Skip to main content
SpringerLink
Log in

Controllability properties of numerical eigenvalue algorithms

  • Conference paper
  • First Online:
Nonlinear control in the Year 2000

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 258))

  • 2878 Accesses

Abstract

We analyze controllability properties of the inverse iteration and the QR-algorithm equipped with a shifting parameter as a control input. In the case of the inverse iteration with real shifts the theory of universally regular controls may be used to obtain necessary and sufficient conditions for complete controllability in terms of the solvability of a matrix equation. Partial results on conditions for the solvability of this matrix equation are given. We discuss an interpretation of the system in terms of control systems on rational functions. Finally, first results on the extension to inverse Rayleigh iteration on Grassmann manifolds using complex shifts is discussed.

This paper was written while Fabian Wirth was a guest at the Centre Automatique et Systèmes, Ecole des Mines de Paris, Fontainebleau, France. The hospitality of all the members of the centre is gratefully acknowledged.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Albertini, F. (1993) Controllability of discrete-time nonlinear systems and some related topics, PhD thesis, Grad. School New Brunswick Rutgers, New Brunswick, New Jersey.

    Google Scholar 

  2. Albertini, F., Sontag, E. (1993) Discrete-time transitivity and accessibility: analytic systems, SIAM J. Contr. & Opt. 31, 1599–1622

    Article  MATH  MathSciNet  Google Scholar 

  3. Batterson, S. (1995) Dynamical analysis of numerical systems. Numer. Linear Algebra Appl. 2, 297–310

    Article  MATH  MathSciNet  Google Scholar 

  4. Batterson, S., Smillie, J. (1990) Rayleigh quotient iteration for nonsymmetric matrices. Math. Comput. 55, 169–178

    Article  MATH  MathSciNet  Google Scholar 

  5. Batterson, S., Smillie, J. (1989) Rayleigh quotient iteration fails for nonsymmetric matrices. Appl. Math. Lett. 2, 19–20

    Article  MATH  MathSciNet  Google Scholar 

  6. Batterson, S., Smillie, J. (1989) The dynamics of Rayleigh quotient iteration. SIAM J. Numer. Anal. 26, 624–636

    Article  MATH  MathSciNet  Google Scholar 

  7. Gelfand, I.M., Serganova, V.V. (1987) Combinatorial geometries and torus strata on homogeneous compact manifolds. Uspekhi Math. Nauk. 42, 107–134

    MathSciNet  Google Scholar 

  8. Gelfand, I.M., Goresky, R.M., MacPherson, R.D., Serganova, V.V. (1987) Combinatorial geometries, convex polyhedra and Schubert cells. Adv. in Math. 63, 301–316

    Article  MathSciNet  Google Scholar 

  9. Brockett, R.W., Krishnaprasad, P.S. (1980) A scaling theory for linear systems, IEEE Trans. Autom. Control AC-25, 197–207

    Article  MathSciNet  Google Scholar 

  10. Colonius, F., Kliemann, W. (1993) Linear control semigroups acting on projective space. J. Dynamics Diff. Equations 5, 495–528

    Article  MATH  MathSciNet  Google Scholar 

  11. Colonius, F., Kliemann, W. (1996) The Lyapunov spectrum of families of time varying matrices. Trans. Amer. Math. Soc. 348, 4389–4408

    Article  MATH  MathSciNet  Google Scholar 

  12. Fuhrmann, P. A. (1996) A Polynomial Approach to Linear Algebra, Springer Publ., New York

    MATH  Google Scholar 

  13. Fuhrmann, P. A., Helmke, U. (2000) On controllability of matrix eigenvalue algorithms: The inverse power method, Syst. Cont. Lett., to appear.

    Google Scholar 

  14. Helmke, U., Wirth, F. (2000) On controllability of the real shifted inverse power iteration, Syst. Cont. Lett., to appear.

    Google Scholar 

  15. Ipsen, I. C. F. (1997) Computing an eigenvector with inverse iteration. SIAM Rev. 39, 254–291

    Article  MATH  MathSciNet  Google Scholar 

  16. Sontag, E. D., Wirth F. R (1998) Remarks on universal nonsingular controls for discrete-time systems, Syst. Cont. Lett. 33, 81–88

    Article  MATH  MathSciNet  Google Scholar 

  17. Suzuki, T. (1992) Inverse iteration method with a complex parameter. Proc. Japan Acad., Ser. A 68, 68–73

    Article  MATH  Google Scholar 

  18. Suzuki, T (1994) Inverse iteration method with a complex parameter. II. in Yajima, K. (ed.), Spectral scattering theory and applications. Proceedings of a conference on spectral and scattering theory held at Tokyo Institute of Technology, Japan, June 30–July 3, 1992 in honour of Shige Toshi Kuroda on the occasion of his 60th birthday. Tokyo: Kinokuniya Company Ltd., Adv. Stud. Pure Math. 23, 307–310

    Google Scholar 

  19. Wirth, F. (1998) Dynamics of time-varying discrete-time linear systems: Spectral theory and the projected system. SIAM J. Contr. & Opt. 36, 447–487

    Article  MATH  MathSciNet  Google Scholar 

  20. Wirth, F. (1998) Dynamics and controllability of nonlinear discrete-time control systems. In 4th IFAC Nonlinear Control Systems Design Symposium (NOLCOS'98) Enschede, The Netherlands, 269–275

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität Würzburg, 97074, Würzburg, Germany

    Uwe Helmke

  2. Zentrum für Technomathematik, Universität Bremen, 28334, Bremen, Germany

    Fabian Wirth

Authors
  1. Uwe Helmke
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fabian Wirth
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Alberto Isidori (Professor)Françoise Lamnabhi-Lagarrigue (Docteur D’état)Witold Respondek (Professor)

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag London Limited

About this paper

Cite this paper

Helmke, U., Wirth, F. (2001). Controllability properties of numerical eigenvalue algorithms. In: Isidori, A., Lamnabhi-Lagarrigue, F., Respondek, W. (eds) Nonlinear control in the Year 2000. Lecture Notes in Control and Information Sciences, vol 258. Springer, London. https://doi.org/10.1007/BFb0110234

Download citation

  • DOI: https://doi.org/10.1007/BFb0110234

  • Published:

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-85233-363-8

  • Online ISBN: 978-1-84628-568-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation