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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 inverse iteration ： ウィキペディア英語版 inverse iteration In numerical analysis, inverse iteration is an iterative eigenvalue algorithm. It allows one to find an approximate eigenvector when an approximation to a corresponding eigenvalue is already known. The method is conceptually similar to the power method and is also known as the inverse power method. It appears to have originally been developed to compute resonance frequencies in the field of structural mechanics. 〔Ernst Pohlhausen, ''Berechnung der Eigenschwingungen statisch-bestimmter Fachwerke'', ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik 1, 28-42 (1921).〕 The inverse power iteration algorithm starts with an approximation $\backslash mu$ for the eigenvalue corresponding to the desired eigenvector and a vector ''b''0, either a randomly selected vector or an approximation to the eigenvector. The method is described by the iteration :$b\_\; =\; \backslash frac,$ where ''Ck'' are some constants usually chosen as $C\_k=\; \backslash |(A\; -\; \backslash mu\; I)^b\_k\; \backslash |.$ Since eigenvectors are defined up to multiplication by constant, the choice of ''Ck'' can be arbitrary in theory; practical aspects of the choice of $C\_k$ are discussed below. At every iteration, the vector ''b''''k'' is multiplied by the inverse of the matrix $(A\; -\; \backslash mu\; I)$ and normalized. It is exactly the same formula as in the power method, except replacing the matrix ''A'' by $(A\; -\; \backslash mu\; I)^.$ The closer the approximation $\backslash mu$ to the eigenvalue is chosen, the faster the algorithm converges; however, incorrect choice of $\backslash mu$ can lead to slow convergence or to the convergence to an eigenvector other than the one desired. In practice, the method is used when a good approximation for the eigenvalue is known, and hence one needs only few (quite often just one) iterations. == Theory and convergence == The basic idea of the power iteration is choosing an initial vector ''b'' (either an eigenvector approximation or a random vector) and iteratively calculating $Ab,\; A^b,\; A^b,...$. Except for a set of zero measure, for any initial vector, the result will converge to an eigenvector corresponding to the dominant eigenvalue. The inverse iteration does the same for the matrix $(A\; -\; \backslash mu\; I)^$, so it converges to eigenvector corresponding to the dominant eigenvalue of the matrix $(A\; -\; \backslash mu\; I)^$. Eigenvalues of this matrix are $(\backslash lambda\_1\; -\; \backslash mu)^,...,(\backslash lambda\_n\; -\; \backslash mu)^,$ where $\backslash lambda\_i$ are eigenvalues of ''A''. The largest of these numbers corresponds to the smallest of $(\backslash lambda\_1\; -\; \backslash mu),...,(\backslash lambda\_n\; -\; \backslash mu).$ It is obvious that the eigenvectors of ''A'' and of $(A\; -\; \backslash mu\; I)^$ are the same. Conclusion: The method converges to the eigenvector of the matrix ''A'' corresponding to the closest eigenvalue to $\backslash mu\; .$ In particular taking $\backslash mu=0$ we see that $(A)^\; b$ converges to the eigenvector corresponding to the eigenvalue of ''A'' with the smallest absolute value. 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「inverse iteration」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.