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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)
65F30 Other matrix algorithms (MSC2010)
93C15 Control/observation systems governed by ordinary differential equations
93B11 System structure simplification
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