Theoretical and Computable Optimal Subspace Expansions for Matrix Eigenvalue Problems
نویسندگان
چکیده
Consider the optimal subspace expansion problem for matrix eigenvalue $Ax=\lambda x$: Which vector $w$ in current $\mathcal{V}$, after multiplied by $A$, provides an approximating a desired eigenvector $x$ sense that has smallest angle with expanded $\mathcal{V}_w=\mathcal{V}+{span}\{Aw\}$, i.e., $w_{opt}=\arg\max_{w\in\mathcal{V}}\cos\angle(\mathcal{V}_w,x)$? This is important as many iterative methods construct nested subspaces successively expand $\mathcal{V}$ to $\mathcal{V}_w$. An expression of $w_{opt}$ Ye [Linear Algebra Appl., 428 (2008), pp. 911--918] $A$ general, but it could not be exploited computable (nearly) optimally subspace. turns deriving maximization characterization $\cos\angle(\mathcal{V}_w,x)$ given $w\in \mathcal{V}$ when Hermitian. We generalize Ye's general case and find its maximizer. Our main contributions consist explicit expressions $w_{opt}$, $(I-P_V)Aw_{opt}$ $\mathcal{V}_{w_{opt}}$ where $P_V$ orthogonal projector onto $\mathcal{V}$. These results are fully obtain within framework standard, harmonic, refined, refined harmonic Rayleigh--Ritz methods. show how efficiently implement proposed approaches. Numerical experiments demonstrate effectiveness our expansions.
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ژورنال
عنوان ژورنال: SIAM Journal on Matrix Analysis and Applications
سال: 2022
ISSN: ['1095-7162', '0895-4798']
DOI: https://doi.org/10.1137/20m1331032