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2008.67: Solving the Generalized Symmetric Eigenvalue Problem

2008.67: M.F. Anjos, S. Hammarling and C.C. Paige (1992) Solving the Generalized Symmetric Eigenvalue Problem. Unpublished. pp. 1-19.

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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.

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Uncontrolled Keywords:Generalized symmetric eigenvalue problem, Cholesky factorization, Jacobi rotations, numerical stability
Subjects:MSC 2000 > 15 Linear and multilinear algebra; matrix theory
MSC 2000 > 65 Numerical analysis
MIMS number:2008.67
Deposited By:Sven Hammarling
Deposited On:29 June 2008

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