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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 rotation ： ウィキペディア英語版 Jacobi rotation In numerical linear algebra, a Jacobi rotation is a rotation, ''Q''''k''ℓ, of a 2-dimensional linear subspace of an ''n-''dimensional inner product space, chosen to zero a symmetric pair of off-diagonal entries of an ''n''×''n'' real symmetric matrix, ''A'', when applied as a similarity transformation: : $A\; \backslash mapsto\; Q\_^T\; A\; Q\_\; =\; A\text{'}\; .\; \backslash ,\backslash !$ : $$ \begin & & & \cdots & & & * \\ & \ddots & & & & & \\ & & a_ & \cdots & a_ & & \\ \vdots & & \vdots & \ddots & \vdots & & \vdots \\ & & a_ & \cdots & a_ & & \\ & & & & & \ddots & \\ & & & \cdots & & & * \end \to \begin & & & \cdots & & & * \\ & \ddots & & & & & \\ & & a'_ & \cdots & 0 & & \\ \vdots & & \vdots & \ddots & \vdots & & \vdots \\ & & 0 & \cdots & a'_ & & \\ & & & & & \ddots & \\ & & & \cdots & & & * \end. It is the core operation in the Jacobi eigenvalue algorithm, which is numerically stable and well-suited to implementation on parallel processors . Only rows ''k'' and ℓ and columns ''k'' and ℓ of ''A'' will be affected, and that ''A''′ will remain symmetric. Also, an explicit matrix for ''Q''''k''ℓ is rarely computed; instead, auxiliary values are computed and ''A'' is updated in an efficient and numerically stable way. However, for reference, we may write the matrix as : $$ Q_ = \begin 1 & & & & & & \\ & \ddots & & & & 0 & \\ & & c & \cdots & s & & \\ & & \vdots & \ddots & \vdots & & \\ & & -s & \cdots & c & & \\ & 0 & & & & \ddots & \\ & & & & & & 1 \end . That is, ''Q''''k''ℓ is an identity matrix except for four entries, two on the diagonal (''q''''kk'' and ''q''ℓℓ, both equal to ''c'') and two symmetrically placed off the diagonal (''q''''k''ℓ and ''q''ℓ''k'', equal to ''s'' and −''s'', respectively). Here ''c'' = cos ϑ and ''s'' = sin ϑ for some angle ϑ; but to apply the rotation, the angle itself is not required. Using Kronecker delta notation, the matrix entries can be written : $q\_\; =$ \delta_ + (\delta_\delta_ + \delta_\delta_)(c-1) + (\delta_\delta_ - \delta_\delta_)s . \,\! Suppose ''h'' is an index other than ''k'' or ℓ (which must themselves be distinct). Then the similarity update produces, algebraically, : $a\text{'}\_\; =\; a\text{'}\_\; =\; c\; a\_\; -\; s\; a\_\; \backslash ,\backslash !$ : $a\text{'}\_\; =\; a\text{'}\_\; =\; c\; a\_\; +\; s\; a\_\; \backslash ,\backslash !$ : $a\text{'}\_\; =\; a\text{'}\_\; =\; (c^2-s^2)a\_\; +\; sc\; (a\_\; -\; a\_)\; =\; 0\; \backslash ,\backslash !$ : $a\text{'}\_\; =\; c^2\; a\_\; +\; s^2\; a\_\; -\; 2\; s\; c\; a\_\; \backslash ,\backslash !$ : $a\text{'}\_\; =\; s^2\; a\_\; +\; c^2\; a\_\; +\; 2\; s\; c\; a\_.\; \backslash ,\backslash !$ == Numerically stable computation == To determine the quantities needed for the update, we must solve the off-diagonal equation for zero . This implies that : $\backslash frac\; =\; \backslash frac\}\; -\; a\_\}\; ,$ so that ρ = tan(ϑ/2). Then the revised update equations are : $a\text{'}\_\; =\; a\text{'}\_\; =\; a\_\; -\; s\; (a\_\; +\; \backslash rho\; a\_)\; \backslash ,\backslash !$ : $a\text{'}\_\; =\; a\text{'}\_\; =\; a\_\; +\; s\; (a\_\; -\; \backslash rho\; a\_)\; \backslash ,\backslash !$ : $a\text{'}\_\; =\; a\text{'}\_\; =\; 0\; \backslash ,\backslash !$ : $a\text{'}\_\; =\; a\_\; -\; t\; a\_\; \backslash ,\backslash !$ : $a\text{'}\_\; =\; a\_\; +\; t\; a\_\; \backslash ,\backslash !$ As previously remarked, we need never explicitly compute the rotation angle ϑ. In fact, we can reproduce the symmetric update determined by ''Q''''k''ℓ by retaining only the three values ''k'', ℓ, and ''t'', with ''t'' set to zero for a null rotation. 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 rotation」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.