On the numerical properties of the Schur approach for solving the matrix Riccati equation
type: Item
instance of: scholarly article
title: On the numerical properties of the Schur approach for solving the matrix Riccati equation
zbMATH Open document ID: 0635.65043
DOI: 10.1016/0167-6911(87)90040-5
author: Petko Hr. Petkov
author: Nicolai D. Christov
author: Mihail M. Konstantinov
published in: Systems \& Control Letters
publication date: 1987-01-01T00:00:00Z
review text: The Schur approach to solving the continuous algebraic matrix Riccati equation, i.e. the reduction of the corresponding Hamiltonian matrix \(H=\left[ \begin{matrix} A\\ -Q\end{matrix} \begin{matrix} -S\\ -A\quad t\end{matrix} \right]\) to upper quasitriangular form \(T=U\) tHU where \(U=\left[ \begin{matrix} U_{11}\\ U_{21}\end{matrix} \begin{matrix} U_{12}\\ U_{22}\end{matrix} \right]\) is an orthogonal matrix, followed by a solution of the matrix equation \(PU_{11}=U_{21}\), is considered. It is shown that the equivalent relative perturbations in the matrices A, Q and S may lead to numerical instability if the norm of some matrix is much smaller than the norm of the other. A numerical example solved by the standard implementation of the Schur approach emphasizes its numerical instability.
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 65K10
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 93C15
zbMATH DE Number: 4034994
zbMATH Keywords: Schur method
zbMATH Keywords: reduction to upper quasitriangular form
zbMATH Keywords: continuous algebraic matrix Riccati equation
zbMATH Keywords: Hamiltonian matrix
zbMATH Keywords: numerical instability
zbMATH Keywords: numerical example
describes a project that uses: RICPAC
describes a project that uses: LINPACK
describes a project that uses: HQR3
describes a project that uses: EISPACK
MaRDI profile type: Publication
full work available at URL: 0167-6911(87)90040-5
OpenAlex ID: W2102575629
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description: scientific article; zbMATH DE number 4034994