On the numerical properties of the Schur approach for solving the matrix Riccati equation (Q1097641)

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.