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Displacement structure: Theory and applications. (English) Zbl 0839.65028

This is a survey of describing how two strands of work from matrix and function theory have come together in interesting ways in some work on fast computational algorithms for matrices with so-called displacement structure. First, the paper reviews some earlier results on matrices with displacement structure and highlights connection with the classical algorithm of Schur and with inverse scattering problems. Then it introduces several generalizations of the notion of displacement structure and some important examples.
A hierarchy of generalized Schur algorithms is derived and exhibited in several different forms. Connections with lossless systems, embedding relations, and transmission zeros are highlighted and shown to be relevant to the solution of interpolation problems. Generalized Schur algorithms are then studied in the presence of state-space structure, with immediate applications to problems in state-space estimation and adaptive filtering.
Finally, it concludes with a brief account of extensions of the notion of displacement structure to time-variant matrices and with even briefer remarks on other results, applications, and some open problems.

MSC:

65F05 Direct numerical methods for linear systems and matrix inversion
15A23 Factorization of matrices
15A06 Linear equations (linear algebraic aspects)
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
30D50 Blaschke products, etc. (MSC2000)
47A57 Linear operator methods in interpolation, moment and extension problems
60G35 Signal detection and filtering (aspects of stochastic processes)
93A99 General systems theory
93C99 Model systems in control theory
93C55 Discrete-time control/observation systems
93E11 Filtering in stochastic control theory
93E24 Least squares and related methods for stochastic control systems
30E05 Moment problems and interpolation problems in the complex plane
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