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## A numerically strongly stable method for computing the Hamiltonian Schur form

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Author(s) : Delin Chu , Xinmin Liu, Volker Mehrmann

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 24-2004

MSC 2000

65F15 Eigenvalues, eigenvectors
93B36 $^\infty$-control

Abstract :
In this paper we solve a long-standing open problem in numerical analysis called 'Van Loan's Curse'. We derive a new numerical method for computing the Hamiltonian Schur form of a Hamiltonian matrix that has no purely imaginary eigenvalues. The proposed method is numerically strongly backward stable, i.e., it computes the exact Hamiltonian Schur form of a nearby Hamiltonian matri x, and it is of complexity O(n^3) and thus Van Loan's curse is lifted. We demonstrate the quality of the new method by showing its performance for the benchmark collection of continuous-time algebraic Riccati equations.

Keywords : Hamiltonian matrix, skew-Hamiltonian matrix, real Hamiltonian Schur form, real skew-Hamiltonian Schur form, symplectic URV-decomposition, stable invariant subspace