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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 Householder's method ： ウィキペディア英語版 Householder's method In mathematics, and more specifically in numerical analysis, Householder's methods are a class of root-finding algorithms that are used for functions of one real variable with continuous derivatives up to some order ''d''+1. Each of these methods is characterized by the number ''d'', which is known as the ''order'' of the method. The algorithm is iterative and has an rate of convergence of ''d''+1. These methods are named after the American mathematician Alston Scott Householder. == Method == Like many root-finding methods, Householder'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 :::$x\_\; =\; x\_n\; +\; d\backslash ;\; \backslash frac$ beginning with an initial guess ''x''0. If ''f'' is a ''(d+1)'' times continuously differentiable function and ''a'' is a zero of ''f'' but not of its derivative, then, in a neighborhood of ''a'', the iterates ''x''''n'' satisfy: :$|\; x\_\; -\; a\; |\; \backslash le\; K\; \backslash cdot\; ^$, for some $K\; >\; 0.\backslash !$ This means that the iterates converge to the zero if the initial guess is sufficiently close, and that the convergence has order ''(d+1)''. Despite their order of convergence, these methods are not widely used because the gain in precision is not commensurate with the rise in effort for large ''d''. The ''Ostrowski index'' expresses the error reduction in the number of function evaluations instead of the iteration count. * For polynomials, the evaluation of the first ''d'' derivatives of ''f'' at ''x''''n'' using the Horner method has an effort of ''d''+''1'' polynomial evaluations. Since ''n''(''d''+''1'') evaluations over ''n'' iterations give an error exponent of (''d''+''1'')''n'', the exponent for one function evaluation is $\backslash sqrt()$, numerically ''1.4142'', ''1.4422'', ''1.4142'', ''1.3797'' for ''d''=''1'',''2'',''3'',''4'', and falling after that. * For general functions the derivative evaluation using the Taylor arithmetic of automatic differentiation requires the equivalent of (''d''+''1'')(''d''+''2'')/''2'' function evaluations. One function evaluation thus reduces the error by an exponent of $\backslash sqrt()\backslash approx\; 1.2599$ for Newton's method, $\backslash sqrt()\backslash approx\; 1.2009$ for Halley's method and falling towards ''1'' or linear convergence for the higher order methods. 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「Householder'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.