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Numerical computation of deflating subspaces of skew-Hamiltonian/Hamiltonian pencils
Benner, Peter ; Byers, Ralph ; Mehrmann, Volker ; Xu, Hongguo
Benner, Peter
Byers, Ralph
Mehrmann, Volker
Xu, Hongguo
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Abstract
We discuss the numerical solution of structured generalized eigenvalue problems that arise from linear- quadratic optimal control problems, H infinity optimization, multibody systems, and many other areas of applied mathematics, physics, and chemistry. The classical approach for these problems requires computing invariant and deflating subspaces of matrices and matrix pencils with Hamiltonian and/ or skew- Hamiltonian structure. We extend the recently developed methods for Hamiltonian matrices to the general case of skew- Hamiltonian/ Hamiltonian pencils. The algorithms circumvent problems with skew- Hamiltonian/ Hamiltonian matrix pencils that lack structured Schur forms by embedding them into matrix pencils that always admit a structured Schur form. The rounding error analysis of the resulting algorithms is favorable. For the embedded matrix pencils, the algorithms use structure- preserving unitary matrix computations and are strongly backwards stable, i. e., they compute the exact structured Schur form of a nearby matrix pencil with the same structure.
Description
Working title was “Numerical Computation of Deflating Subspaces of Embedded Hamiltonian and Symplectic Pencils"
Date
2002-07-09
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SIAM PUBLICATIONS
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Keywords
Eigenvalue problem, Deflating subspace, Hamiltonian matrix, Skew-hamiltonian matrix, Skew-hamiltonian/hamiltonian matrix pencil
Citation
Benner, P; Byers, R; Mehrmann, V; Xu, HG. Numerical computation of deflating subspaces of skew-Hamiltonian/Hamiltonian pencils. SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS. July 9 2002. 24(1):165-190.