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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 Nonlinear eigenproblem ： ウィキペディア英語版 Nonlinear eigenproblem A nonlinear eigenproblem is a generalization of an ordinary eigenproblem to equations that depend nonlinearly on the eigenvalue. Specifically, it refers to equations of the form: :$A(\backslash lambda)\; \backslash mathbf\; =\; 0\; ,\; \backslash ,$ where x is a vector (the nonlinear "eigenvector") and ''A'' is a matrix-valued function of the number $\backslash lambda$ (the nonlinear "eigenvalue"). (More generally, $A(\backslash lambda)$ could be a linear map, but most commonly it is a finite-dimensional, usually square, matrix.) ''A'' is usually required to be a holomorphic function of $\backslash lambda$ (in some domain). For example, an ordinary linear eigenproblem $B\backslash mathbf\; =\; \backslash lambda\; \backslash mathbf$, where ''B'' is a square matrix, corresponds to $A(\backslash lambda)\; =\; B\; -\; \backslash lambda\; I$, where ''I'' is the identity matrix. One common case is where ''A'' is a polynomial matrix, which is called a polynomial eigenvalue problem. In particular, the specific case where the polynomial has degree two is called a quadratic eigenvalue problem, and can be written in the form: :$A(\backslash lambda)\; \backslash mathbf\; =\; (\; A\_2\; \backslash lambda^2\; +\; A\_1\; \backslash lambda\; +\; A\_0)\; \backslash mathbf\; =\; 0\; ,\; \backslash ,$ in terms of the constant square matrices ''A''0,1,2. This can be converted into an ordinary linear generalized eigenproblem of twice the size by defining a new vector $\backslash mathbf\; =\; \backslash lambda\; \backslash mathbf$. In terms of x and y, the quadratic eigenvalue problem becomes: : \begin A_1 & A_2 \\ I & 0 \end \begin \mathbf \\ \mathbf \end , where ''I'' is the identity matrix. More generally, if ''A'' is a matrix polynomial of degree ''d'', then one can convert the nonlinear eigenproblem into a linear (generalized) eigenproblem of ''d'' times the size. Besides converting them to ordinary eigenproblems, which only works if ''A'' is polynomial, there are other methods of solving nonlinear eigenproblems based on the Jacobi-Davidson algorithm or based on Newton's method (related to inverse iteration). ==References== * Françoise Tisseur and Karl Meerbergen, "The quadratic eigenvalue problem," ''SIAM Review'' 43 (2), 235-286 (2001). * Gene H. Golub and Henk A. van der Vorst, "Eigenvalue computation in the 20th century," ''Journal of Computational and Applied Mathematics'' 123, 35-65 (2000). * Philippe Guillaume, "Nonlinear eigenproblems," ''SIAM J. Matrix. Anal. Appl.'' 20 (3), 575-595 (1999). * Axel Ruhe, "Algorithms for the nonlinear eigenvalue problem," ''SIAM Journal on Numerical Analysis'' 10 (4), 674-689 (1973). 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「Nonlinear eigenproblem」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.