Numerische Mathematik

, Volume 105, Issue 3, pp 375–412 | Cite as

A numerical method for computing the Hamiltonian Schur form

  • Delin Chu
  • Xinmin Liu
  • Volker MehrmannEmail author


We derive a new numerical method for computing the Hamiltonian Schur form of a Hamiltonian matrix \({\mathcal{M}\in \mathbb{R}^{2n\times 2n}}\) that has no purely imaginary eigenvalues. We demonstrate the properties of the new method by showing its performance for the benchmark collection of continuous-time algebraic Riccati equations. Despite the fact that no complete error analysis for the method is yet available, the numerical results indicate that if no eigenvalues of \({\mathcal{M}}\) are close to the imaginary axis then the method computes the exact Hamiltonian Schur form of a nearby Hamiltonian matrix and thus is numerically strongly backward stable. The new method is of complexity \({\mathbf{O}(n^{3})}\) and hence it solves a long-standing open problem in numerical analysis.

Mathematics Subject Classification

65F15 93B36 93B40 93C60 


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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of MathematicsNational University of SingaporeSingaporeSingapore
  2. 2.Department of Electrical and Computer EngineeringUniversity of VirginiaCharlottesvilleUSA
  3. 3.Institut für Mathematik MA 4-5TU BerlinBerlinGermany

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