Advertisement

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
Article

Abstract

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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ammar G.S., Mehrmann V. (1991): On Hamiltonian and symplectic Hessenberg forms. Linear Algebra Appl. 149, 55–72CrossRefMathSciNetGoogle Scholar
  2. 2.
    Ammar G.S., Benner P., Mehrmann V. (1993): A multishift algorithm for the numerical solution of algebraic Riccati equations. Electr. Trans. Numer. Anal. 1, 33–48MathSciNetGoogle Scholar
  3. 3.
    Arnold, W.F., III, Laub, A.J.: Generalized eigenproblem algorithms and software for algebraic Riccati equations. Proc. IEEE 72, 1746–1754 (1984)Google Scholar
  4. 4.
    Benner, P., Kressner, D.: Fortran 77 subroutines for computing the eigenvalues of Hamiltonian matrices II. Submitted. See also http://www.math.tu-berlin.de/~kressner/hapack/ (2004)Google Scholar
  5. 5.
    Benner, P., Laub, A., Mehrmann, V.: A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: Continuous-time case. Technical Report SPC 95_22, Fak. f. Mathematik, TU Chemnitz–Zwickau, 09107 Chemnitz, FRG. http://www.mathematik. tu-chemnitz.de (1995)Google Scholar
  6. 6.
    Benner P., Laub A.J., Mehrmann V. (1997): Benchmarks for the numerical solution of algebraic Riccati equations. IEEE Contr. Syst. Mag. 7, 18–28CrossRefGoogle Scholar
  7. 7.
    Benner P., Mehrmann V., Xu H. (1997): A new method for computing the stable invariant subspace of a real Hamiltonian matrix. J. Comput. Appl. Math. 86, 17–43CrossRefMathSciNetGoogle Scholar
  8. 8.
    Benner P., Mehrmann V., Xu H. (1998): A numerically stable, structure preserving method for computing the eigenvalues of real Hamiltonian or symplectic pencils. Numer. Math. 78(3):329–358CrossRefMathSciNetGoogle Scholar
  9. 9.
    Benner P., Mehrmann V., Xu H. (1999): A note on the numerical solution of complex Hamiltonian and skew-Hamiltonian eigenvalue problems. Electr. Trans. Numer. Anal. 8, 115–126MathSciNetGoogle Scholar
  10. 10.
    Benner P., Mehrmann V., Xu H. (2002): Perturbation analysis for the eigenvalue problem of a formal product of matrices. BIT 42(1): 1–43CrossRefMathSciNetGoogle Scholar
  11. 11.
    Benner, P., Kreß ner, D.: HAPACK, software for (skew-)Hamiltonian eigenvalue problems (2003)Google Scholar
  12. 12.
    Bojanczyk A., Golub G.H., Van Dooren P. (1992): The periodic Schur decomposition; algorithms and applications. Proc. SPIE Conf. 1770, 31–42Google Scholar
  13. 13.
    Bora, S., Mehrmann, V.: Perturbation theory for structured matrix pencils arising in control theory. SIAM J. Matrix Anal. Appl. (2006) (to appear)Google Scholar
  14. 14.
    Bunch J.R. (1987): The weak and strong stability of algorithms in numerical algebra. Linear Algebra Appl. 88, 49–66CrossRefMathSciNetGoogle Scholar
  15. 15.
    Bunse-Gerstner, A., Byers, R., Mehrmann, V.: Numerical methods for algebraic Riccati equations. In: Bittanti, S. (ed.) Proc. Workshop on the Riccati Equation in Control, Systems, and Signals pp. 107–116. Como, Italy (1989)Google Scholar
  16. 16.
    Bunse-Gerstner A., Mehrmann V. (1986): A symplectic QR-like algorithm for the solution of the real algebraic Riccati equation. IEEE Trans. Automat. Control AC-31, 1104–1113CrossRefMathSciNetGoogle Scholar
  17. 17.
    Byers, R.: Hamiltonian and symplectic algorithms for the algebraic Riccati equation. PhD thesis, Cornell University, Dept. Comp. Sci., Ithaca (1983)Google Scholar
  18. 18.
    Byers R. (1986): A Hamiltonian QR-algorithm. SIAM J. Sci. Statist. Comput 7, 212–229CrossRefMathSciNetGoogle Scholar
  19. 19.
    Chu, D., Liu, X., Mehrmann, V.: A numerically backwards stable method for computing the Hamiltonian schur form. Preprint 2004/24. url: http://www.math.tu-berlin.de/preprints/Google Scholar
  20. 20.
    Demmel J.W. (1987): Three methods for refining estimates of invariant subspaces. Computing 38, 43–57CrossRefMathSciNetGoogle Scholar
  21. 21.
    Freiling G., Mehrmann V., Xu H. (2002): Existence, uniqueness and parametrization of Lagrangian invariant subspaces. SIAM J. Matrix Anal. Appl. 23, 1045–1069CrossRefMathSciNetGoogle Scholar
  22. 22.
    Golub G.H., Van Loan C.F. (1996): Matrix Computations, 3rd edn. Johns Hopkins University Press, BaltimorezbMATHGoogle Scholar
  23. 23.
    Hench J.J., Laub A.J. (1994): Numerical solution of the discrete-time periodic Riccati equation. IEEE Trans. Automat. Control 39, 1197–1210CrossRefMathSciNetGoogle Scholar
  24. 24.
    Higham N.J. (1996): Accuracy and Stability of Numerical Algorithms. SIAM Publications, PhiladelphiazbMATHGoogle Scholar
  25. 25.
    Abou Kandil H., Freiling G., Ionescu V., Jank G. (2003): Matrix Riccati Equations in Control and Systems Theory. Birkhäuser, Verlag, BaselzbMATHGoogle Scholar
  26. 26.
    Konstantinov M.M., Mehrmann V., Petkov P.Hr. (2002): Perturbation analysis for the Hamiltonian Schur form. SIAM J. Matrix Anal. Appl. 23, 387–424CrossRefMathSciNetGoogle Scholar
  27. 27.
    Kreßner, D.: Numerical methods and software for general and structured eigenvalue problems. PhD thesis, TU Berlin, Institut für Mathematik, Berlin (2004)Google Scholar
  28. 28.
    Kreßner D. (2004): The periodic QR algorithm is a disguised QR algorithm. Linear Algebra Appl. 417, 423–433CrossRefGoogle Scholar
  29. 29.
    Kressner D. (2005): Perturbation bounds for isotropic invariant subspaces of skew-Hamiltonian matrices. SIAM J. Matrix Anal. Appl. 26(4): 947–961CrossRefMathSciNetGoogle Scholar
  30. 30.
    Lancaster P., Rodman L. (1995): The Algebraic Riccati Equation. Oxford University Press, OxfordGoogle Scholar
  31. 31.
    Laub, A.J.: A Schur method for solving algebraic Riccati equations. IEEE Trans. Automat. Control AC-24, 913–921 (1979) (See also Proc. 1978 CDC (Jan. 1979) pp. 60–65).Google Scholar
  32. 32.
    Laub A.J. (1991): Invariant subspace methods for the numerical solution of Riccati equations. In: Bittanti S., Laub A.J., Willems J.C. (eds) The Riccati Equation. Springer, Berlin Heidelberg New York, pp. 163–196Google Scholar
  33. 33.
    Lin, W.-W., Mehrmann, V., Xu, H.: Canonical forms for Hamiltonian and symplectic matrices and pencils. Linear Algebra Appl. 301–303, 469–533 (1999)Google Scholar
  34. 34.
    The MathWorks, Inc., Cochituate Place, 24 Prime Park Way, Natick, Mass, 01760. MATLAB Version 6.5 (2002)Google Scholar
  35. 35.
    Mehrmann V. (1991): The Autonomous Linear Quadratic Control Problem, Theory and Numerical Solution. Lecture Notes in Control and Information Sciences, vol. 163. Springer, Berlin Heidelberg New YorkzbMATHCrossRefGoogle Scholar
  36. 36.
    Mehrmann V., Watkins D. (2001): Structure-preserving methods for computing eigenpairs of large sparse skew-hamiltonian/hamiltonian pencils. SIAM J. Sci. Comput. 22, 1905–1925CrossRefMathSciNetGoogle Scholar
  37. 37.
    Paige C.C., Van Loan C.F. (1981): A Schur decomposition for Hamiltonian matrices. Linear Algebra Appl. 14, 11–32CrossRefGoogle Scholar
  38. 38.
    Ran A.C.M., Rodman L. (1988): Stability of invariant Lagrangian subspaces I. In: Gohberg I. (eds) Operator Theory: Advances and Applications vol. 32. Birkhäuser-Verlag, Basel, pp. 181–218Google Scholar
  39. 39.
    Ran A.C.M., Rodman L. (1989): Stability of invariant Lagrangian subspaces II. In: Dym H.,S., Kaashoek M.A., Lancaster P. (eds) Operator Theory: Advances and Applications vol 40. Birkhäuser-Verlag, Basel, pp. 391–425Google Scholar
  40. 40.
    Sreedhar J., Van Dooren P. (1994): Periodic Schur forms and some matrix equations. Systems Netw. Math. Theory Appl. 77, 339–362Google Scholar
  41. 41.
    Stewart G.W. (1976): Simultaneous iteration for computing invariant subspaces of non Hermitian matrices. Numer. Math. 25, 123–136CrossRefGoogle Scholar
  42. 42.
    Van Dooren P. (1981): A generalized eigenvalue approach for solving Riccati equations. SIAM J. Sci. Statist. Comput. 2, 121–135CrossRefMathSciNetGoogle Scholar
  43. 43.
    Van Loan C.F. (1984): A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix. Linear Algebra Appl. 16, 233–251CrossRefGoogle Scholar
  44. 44.
    Van Loan C.F. (1985): Computing the CS and the generalized singular value decomposition. Numer. Math. 46, 479–491CrossRefMathSciNetGoogle Scholar
  45. 45.
    Watkins, D.S.: On the reduction of a hamiltonian matrix to hamiltonian schur form. Preprint, Washington State University, Pullman 2005. http://www.sci.wsu.edu/math/faculty/watkins/ refpub.html. Elect. Trans. Numer. Anal. 23, 141–157 (2006)Google Scholar
  46. 46.
    Xu, H.: Solving algebraic Riccati equations via Skew-Hamiltonian matrices. PhD thesis, Inst. of Math., Fudan University, Shanghai (1993)Google Scholar
  47. 47.
    Xu H., Lu L. (1995): Properties of a quadratic matrix equation and the solution of the continuous-time algebraic Riccati equation. Linear Algebra Appl. 222, 127–146CrossRefMathSciNetGoogle Scholar
  48. 48.
    Zhou K., Doyle J.C., Glover K. (1995): Robust and Optimal Control. Prentice-Hall, Upper Saddle RiverGoogle Scholar

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

Personalised recommendations