翻訳と辞書 Words near each other ・ Matrix planting ・ Matrix polynomial ・ Matrix population models ・ Matrix Powertag ・ Matrix product state ・ Matrix Quest ・ Matrix regularization ・ Matrix representation ・ Matrix representation of conic sections ・ Matrix representation of Maxwell's equations ・ Matrix Requirements Medical ・ Matrix ring ・ Matrix scheme ・ Matrix similarity ・ Matrix Software ・ Matrix splitting ・ Matrix string theory ・ Matrix t-distribution ・ Matrix Template Library ・ Matrix theory (physics) ・ Matrix Toolkit Java ・ Matrix unit ・ Matrix vote ・ Matrix-assisted laser desorption electrospray ionization ・ Matrix-assisted laser desorption/ionization ・ Matrix-core ・ Matrix-exponential distribution ・ Matrix-free methods ・ Matrix-supported rock ・ MatrixDB Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Matrix splitting ： ウィキペディア英語版 Matrix splitting In the mathematical discipline of numerical linear algebra, a matrix splitting is an expression which represents a given matrix as a sum or difference of matrices. Many iterative methods (e.g., for systems of differential equations) depend upon the direct solution of matrix equations involving matrices more general than tridiagonal matrices. These matrix equations can often be solved directly and efficiently when written as a matrix splitting. The technique was devised by Richard S. Varga in 1960. ==Regular splittings== We seek to solve the matrix equation : $\backslash bold\; A\; \backslash bold\; x\; =\; \backslash bold\; k,\; \backslash quad\; (1)$ where A is a given ''n'' × ''n'' non-singular matrix, and k is a given column vector with ''n'' components. We split the matrix A into :$\backslash bold\; A\; =\; \backslash bold\; B\; -\; \backslash bold\; C,\; \backslash quad\; (2)$ where B and C are ''n'' × ''n'' matrices. If, for an arbitrary ''n'' × ''n'' matrix M, M has nonnegative entries, we write M ≥ 0. If M has only positive entries, we write M > 0. Similarly, if the matrix M1 − M2 has nonnegative entries, we write M1 ≥ M2. Definition: A = B − C is a regular splitting of A if and only if B−1 ≥ 0 and C ≥ 0. We assume that matrix equations of the form : $\backslash bold\; B\; \backslash bold\; x\; =\; \backslash bold\; g,\; \backslash quad\; (3)$ where g is a given column vector, can be solved directly for the vector x. If (2) represents a regular splitting of A, then the iterative method : $\backslash bold\; B\; \backslash bold\; x^\; =\; \backslash bold\; C\; \backslash bold\; x^\; +\; \backslash bold\; k,\; \backslash quad\; m\; =\; 0,\; 1,\; 2,\; \backslash ldots\; ,\; \backslash quad\; (4)$ where x(0) is an arbitrary vector, can be carried out. Equivalently, we write (4) in the form : $\backslash bold\; x^\; =\; \backslash bold\; B^\; \backslash bold\; C\; \backslash bold\; x^\; +\; \backslash bold\; B^\; \backslash bold\; k,\; \backslash quad\; m\; =\; 0,\; 1,\; 2,\; \backslash ldots\; \backslash quad\; (5)$ The matrix D = B−1C has nonnegative entries if (2) represents a regular splitting of A. It can be shown that if A−1 > 0, then $\backslash rho\; (\backslash bold\; D)$ < 1, where $\backslash rho\; (\backslash bold\; D)$ represents the spectral radius of D, and thus D is a convergent matrix. As a consequence, the iterative method (5) is necessarily convergent. If, in addition, the splitting (2) is chosen so that the matrix B is a diagonal matrix (with the diagonal entries all non-zero, since B must be invertible), then B can be inverted in linear time (see Time complexity). 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「Matrix splitting」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.