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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 Halley's method ： ウィキペディア英語版 Halley's method In numerical analysis, Halley’s method is a root-finding algorithm used for functions of one real variable with a continuous second derivative, i.e., C2 functions. It is named after its inventor Edmond Halley. The algorithm is second in the class of Householder's methods, right after Newton's method. Like the latter, it produces iteratively a sequence of approximations to the root; their rate of convergence to the root is cubic. Multidimensional versions of this method exist. Halley's method can be viewed as exactly finding the roots of a linear-over-linear Padé approximation to the function, in contrast to Newton's method/Secant method (approximates/interpolates the function linearly) or Cauchy's method/Muller's method (approximates/interpolates the function quadratically). ==Method== Halley’s method is a numerical algorithm for solving the nonlinear equation ''f''(''x'') = 0. In this case, the function ''f'' has to be a function of one real variable. The method consists of a sequence of iterations: :$x\_\; =\; x\_n\; -\; \backslash frac$ beginning with an initial guess ''x''0. If ''f'' is a three times continuously differentiable function and ''a'' is a zero of ''f'' but not of its derivative, then, in a neighborhood of ''a'', the iterates ''xn'' satisfy: :$|\; x\_\; -\; a\; |\; \backslash le\; K\; \backslash cdot\; ^3,\backslash textK\; >\; 0.$ This means that the iterates converge to the zero if the initial guess is sufficiently close, and that the convergence is cubic. The following alternative formulation shows the similarity between Halley’s method and Newton’s method. The expression $f(x\_n)/f\text{'}(x\_n)$ is computed only once, and it is particularly useful when $f\text{'}\text{'}(x\_n)/f\text{'}(x\_n)$ can be simplified. :$x\_\; =\; x\_n\; -\; \backslash frac\; \backslash left(-\; \backslash frac\; \backslash cdot\; \backslash frac\; \backslash right)^$ A further alternative is as below, in which case the technique is sometimes referred to as Hailey’s method.〔See for example the Bond Exchange of South Africa’s (''Bond Pricing Formula Specifications'' ), where Bailey’s method is suggested when solving for a bond’s Yield to maturity.〕 :$x\_\; =\; x\_n\; -\; \backslash frac\}$ Using any variation, when the second derivative is very close to zero, the iteration is almost the same as under Newton’s method. 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「Halley'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.