The Kähler Structure of the Total Least Squares Problem, Brockett’s Steepest Descent Equations, and Constrained Flows

  • Anthony M. Bloch
Part of the Progress in Systems and Control Theory book series (PSCT, volume 3)


In this paper we show how the Total Least Squares identification problem may be viewed as a steepest descent problem on a Kähler manifold. The Kähler structure gives a method of explicitly deriving the steepest descent equations from a corresponding Hamiltonian flow associated with the problem. In the line-fitting case the steepest descent flow itself is shown also to be equivalent to the flow of a constrained Hamiltonian system — the Toda system.


Steep Descent Symplectic Form Gradient Flow Toda Lattice Grassmann Manifold 
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  1. 1.
    R. A. Abraham and J. E. Marsden, “Foundation of Mechanics,” Benjamin Cummings, 1978.Google Scholar
  2. 2.
    A. Bloch, Estimation, Principal Components and Hamiltonian Systems, Systems and Control Letters 6 (1985), 103–108.CrossRefGoogle Scholar
  3. 3.
    A. Bloch, An Infinite-Dimensional Hamiltonian System on Projective Hilbert Space, Trans. A.M.S. 302 (1987), 787–796.CrossRefGoogle Scholar
  4. 4.
    A. Bloch, Steepest Descent, Linear Programming and Hamiltonian Flows, to appear in Cont Math. A.M.S.Google Scholar
  5. 5.
    A. 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, to appear.Google Scholar
  6. 6.
    R. W. Brockett, A Geometrical Matching Problem, to appear in J. of Linear Algebra and its Applications.Google Scholar
  7. 7.
    R. W. Brockett, Dynamical Systems that Sort Lists and Solve Linear Programming Problems, in, Proc 27th IEEE Conf. on Decision and Control, IEEE, 1988, 799–803.CrossRefGoogle Scholar
  8. 8.
    C. I. Byrnes and J. C. Willems, Least Squares Estimation, Linear Programming and Momentum, preprint.Google Scholar
  9. 9.
    P. Deift, T. Nanda and C. Tomei, Differential Equations for the Symmetric Eigenvalue Problem, SIAM J. on Numerical Analysis 20 (1983), 1–22.CrossRefGoogle Scholar
  10. 10.
    P. Deift, F. Lund and E. Trubowitz, Nonlinear Wave Equations and Constrained Harmonic Motion, Communications in Math. Physics 74 (1980), 141–188.CrossRefGoogle Scholar
  11. 11.
    H. Flaschka, The Toda Lattice, Phys. Rev. B 9 (1976), 1924–1925.CrossRefGoogle Scholar
  12. 12.
    J. Moser, Finitely Many Mass Points on the Line under the Influence of an Exponential Potential, Battelles Recontres, Springer Notes in Physics (1976), 417–497.Google Scholar
  13. 13.
    A. Pressley and G. Segal, Loop Groups, Oxford University Press, 1986.Google Scholar

Copyright information

© Birkhäuser Boston 1990

Authors and Affiliations

  • Anthony M. Bloch

There are no affiliations available

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