Improved perturbation estimates for the matrix equations \(X \pm A^{*} X^{-1} A=Q\). (Q1426295)

The authors give new and improved perturbation estimates for the solutions of the matrix quadratic equations \(X\pm A^*X^{-1}A=Q\) (\(X\) and \(Q\) are positive definite Hermitian matrices). Some of the estimates depend and some do not depend on the knowledge of the exact solution \(X\). The results are compared with those of \textit{S. F. Xu} [Linear Algebra Appl. 336, 61--70 (2001; Zbl 0992.15013)] and of \textit{A. C. M. Ran} and \textit{M. C. B. Reurings} [Linear Algebra Appl. 346, 15--26 (2002; Zbl 1086.15013)]. The paper contains a presentation of numerical experiments.