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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 Jacobi eigenvalue algorithm ： ウィキペディア英語版 Jacobi eigenvalue algorithm In numerical linear algebra, the Jacobi eigenvalue algorithm is an iterative method for the calculation of the eigenvalues and eigenvectors of a real symmetric matrix (a process known as diagonalization). It is named after Carl Gustav Jacob Jacobi, who first proposed the method in 1846, but only became widely used in the 1950s with the advent of computers. == Description == Let ''S'' be a symmetric matrix, and ''G'' = ''G''(''i'',''j'',''θ'') be a Givens rotation matrix. Then: :$S\text{'}=G\; S\; G^\backslash top\; \backslash ,$ is symmetric and similar to ''S''. Furthermore, ''S′'' has entries: :$\backslash begin$ S'_ &= c^2\, S_ - 2\, s c \,S_ + s^2\, S_ \\ S'_ &= s^2 \,S_ + 2 s c\, S_ + c^2 \, S_ \\ S'_ &= S'_ = (c^2 - s^2 ) \, S_ + s c \, (S_ - S_ ) \\ S'_ &= S'_ = c \, S_ - s \, S_ & k \ne i,j \\ S'_ &= S'_ = s \, S_ + c \, S_ & k \ne i,j \\ S'_ &= S_ &k,l \ne i,j \end where ''s'' = sin(''θ'') and ''c'' = cos(''θ''). Since ''G'' is orthogonal, ''S'' and ''S''′ have the same Frobenius norm ||·||F (the square-root sum of squares of all components), however we can choose ''θ'' such that ''S''′''ij'' = 0, in which case ''S''′ has a larger sum of squares on the diagonal: :$S\text{'}\_\; =\; \backslash cos(2\backslash theta)\; S\_\; +\; \backslash tfrac\; \backslash sin(2\backslash theta)\; (S\_\; -\; S\_)$ Set this equal to 0, and rearrange: :$\backslash tan(2\backslash theta)\; =\; \backslash frac\; -\; S\_\}$ if $S\_\; =\; S\_$ :$\backslash theta\; =\; \backslash frac$ In order to optimize this effect, ''S''''ij'' should be the off-diagonal component with the largest absolute value, called the ''pivot''. The Jacobi eigenvalue method repeatedly performs rotations until the matrix becomes almost diagonal. Then the elements in the diagonal are approximations of the (real) eigenvalues of ''S''. 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「Jacobi eigenvalue algorithm」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.