For adaptive extraction of generalized eigensubspace, Nguyen, Takahashi and Yamada proposed a scheme for solving generalized Hermitian eigenvalue problem based on nested orthogonal complement structure. This scheme ca...
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For adaptive extraction of generalized eigensubspace, Nguyen, Takahashi and Yamada proposed a scheme for solving generalized Hermitian eigenvalue problem based on nested orthogonal complement structure. This scheme can extract multiple generalized eigenvectors by combining with any algorithm designed for estimation of the first minor generalized eigenvector. In this paper, we carefully analyse the effect of a discontinuous function employed in the scheme, and show that the discontinuous function can cause unsmooth changes of the estimates by the scheme in its adaptive implementation. To remedy the weakness, we newly introduce a projection step, for smoothing, without increasing the order of the computational complexity. Numerical experiments show that the learning curves of the non-first generalized eigenvectors are improved drastically through the proposed smoothing even when the original scheme results in unexpected performance degradation.
Nguyen and Yamada [NY'13] proposed an adaptive algorithm for fast and stable extraction of the first generalized Hermitian eigen-vector and mentioned the extension to the first r generalized eigenvector extraction...
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ISBN:
(纸本)9781479999880
Nguyen and Yamada [NY'13] proposed an adaptive algorithm for fast and stable extraction of the first generalized Hermitian eigen-vector and mentioned the extension to the first r generalized eigenvector extraction based on the nested orthogonal complement structure [NTY'12]. However, we recently found that the estimates of the eigenvectors are not expressed ideally in the time-varying coordinate system and can change drastically in a certain situation, which may cause numerical instability. In this paper, we propose a new expression of the estimates along with time-varying coordinate system. This modification can be done efficiently with additional multiplications of orthogonalcomplement matrices. Numerical experiments show that the modified scheme has better stability compared with the original scheme [NTY'12].
Nguyen and Yamada [NY'13] proposed an adaptive algorithm for fast and stable extraction of the first generalized Hermitian eigenvector and mentioned the extension to the first r generalized eigenvector extraction ...
详细信息
ISBN:
(纸本)9781479999897
Nguyen and Yamada [NY'13] proposed an adaptive algorithm for fast and stable extraction of the first generalized Hermitian eigenvector and mentioned the extension to the first r generalized eigenvector extraction based on the nested orthogonal complement structure [NTY'12]. However, we recently found that the estimates of the eigenvectors are not expressed ideally in the time-varying coordinate system and can change drastically in a certain situation, which may cause numerical instability. In this paper, we propose a new expression of the estimates along with time-varying coordinate system. This modification can be done efficiently with additional multiplications of orthogonalcomplement matrices. Numerical experiments show that the modified scheme has better stability compared with the original scheme [NTY' 12].
The contribution of this paper is three-fold: first, we propose a novel scheme for generalized minor subspace extraction by extending an idea of dimension reduction technique. The key of this scheme is the reduction o...
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The contribution of this paper is three-fold: first, we propose a novel scheme for generalized minor subspace extraction by extending an idea of dimension reduction technique. The key of this scheme is the reduction of the problem for extracting the ith (i a parts per thousand yen 2) minor generalized eigenvector of the original matrix pencil to that for extracting the first minor generalized eigenvector of a matrix pencil of lower dimensionality. The proposed scheme can employ any algorithm capable of estimating the first minor generalized eigenvector. Second, we propose a pair of such iterative algorithms and analyze their convergence properties in the general case where the generalized eigenvalues are not necessarily distinct. Third, by using these algorithms inductively, we present adaptive implementations of the proposed scheme for estimating an orthonormal basis of the generalized minor subspace. Numerical examples show that the proposed adaptive subspace extraction algorithms have better numerical stability than conventional algorithms.
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