Optimal Subspace Expansion for Matrix Eigenvalue Problems
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In this paper, we consider the optimal subspace expansion problem for the matrix eigenvalue problem Ax=λ x: Which vector w in the current subspace V, after multiplied by A, provides an optimal subspace expansion for approximating a desired eigenvector x in the sense that x has the smallest angle with the expanded subspace V_w=V+ span{Aw}? Our research motivation is that many iterative methods construct nested subspaces that successively expands V to V_w. Ye (Linear Algebra Appl., 428 (2008), pp. 911–918) studies the maximization characterization of cosine between x and V_w but does not obtain the maximizer. He shows how to approximately maximize the cosine so as to find approximate solutions of the subspace expansion problem for A Hermitian. However, his approach and analysis cannot extend to the non-Hermitian case. We study the optimal expansion problem in the general case and derive explicit expressions of the optimal expansion vector w_opt. By a careful analysis on the theoretical results, we obtain computable nearly optimal choices of w_opt for the standard, harmonic and refined (harmonic) Rayleigh–Ritz methods.
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