Orthogonal Transformation (Square-Root) Implementations of the Generalized Chandrasekhar and Generalized Levinson Algorithms

  • T. Kailath
  • A. Vieira
  • M. Morf


In recent work, we have shown how least-squares estimation problems for arbitrary nonstationary processes can be speeded up by using the notion of an index of nonstationarity and a corresponding generalized Levinson algorithm. It was also shown that when the nonstationary processes have constant-parameter state-space models, the generalized Levinson algorithms reduce to the generalized Chandrasekhar equations. In this paper we shall show that the explicit equations of the above algorithms can be replaced by certain implicitly defined J-orthogonal transformation procedures, where J is a signature matrix (zero everywhere except for ±l’s on the diagonal). In the state-space case, these methods yield the previously-derived fast square-root algorithms of Morf and Kailath.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    T. Kailath, M. Morf and G. Sidhu, “Some new algorithms for recursive estimation in constant discrete-time linear systems,”IEEE Trans. Automat. Contr., Vol. AC-19 Aug. 1974, pp.315–323.CrossRefMATHGoogle Scholar
  2. [2]
    M. Morf, T. Kailath, “Square-root algorithms for least-squares estimation,”IEEE Trans. on Auto.Control, Vol. AC-20, no. 4, Aug. 1975, pp.487–497.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    L.M. Silverman, “Discrete Riccati equations: alternative algorithms, asymptotic properties and system theory interpretations,” inAdvances in Control and Dynamic Systems: Theory and Applications, Vol. 12, L. Leondes, editor, Academic Press, 1975.Google Scholar
  4. [4]
    V. Belevitch,Classical Network Synthesis, San Francisco: Holden-Day, 1966.Google Scholar
  5. [5]
    A.V. Efimov and V.P. Potapov, “J-expanding matrix functions and their role in the analytical theory of electrical circuits,”Russian Math. Surveys, vol. 28, no. 1, pp.69–140, 1973.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    P. Dewilde, A. Vieira and T. Kailath, “On a generalized Szegö-Levinson realization algorithm for optimal linear predictors based on a network synthesis approach,”IEEE Trans. on Circuits and Systems, Sept. 1978.Google Scholar
  7. [7]
    J. Makhoul, “Linear prediction: a tutorial review,”Proc. IEEE, vol. 63, pp. 561–580, Aprl. 1975.CrossRefGoogle Scholar
  8. [8]
    B. Friedlander, T. Kailath, M. Morf and L. Ljung, “Extended Levinson and Chan-drasekhar equations for general discrete-time linear estimation problems,”IEEE Trans. Auto. Control, vol. AC-23, pp. 653–659, Aug. 1978.CrossRefMATHGoogle Scholar
  9. [9]
    R.E. Kaiman, “A new approach to linear filtering and prediction problems,”Trans. ASME. (J. Basic Eng.), Vol. 82D, pp.34–45, March 1960.Google Scholar
  10. [10]
    T. Kailath,Lectures on Linear Least Squares Estimation, Wien: Springer-Verlag, 1978.Google Scholar
  11. [11]
    T. Kailath and L. Ljung, “A scattering theory framework for fast least-squares algorithms,” inMultivariable Analysis — IV, P.R. Krishnaiah, editor, Amsterdam: North Holland Publishing Co., 1977. (Original symposium in Dayton, Ohio, June 1975).Google Scholar
  12. [12]
    B. Friedlander, T. Kailath and L. Ljung, “Scattering theory and least squares estimation — II: Discrete-time Problems,”J. Franklin Inst., Vol. 301, nos.1–2, Jan.-Feb. 1976, pp.71–82.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    G. Verghese, B. Friedlander and T. Kailath, “Scattering theory and least squares estimation, Pt. Ill — The Estimates,”IEEE Trans. Auto. Control, Vol. AC-24, 1979, to appear.Google Scholar
  14. [14]
    I.R. Bunch and B.N. Parlett, “Direct method for solving symmetric indefinite systems of linear equations”,SIAM J. Numer.Anal., Vol. 8, pp.639–655, 1971.MathSciNetCrossRefGoogle Scholar
  15. [15]
    G. Stewart,Introduction to Matrix Computations, New York: Academic Press, 1973.MATHGoogle Scholar
  16. [16]
    N. Levinson, “The Wiener RMS (root-mean-square) error criterion in filter design and prediction,”J. Math. Phys., Vol. 25, pp.261–278, Jan. 1947.MathSciNetGoogle Scholar
  17. [17]
    T. Kailath, M. Morf and G. Sidhu, “Some new algorithms for recursive estimation in constant discrete-time linear systems,” Proc. 7th Princeton Symposium Information and System Sciences, pp. 344–352, April, 1973.Google Scholar
  18. [18]
    A. Lindquist, “A new algorithm for optimal filtering of discrete-time stationary processes,”SIAM J. Control, Vol. 12, 1974, pp.736–746.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    T. Kailath, S. Kung and M. Morf, “Displacement rank of matrices and linear operators,”J. Math. Anal. and Applns., to appear.Google Scholar
  20. [20]
    W. Greub,Linear Algebra, New York: Springer-Verlag, 3rd edition, 1973.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1979

Authors and Affiliations

  • T. Kailath
    • 1
  • A. Vieira
    • 1
  • M. Morf
    • 1
  1. 1.Information Systems Laboratory, Department of Electrical EngineeringStanford UniversityStanfordUSA

Personalised recommendations