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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 Aberth method ： ウィキペディア英語版 Aberth method The Aberth method, or Aberth–Ehrlich method, named after Oliver Aberth and Louis W. Ehrlich, is a root-finding algorithm for simultaneous approximation of all the roots of a univariate polynomial. The fundamental theorem of algebra states that for each polynomial with complex coefficients there are as many roots as the degree of the polynomial. This method converges cubically, an improvement over the Weierstrass–(Durand–Kerner) method, another numerical algorithm that approximates all roots at once, which converges quadratically.〔〔 (However, both algorithms converge linearly at multiple zeros.〔) ==Description== Let $p(x)=p\_nx^n+p\_x^+\backslash cdots+p\_1x+p\_0$ be a univariate polynomial of degree ''n'' with real or complex coefficients. Then there exist complex numbers $z^$ *_1,\,z^ *_2,\dots,z^ *_n, the roots of ''p(x)'', that give the factorisation: :$p(x)=p\_n\backslash cdot(x-z^$ *_1)\cdot(x-z^ *_2)\cdots(x-z^ *_n). Although those numbers are unknown, upper and lower bounds for their absolute values are computable from the coefficients of the polynomial. Now one can pick ''n'' distinct numbers in the complex plane—randomly or evenly distributed—such that their absolute values are within the same bounds. (Also, if the zeros are symmetrical, the starting points must not be exactly symmetrical along the same axis, as this can prevent convergence.)〔 A set of such numbers is called an initial approximation of the set of roots of ''p(x)''. This approximation can be iteratively improved using the following procedure. Let $z\_1,\backslash dots,z\_n\backslash in\backslash mathbb\; C$ be the current approximations of the zeros of ''p(x)''. Then offset numbers $w\_1,\backslash dots,w\_n\backslash in\backslash mathbb\; C$ are computed as :$w\_k=-\backslash frac\}\backslash cdot\; \backslash sum\_\backslash frac1\},$ where ''p'(z)'' is the polynomial derivative of ''p'' evaluated in the point ''z''. The next set of approximations of roots of ''p(x)'' is then $z\_1+w\_1,\backslash dots,z\_n+w\_n$. One can measure the quality of the current approximation by the values of the polynomial or by the size of the offsets. Conceptually, this method uses an electrostatic analogy, modeling the approximated zeros as movable negative point charges, which converge toward the true zeros, represented by fixed positive point charges.〔 A direct application of Newton's method to each approximated zero will often cause multiple starting points to incorrectly converge to the same root. The Aberth method avoids this by also modeling the repulsive effect the movable charges have on each other. In this way, when a movable charge has converged on a zero, their charges will cancel out, so that other movable charges are no longer attracted to that location, encouraging them to converge to other "unoccupied" zeros. (Stieltjes also modeled the positions of zeros of polynomials as solutions to electrostatic problems.) Inside the formula of the Aberth method one can find elements of Newton's method and the Weierstrass–(Durand–Kerner) method. Details for an efficient implementation, esp. on the choice of good initial approximations, can be found in Bini (1996). 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「Aberth 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.