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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 Modified Richardson iteration ： ウィキペディア英語版 Modified Richardson iteration Modified Richardson iteration is an iterative method for solving a system of linear equations. Richardson iteration was proposed by Lewis Richardson in his work dated 1910. It is similar to the Jacobi and Gauss–Seidel method. We seek the solution to a set of linear equations, expressed in matrix terms as :$A\; x\; =\; b.\backslash ,$ The Richardson iteration is :$$ x^ = x^ + \omega \left( b - A x^ \right), where $\backslash omega$ is a scalar parameter that has to be chosen such that the sequence $x^$ converges. It is easy to see that the method has the correct fixed points, because if it converges, then $x^\; \backslash approx\; x^$ and $x^$ has to approximate a solution of $A\; x\; =\; b$. == Convergence == Subtracting the exact solution $x$, and introducing the notation for the error $e^\; =\; x^-x$, we get the equality for the errors :$$ e^ = e^ - \omega A e^ = (I-\omega A) e^. Thus, :$$ \|e^\| = \|(I-\omega A) e^\|\leq \|I-\omega A\| \|e^\|, for any vector norm and the corresponding induced matrix norm. Thus, if , the method converges. Suppose that $A$ is diagonalizable and that $(\backslash lambda\_j,\; v\_j)$ are the eigenvalues and eigenvectors of $A$. The error converges to $0$ if for all eigenvalues $\backslash lambda\_j$. If, e.g., all eigenvalues are positive, this can be guaranteed if $\backslash omega$ is chosen such that . The optimal choice, minimizing all $|\; 1\; -\; \backslash omega\; \backslash lambda\_j\; |$, is $\backslash omega\; =\; 2/(\backslash lambda\_(A)+\backslash lambda\_(A))$, which gives the simplest Chebyshev iteration. If there are both positive and negative eigenvalues, the method will diverge for any $\backslash omega$ if the initial error $e^$ has nonzero components in the corresponding eigenvectors. 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「Modified Richardson iteration」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.