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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 Kantorovich theorem ： ウィキペディア英語版 Kantorovich theorem The Kantorovich theorem is a mathematical statement on the convergence of Newton's method. It was first stated by Leonid Kantorovich in 1940. Newton's method constructs a sequence of points that—with good luck—will converge to a solution $x$ of an equation $f(x)=0$ or a vector solution of a system of equation $F(x)=0$. The Kantorovich theorem gives conditions on the initial point of this sequence. If those conditions are satisfied then a solution exists close to the initial point and the sequence converges to that point. == Assumptions == Let $X\backslash subset\backslash R^n$ be an open subset and $F:\backslash R^n\backslash supset\; X\backslash to\backslash R^n$ a differentiable function with a Jacobian $F^(x)$ that is locally Lipschitz continuous (for instance if it is twice differentiable). That is, it is assumed that for any open subset $U\backslash subset\; X$ there exists a constant $L>0$ such that for any $\backslash mathbf\; x,\backslash mathbf\; y\backslash in\; U$ :$\backslash |F\text{'}(\backslash mathbf\; x)-F\text{'}(\backslash mathbf\; y)\backslash |\backslash le\; L\backslash ;\backslash |\backslash mathbf\; x-\backslash mathbf\; y\backslash |$ holds. The norm on the left is some operator norm that is compatible with the vector norm on the right. This inequality can be rewritten to only use the vector norm. Then for any vector $v\backslash in\backslash R^n$ the inequality :$\backslash |F\text{'}(\backslash mathbf\; x)(v)-F\text{'}(\backslash mathbf\; y)(v)\backslash |\backslash le\; L\backslash ;\backslash |\backslash mathbf\; x-\backslash mathbf\; y\backslash |\backslash ,\backslash |v\backslash |$ must hold. Now choose any initial point $\backslash mathbf\; x\_0\backslash in\; X$. Assume that $F\text{'}(\backslash mathbf\; x\_0)$ is invertible and construct the Newton step $\backslash mathbf\; h\_0=-F\text{'}(\backslash mathbf\; x\_0)^F(\backslash mathbf\; x\_0).$ The next assumption is that not only the next point $\backslash mathbf\; x\_1=\backslash mathbf\; x\_0+\backslash mathbf\; h\_0$ but the entire ball $B(\backslash mathbf\; x\_1,\backslash |\backslash mathbf\; h\_0\backslash |)$ is contained inside the set ''X''. Let $M\backslash le\; L$ be the Lipschitz constant for the Jacobian over this ball. As a last preparation, construct recursively, as long as it is possible, the sequences $(\backslash mathbf\; x\_k)\_k$, $(\backslash mathbf\; h\_k)\_k$, $(\backslash alpha\_k)\_k$ according to :$\backslash begin$ \mathbf h_k&=-F'(\mathbf x_k)^F(\mathbf x_k)\\() \alpha_k&=M\,\|F'(\mathbf x_k)^\|\,\|h_k\|\\() \mathbf x_&=\mathbf x_k+\mathbf h_k. \end 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「Kantorovich theorem」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.