Solving the algebraic Riccati equation with the matrix sign function (Q1087314)

The algebraic Riccati equation \(G+A^ TX+XA-XFX=0\) is reduced to a linear matrix equation of the form \(MX=N\) where the matrices M and N are defined by the sign function, Sign(K), of the Hamiltonian matrix \(K=\left[ \begin{matrix} A^ T\quad G\\ F\quad -A\end{matrix} \right]\). An iterative refinement of the matrix-sign-function algorithm and a stopping criterion limiting the effects of rounding errors lead to a stable numerical procedure which compares favorably with current Schur vector- based algorithms [\textit{A. Laub}, IEEE Trans. Autom. Control AC-24, 913- 921 (1979; Zbl 0424.65013)]. Comparative numerical experiments on three examples are also presented.