Inhalt des Dokuments
Preprint 24-2004
A numerically strongly stable method for computing the Hamiltonian Schur form
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