Abstract
Numerical matrix eigenvalue methods such as the inverse power iteration or the QR-algorithm can be reformulated as inverse power iterations on homogeneous spaces. In this paper we survey some recent results on controllability properties of the shifted inverse power iteration on flag manifolds. It is shown that the reachable sets are orbits for a semigroup action on the flag manifold. Except for the special case of projective spaces, the algorithm is never controllable. This implies in particular the non-controllability of the shifted QR-algorithm on isospectral matrices. Controllability results for the inverse power iteration on projective space for real or complex shifts are presented, following [20, 22], and a connection with output feedback pole assignability is mentioned. Controllability of the algorithm on Hessenberg flags is shown. This implies controllability of the shifted QR-algorithm on Hessenberg matrices.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
P.A. Absil, R. Mahony, R. Sepulchre and P. Van Dooren, A Grassmann-Rayleigh quotient iteration for computing invariant subspaces, SIAM Review, 44 (2002), pp. 57–73.
F. Albertini and E. Sontag, Discrete-time transitivity and accessibility: analytic systems, SIAM J. Contr. & Opt., 31 (1993), pp. 1599–1622.
G. Ammar and C. Martin, The Geometry of Matrix Eigenvalue Methods, Acta Applicandae Mathematicae, 5 (1986), pp. 239–278.
S. Batterson, Dynamical analysis of numerical systems, Numer. Linear Algebra Appl., 2 (1995), pp. 297–310.
S. Batterson and J. Smillie, Rayleigh quotient iteration for nonsymmetric matrices, Math. Comput., 55 (1990), pp. 169–178.
S. Batterson and J. Smillie, The dynamics of Rayleigh quotient iteration, SIAM J. Numer. Anal., 26 (1989), pp. 624–636.
A.M. Bloch, R.W. Brockett and T. Ratiu, A new formulation of the generalized Toda lattice equations and their fixed point analysis via the moment map, Bull. Amer. Math. Soc., 23 (1990), pp. 447–456.
A.M. Bloch, H. Flaschka and T. Ratiu, A convexity theorem for isospectral sets of Jacobi matrices in a compact Lie algebra, Duke Math J., 61 (1990), pp. 41–65.
R.W. Brockett, Dynamical systems that sort lists and solve linear programming problems, Proc. 27th IEEE Conference on Decision and Control, Austin, TX, 779–803. See also Linear Algebra Appl., 146 (1991), pp. 79–91.
R.W. Brockett, Smooth dynamical systems which realize arithmetical and logical operations, Three Decades of Mathematical System Theory, 135, Lecture Notes in Control and Information Sciences, Springer-Verlag, 1989, pp. 19–30.
R.W. Brockett, Differential geometry and the design of gradient algorithms. Proceedings of Symposia in Pure Mathematics, 54 (1993), pp. 69–91.
P. Deift, T. Nanda and C. Tomei, Ordinary differential equations for the symmetric eigenvalue problem, SIAM J. Numer. Anal., 20 (1983), pp. 1–22.
F. De Mari, C. Procesi and M. Shayman, Hessenberg varieties, Transactions of the American Mathematical Society 332 (1992), pp. 529–534.
F. De Mari and M. Shayman, Generalized Eulerian numbers and the topology of the Hessenberg variety of a matrix, Acta Applicandae Mathematicae, 12 (1988), pp. 213–235.
A. Edelman, T. Arias and S.T. Smith, The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Analysis and Applications, 20 (1998), pp. 303–353.
H. Flaschka, The Toda lattice, I, Phys. Rev., B9 (1974), pp. 1924–1925.
I.M. Gelfand, R.M. Goresky, R.D. MacPherson and V.V. Serganova, Combinatorial Geometries, Convex Polyhedra and Schubert Cells, Advances in Mathematics, 63 (1987), pp. 301–316.
G. M. L. Gladwell, Total Positivity and the QR-Algorithm, Linear Algebra and its Applications, 271 (1998), pp 257–272.
K. Gustafsson, Stepsize control in ODE Solvers — Analysis and Synthesis, Ph.D. Thesis, Lund Institute of Technology, (1998).
U. Helmke and P.A. Fuhrmann, Controllability of matrix eigenvalue algorithms: the inverse power method. Systems and Control Letters, 41 (2000), pp. 57–66.
U. Helmke and J.B. Moore, Optimization and Dynamical Systems, Communication and Control Engineering Series, Springer Publ. London, 1994.
U. Helmke and F. Wirth, On controllability of the real shifted inverse power iteration, Systems and Control Letters, 43 (2001), pp. 9–23.
K. Hüper, A calculus approach to matrix eigenvalue algorithms. Habilitation Address, July 2002. Department of Mathematics, Würzburg University, Germany.
K. Hüper, A Dynamical System Approach to Matrix Eigenvalue Algorithms. This volume.
M. Shub and A.T. Vasquez, Some linearly induced Morse-Smale systems, the QR-algorithm and the Toda lattice, in: The Legacy of Sonya Kovaleskaya (L. Keen, ed) Contemporary Mathematics 64, A.M.S., Providence (1987), pp. 181–194.
P. Van Dooren and R. Sepulchre, Shift policies in QR-like algorithms and feedback control of self-similar flows. Open Problems in Mathematical Systems and Control Theory (V.D. Blondel, E.D. Sontag, M. Vidyasagar and J.C. Willems, Eds.), Communications and Control Engineering Series. Springer Publ., London, 1999, pp. 245–249.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag New York, Inc.
About this paper
Cite this paper
Helmke, U., Jordan, J. (2003). Numerics Versus Control. In: Rosenthal, J., Gilliam, D.S. (eds) Mathematical Systems Theory in Biology, Communications, Computation, and Finance. The IMA Volumes in Mathematics and its Applications, vol 134. Springer, New York, NY. https://doi.org/10.1007/978-0-387-21696-6_7
Download citation
DOI: https://doi.org/10.1007/978-0-387-21696-6_7
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-2326-4
Online ISBN: 978-0-387-21696-6
eBook Packages: Springer Book Archive