On the Convergence of a New Rayleigh Quotient Method with Applications
to Large Eigenproblems
On the Convergence of a New Rayleigh Quotient Method with Applications
to Large Eigenproblems
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Date
1998-10-15
Authors
O'Leary, D. P.
Stewart, G. W.
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Abstract
In this paper we propose a variant of the Rayleigh quotient method to
compute an eigenvalue and corresponding eigenvectors of a matrix. It
is based on the observation that eigenvectors of a matrix with
eigenvalue zero are also singular vectors corresponding to zero
singular values. Instead of computing eigenvector approximations by
the inverse power method, we take them to be the singular vectors
corresponding to the smallest singular value of the shifted matrix.
If these singular vectors are computed exactly the method is
quadratically convergent. However, exact singular vectors are not
required for convergence, and the resulting method combined with
Golub--Kahan--Krylov bidiagonalization looks promising for
enhancement/refinement methods for large eigenvalue problems.
(Also cross-referenced as UMIACS-97-74)