Solving the Generalized Symmetric Eigenvalue Problem

Anjos, M.F. and Hammarling, S. and Paige, C.C. (1992) Solving the Generalized Symmetric Eigenvalue Problem. Unpublished. pp. 1-19. (Unpublished)

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Abstract

The generalized symmetric eigenvalue problem (GSEVP) $A x = \lambda B x$, $A$ symmetric, $B$ symmetric positive definite, occurs in many practical problems, but there is as yet no numerically stable algorithm which takes full advantage of its structure. The \textit{standard method} (factor $B = G G^T$, solve the ordinary symmetric eigenvalue problem $G^{-1} A G^{-T} (G^T x) = \lambda (G^T x)) is not numerically stable, while the backward stable $QZ$ algorithm takes no account of the special structure of the GSEVP. We show by example a previously unrecognized deficiency in the eigenvectors produced by the $QZ$ algorithm for some cases of this special class of problems. We suggest a new method that takes full advantage of the structure of the GSEVP yet appears not to suffer from either the eigenvector deficiency of the $QZ$ algorithm, nor the lack of accuracy that can be introduced by the standard method when $B$ has a large condition number $\kappa(B)$. The new method first reduces the problem to an equivalent one $A_c y = \lambda D^{2}_{c} y$, $A_c$ symmetric, $D_c$ diagonal. It then implicitly applies Jacobi's method to $D^{-1}_{c} A_c D^{-1}_c$, while maintaining the symmetric and diagonal forms. The results of numerical tests indicate the benefits of this approach. A variant of the approach is amenable to parallel computation.

Item Type: Article
Additional Information: See also final item at: http://cheetah.vlsi.uwaterloo.ca/~anjos/MFA_publications.html
Uncontrolled Keywords: Generalized symmetric eigenvalue problem, Cholesky factorization, Jacobi rotations, numerical stability
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Sven Hammarling
Date Deposited: 29 Jun 2008
Last Modified: 20 Oct 2017 14:12
URI: http://eprints.maths.manchester.ac.uk/id/eprint/1120

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