Self-equivalent flows associated with the generalized eigenvalue problem. (English) Zbl 0684.65037
The authors use elementary Lie theory to develop a family of algorithms for solving the generalized eigenvalue problem \(Ax=\lambda Bx\) for arbitrary pairs of nonsingular matrices. The family (referred as FGZ algorithms) includes the well known LZ algorithm of L. Kaufman [SIAM J. Numer. Anal. 11, 997-1024 (1974; Zbl 0294.65025)] and the QZ algorithm of C. B. Moler and G. W. Stewart [SIAM J. Numer. Anal. 10, 241-256 (1973; Zbl 0225.65046)], as well as new algorithms called SZ and HZ. The continuous analogues of the unshifted and shifted LZ and QZ algorithms are presented. For each considered algorithm a family of associated flows is constructed.
Reviewer: D.Herceg
MSC:
| 65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |
| 37C10 | Dynamics induced by flows and semiflows |
Keywords:
self-equivalent flows; Lie theory; generalized eigenvalue problem; FGZ algorithms; LZ algorithm; QZ algorithmReferences:
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