2 provided that $f$ satisfies the regularity conditions described below. == Algorithm == We call $\backslash alpha$ the root of $f$, that is, $f(\backslash alpha)=0$. Sidi's method is an iterative method which generates a sequence $\backslash$ of approximations of $\backslash alpha$. Starting with ''k'' + 1 initial approximations $x\_1\; ,\; \backslash dots\; ,\; x\_$, the approximation $x\_$ is calculated in the first iteration, the approximation $x\_$ is calculated in the second iteration, etc. Each iteration takes as input the last ''k'' + 1 approximations and the value of $f$ at those approximations. Hence the ''n''th iteration takes as input the approximations $x\_n\; ,\; \backslash dots\; ,\; x\_$ and the values $f(x\_n)\; ,\; \backslash dots\; ,\; f(x\_)$. The number ''k'' must be 1 or larger: ''k'' = 1, 2, 3, .... It remains fixed during the execution of the algorithm. In order to obtain the starting approximations $x\_1\; ,\; \backslash dots\; ,\; x\_$ one could carry out a few initializing iterations with a lower value of ''k''. The approximation $x\_$ is calculated as follows in the ''n''th iteration. A polynomial of interpolation $p\_\; (x)$ of degree ''k'' is fitted to the ''k'' + 1 points $(x\_n,\; f(x\_n)),\; \backslash dots\; (x\_,\; f(x\_))$. With this polynomial, the next approximation $x\_$ of $\backslash alpha$ is calculated as with $p\_\text{'}(x\_)$ the derivative of $p\_$ at $x\_$. Having calculated $x\_$ one calculates $f(x\_)$ and the algorithm can continue with the (''n'' + 1)th iteration. Clearly, this method requires the function $f$ to be evaluated only once per iteration; it requires no derivatives of $f$. The iterative cycle is stopped if an appropriate stop-criterion is met. Typically the criterion is that the last calculated approximation is close enough to the sought-after root $\backslash alpha$. To execute the algorithm effectively, Sidi's method calculates the interpolating polynomial $p\_\; (x)$ in its Newton form. 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「Sidi's generalized secant 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.