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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 LOBPCG ： ウィキペディア英語版 LOBPCG Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is a matrix-free method for finding the largest (or smallest) eigenvalues and the corresponding eigenvectors of a symmetric positive definite generalized eigenvalue problem :$A\; x=\; \backslash lambda\; B\; x,$ for a given pair $(A,\; B)$ of complex Hermitian or real symmetric matrices, where the matrix $B$ is also assumed positive-definite. ==Algorithm== The method performs an iterative maximization (or minimization) of the generalized Rayleigh quotient :$\backslash rho(x)\; :=\; \backslash rho(A,B;\; x)\; :=\backslash frac,$ which results in finding largest (or smallest) eigenpairs of $A\; x=\; \backslash lambda\; B\; x.$ The direction of the steepest ascent, which is the gradient, of the generalized Rayleigh quotient is positively proportional to the vector :$r\; :=\; Ax\; -\; \backslash rho(x)\; Bx,$ called the eigenvector residual. If a preconditioner $T$ is available, it is applied to the residual giving vector :$w\; :=\; Tr,$ called the preconditioned residual. Without preconditioning, we set $T\; :=\; I$ and so $w\; :=\; r,$. An iterative method :$x^\; :=\; x^i\; +\; \backslash alpha^i\; T(Ax^i\; -\; \backslash rho(x^i)\; Bx^i),$ or, in short, :$x^\; :=\; x^i\; +\; \backslash alpha^i\; w^i,\backslash ,$ :$w^i\; :=\; Tr^i,\backslash ,$ :$r^i\; :=\; Ax^i\; -\; \backslash rho(x^i)\; Bx^i,$ is known as preconditioned steepest ascent (or descent), where the scalar $\backslash alpha^i$ is called the step size. The optimal step size can be determined by maximizing the Rayleigh quotient, i.e., :$x^\; :=\; \backslash arg\backslash max\_\; :=\; \backslash arg\backslash max\_\}\; \backslash rho(y)$ (use $\backslash arg\backslash min$ in case of minimizing). The maximization/minimization of the Rayleigh quotient in a 3-dimensional subspace can be performed numerically by the Rayleigh–Ritz method. As the iterations converge, the vectors $x^i$ and $x^$ become nearly linearly dependent, making the Rayleigh–Ritz method numerically unstable in the presence of round-off errors. It is possible to substitute the vector $x^$ with an explicitly computed difference $p^i=x^-x^i$ making the Rayleigh–Ritz method more stable; see. This is a single-vector version of the LOBPCG method. It is one of possible generalization of the preconditioned conjugate gradient linear solvers to the case of symmetric eigenvalue problems. Even in the trivial case $T=I$ and $B=I$ the resulting approximation with $i>3$ will be different from that obtained by the Lanczos algorithm, although both approximations will belong to the same Krylov subspace. Iterating several approximate eigenvectors together in a block in a similar locally optimal fashion, gives the full block version of the LOBPCG. It allows robust computation of eigenvectors corresponding to nearly-multiple eigenvalues. 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「LOBPCG」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.