Backward error for the discrete-time algebraic Riccati equation (Q1362633)

Consider the discrete-time matrix algebraic Riccati equation \[ \Phi (X;F,G,H): =X- F^TX (I_n+GX)^{-1} F-H=0, \tag{1} \] where all involved matrices are real \(n\times n\) with \(G,H\) symmetric and non-negative. It is supposed that the pair \((F,G)\) is stabilizable and the pair \((H,F)\) is detectable which guarantees the existence of an unique symmetric nonnegative solution \(X\) such that the closed-loop system matrix \((I_n+ GX)^{-1}F\) is convergent, i.e. has its spectrum in the open unit disc in \({\mathcal C}\). Let \(\Delta F\), \(\Delta G\) and \(\Delta H\) be perturbations in \(F,G\) and \(H\) with \(\Delta G\), \(\Delta H\) symmetric and nonnegative definite. Let \(X^*\) be an approximate solution to equation (1) and \(\alpha_F\), \(\alpha_G\), \(\alpha_H\) be positive constants. The normwise backward error of \(X^*\) is defined as \(\eta= \eta (X^*, \alpha_F, \alpha_G, \alpha_H) =\min \{\varepsilon: |\Delta Z |_F \leq \varepsilon \alpha_Z; Z=F,G,H\}\) under the constraint \[ \Phi (X^*, F+ \Delta F,G+ \Delta G,H +\Delta H)=0. \tag{2} \] The author gives lower and upper bounds for \(\eta\). For this purpose equation (2) is first rewritten as an equivalent operator equation for the perturbations in the coefficient matrices. After that it is shown that the equivalent operator satisfies the Schauder fixed point principle in a certain (small) compact which results in the desired estimates.