×

zbMATH — the first resource for mathematics

Riccati-like flows and matrix approximations. (English) Zbl 0802.65058
Authors’ summary: A classical problem in matrix analysis, total least squares estimation and model reduction theory is that of finding a best approximant of a given matrix by lower rank ones. It is common believe that behind every such least squares problem there is an algebraic Riccati equation.
In this paper, we consider the task of minimizing the distance function \(f_ A(X)= \| A- X\|^ 2\) on varieties of fixed rank symmetric matrices, using gradient-like flows for the distance function \(f_ A\). These flows turn out to have similar properties as the dynamic Riccati equation and are thus termed Riccati-like flows.
A complete phase portrait analysis of these Riccati-like flows is presented, with special emphasis on positive semidefinite solutions. A variable step-size discretization of the flows is considered. The results may be viewed as a prototype for similar investigations one would like to pursue in model reduction theory of linear control systems.
Reviewer: D.Braess (Bochum)
MSC:
65F30 Other matrix algorithms (MSC2010)
93C15 Control/observation systems governed by ordinary differential equations
93B11 System structure simplification
PDF BibTeX XML Cite
Full Text: Link EuDML
References:
[1] G. Eckart, G. Young: The approximation of one matrix by another of lower rank. Psychometrika / (1936), 211-218. · JFM 62.1075.02
[2] G. H. Golub, C. Van Loan: An analysis of the total least squares problem. SIAM J. Numer. Anal. 17 (1980), 883-843. · Zbl 0468.65011 · doi:10.1137/0717073
[3] G. H. Golub A. Hoffmann, G. W. Stewart: A generalization of the Eckart-Young-Mirsky matrix approximation theorem. Linear Algebra Appl. 88/89 (1987), 317-327. · Zbl 0623.15020 · doi:10.1016/0024-3795(87)90114-5
[4] U. Helmke, J. B. Moore: Optimization and Dynamical Systems. Springer-Verlag, Berlin 1993. · Zbl 0943.93001
[5] U. Helmke, M. A. Shayman: Critical points of matrix least squares distance functions. Linear Algebra Appl., to appear. · Zbl 0816.15026 · doi:10.1016/0024-3795(93)00070-G
[6] N. J. Higham: Computing a nearest symmetric positive semidefinite matrix. Linear Algebra Appl. 103 (1988), 103-118. · Zbl 0649.65026 · doi:10.1016/0024-3795(88)90223-6
[7] B. De Moor, J. David: Total linear least squares and the algebraic Riccati equation. Systems Control Lett. 5 (1992), 329-337. · Zbl 0763.93085 · doi:10.1016/0167-6911(92)90022-K
[8] J. B. Moore R. E. Mahony, U. Helmke: Recursive gradient algorithms for eigenvalue and singular value decomposition. SIAM J. Matrix Anal. Appl., to appear. · Zbl 0855.65033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.