翻訳と辞書 Words near each other ・ Generalized Korteweg–de Vries equation ・ Generalized Lagrangian mean ・ Generalized least squares ・ Generalized lentiginosis ・ Generalized lifting ・ Generalized linear array model ・ Generalized linear mixed model ・ Generalized linear model ・ Generalized logistic distribution ・ Generalized Lotka–Volterra equation ・ Generalized lymphadenopathy ・ Generalized map ・ Generalized Maxwell model ・ Generalized mean ・ Generalized method of moments ・ Generalized minimal residual method ・ Generalized minimum-distance decoding ・ Generalized Multi-Protocol Label Switching ・ Generalized multidimensional scaling ・ Generalized multivariate log-gamma distribution ・ Generalized Music Plug-in Interface ・ Generalized Newtonian fluid ・ Generalized nondeterministic finite automaton ・ Generalized normal distribution ・ Generalized other ・ Generalized Ozaki cost function ・ Generalized p-value ・ Generalized Pareto distribution ・ Generalized permutation matrix ・ Generalized Petersen graph 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 Generalized minimal residual method ： ウィキペディア英語版 Generalized minimal residual method In mathematics, the generalized minimal residual method (usually abbreviated GMRES) is an iterative method for the numerical solution of a nonsymmetric system of linear equations. The method approximates the solution by the vector in a Krylov subspace with minimal residual. The Arnoldi iteration is used to find this vector. The GMRES method was developed by Yousef Saad and Martin H. Schultz in 1986.〔Saad and Schultz〕 GMRES is a generalization of the MINRES method developed by Chris Paige and Michael Saunders in 1975. GMRES also is a special case of the DIIS method developed by Peter Pulay in 1980. DIIS is also applicable to non-linear systems. == The method == Denote the Euclidean norm of any vector ''v'' by $\backslash |v\backslash |$. Denote the system of linear equations to be solved by :$Ax\; =\; b.\; \backslash ,$ The matrix ''A'' is assumed to be invertible of size ''m''-by-''m''. Furthermore, it is assumed that ''b'' is normalized, i.e., that ||''b''|| = 1. The ''n''th Krylov subspace for this problem is :$K\_n\; =\; K\_n(A,b)\; =\; \backslash operatorname\; \backslash ,\; \backslash .\; \backslash ,$ GMRES approximates the exact solution of ''Ax'' = ''b'' by the vector ''x''''n'' ∈ ''K''''n'' that minimizes the Euclidean norm of the residual ''r''''n'' = ''Ax''''n'' − ''b''. The vectors ''b'', ''Ab'', …, ''A''''n''−1''b'' might be almost linearly dependent, so instead of this basis, the Arnoldi iteration is used to find orthonormal vectors :$q\_1,\; q\_2,\; \backslash ldots,\; q\_n\; \backslash ,$ which form a basis for ''K''''n''. Hence, the vector ''x''''n'' ∈ ''K''''n'' can be written as ''x''''n'' = ''Q''''n''''y''''n'' with ''y''''n'' ∈ R''n'', where ''Q''''n'' is the ''m''-by-''n'' matrix formed by ''q''1, …, ''q''n. The Arnoldi process also produces an (''n''+1)-by-''n'' upper Hessenberg matrix $\backslash tilde\_n$ with :$AQ\_n\; =\; Q\_\; \backslash tilde\_n.\; \backslash ,$ Because columns of $Q\_n$ are orthogonal, we have :$\backslash |\; Ax\_n\; -\; b\; \backslash |\; =\; \backslash |\; \backslash tilde\_ny\_n\; -\; \backslash beta\; e\_1\; \backslash |,\; \backslash ,$ where :$e\_1\; =\; (1,0,0,\backslash ldots,0)^T\; \backslash ,$ is the first vector in the standard basis of R''n''+1, and :$\backslash beta\; =\; \backslash |b-Ax\_0\backslash |\; \backslash ,\; ,$ $x\_0$ being the first trial vector (usually zero). Hence, $x\_n$ can be found by minimizing the Euclidean norm of the residual :$r\_n\; =\; \backslash tilde\_n\; y\_n\; -\; \backslash beta\; e\_1.$ This is a linear least squares problem of size ''n''. This yields the GMRES method. On the ''n''-th iteration: # calculate $q\_n$ with the Arnoldi method; # find the $y\_n$ which minimizes ||''r''''n''||; # compute $x\_n\; =\; Q\_n\; y\_n$; # repeat if the residual is not yet small enough. At every iteration, a matrix-vector product ''Aq''''n'' must be computed. This costs about 2''m''2 floating-point operations for general dense matrices of size ''m'', but the cost can decrease to O(''m'') for sparse matrices. In addition to the matrix-vector product, O(''n'' ''m'') floating-point operations must be computed at the ''n''th iteration. 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「Generalized minimal residual 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.