# 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 Unpublished.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 (Unpublished)

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.