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Unpublished.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(Unpublished)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 |
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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: | https://eprints.maths.manchester.ac.uk/id/eprint/1120 |

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