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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 Gauss–Seidel ： ウィキペディア英語版 Gauss–Seidel method In numerical linear algebra, the Gauss–Seidel method, also known as the Liebmann method or the method of successive displacement, is an iterative method used to solve a linear system of equations. It is named after the German mathematicians Carl Friedrich Gauss and Philipp Ludwig von Seidel, and is similar to the Jacobi method. Though it can be applied to any matrix with non-zero elements on the diagonals, convergence is only guaranteed if the matrix is either diagonally dominant, or symmetric and positive definite. It was only mentioned in a private letter from Gauss to his student Gerling in 1823.〔 ; (direct link ).〕 A publication was not delivered before 1874 by Seidel. == Description == The Gauss–Seidel method is an iterative technique for solving a square system of ''n'' linear equations with unknown x: :$A\backslash mathbf\; x\; =\; \backslash mathbf\; b$. It is defined by the iteration :$L\_$ * \mathbf^ = \mathbf - U \mathbf^, where $\backslash mathbf^$ is the ''k''th approximation or iteration of $\backslash mathbf,\backslash ,\backslash mathbf^$ is the next or ''k'' + 1 iteration of $\backslash mathbf$, and the matrix ''A'' is decomposed into a lower triangular component $L\_$ *, and a strictly upper triangular component ''U'': $A\; =\; L\_$ * + U .〔.〕 In more detail, write out ''A'', x and b in their components: : Then the decomposition of ''A'' into its lower triangular component and its strictly upper triangular component is given by: :$A=L\_$ *+U \qquad \text \qquad L_ * = \begin a_ & 0 & \cdots & 0 \\ a_ & a_ & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\a_ & a_ & \cdots & a_ \end, \quad U = \begin 0 & a_ & \cdots & a_ \\ 0 & 0 & \cdots & a_ \\ \vdots & \vdots & \ddots & \vdots \\0 & 0 & \cdots & 0 \end. The system of linear equations may be rewritten as: :$L\_$ * \mathbf = \mathbf - U \mathbf The Gauss–Seidel method now solves the left hand side of this expression for x, using previous value for x on the right hand side. Analytically, this may be written as: :$\backslash mathbf^\; =\; L\_$ *^ (\mathbf - U \mathbf^). However, by taking advantage of the triangular form of $L\_$ *, the elements of x(''k''+1) can be computed sequentially using forward substitution: :$x^\_i\; =\; \backslash fraca\_x^\_j\; -\; \backslash sum\_a\_x^\_j\; \backslash right),\backslash quad\; i,j=1,2,\backslash ldots,n.$ 〔.〕 The procedure is generally continued until the changes made by an iteration are below some tolerance, such as a sufficiently small residual. 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「Gauss–Seidel 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.