James Demmel
Bo Kågström
Abstract
Robust software with error bounds for computing the generalized Schur decomposition of an arbitrary matrix pencil A – λB (regular or singular) is presented. The decomposition is a generalization of the Schur canonical form of A – λI to matrix pencils and reveals the Kronecker structure of a singular pencil. Since computing the Kronecker structure of a singular pencil is a potentially ill-posed problem, it is important to be able to compute rigorous and reliable error bounds for the computed features. The error bounds rely on perturbation theory for reducing subspaces and generalized eigenvalues of singular matrix pencils. The first part of this two-part paper presents the theory and algorithms for the decomposition and its error bounds, while the second part describes the computed generalized Schur decomposition and the software, and presents applications and an example of its use.
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Index Terms
- The generalized Schur decomposition of an arbitrary pencil A–λB—robust software with error bounds and applications. Part I: theory and algorithms
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Reviews
James Martin Varah
The authors present algorithms for computing the generalized Schur decomposition of an arbitrary matrix pencil A- l B , as a useful stable alternative to the Kronecker canonical form. The key element is the unitary reduction to GUPTRI (generalized upper triangular) form. Using perturbation results for reducing subspaces and generalized eigenvalues, they also compute error bounds for these computed quantities. Part 1 discusses the algorithms used, and Part 2 gives the details of the software implementation and some special cases . The authors also mention the application of this software to problems in control theory, and make a comparison with the work of van Dooren in this context.
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