Orthogonal Transformation (Square-Root) Implementations of the Generalized Chandrasekhar and Generalized Levinson Algorithms
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.
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- 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
- V. Belevitch,Classical Network Synthesis, San Francisco: Holden-Day, 1966.Google Scholar
- 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
- 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
- T. Kailath,Lectures on Linear Least Squares Estimation, Wien: Springer-Verlag, 1978.Google Scholar
- 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
- 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
- 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
- T. Kailath, S. Kung and M. Morf, “Displacement rank of matrices and linear operators,”J. Math. Anal. and Applns., to appear.Google Scholar
- W. Greub,Linear Algebra, New York: Springer-Verlag, 3rd edition, 1973.Google Scholar