*(English)*Zbl 0992.15013

The nonlinear matrix equation $X+{A}^{*}{X}^{-1}A=P$ is considered where $A$, $P$ are $n\times n$ complex matrices with $P$ Hermitian positive definite and ${A}^{*}$ denoting the conjugate transpose of a matrix $A$. Such equations arise in control theory, ladder networks, dynamic programming, stochastic filtering, statistics etc. Hermitian positive definite solutions $X$ are of particular interest. The inequality $X\ge Y$ means that $X-Y$ is positive semidefinite. One defines the maximal and minimal solutions ${X}_{L}$ and ${X}_{S}$ such that ${X}_{S}\le X\le {X}_{L}$ for any Hermitian positive definite solution $X$.

The paper contains a perturbation bound for the maximal solution of the above matrix equation and a computable error bound for approximate solutions; they are derived using a property of the maximal solution expressed in terms of matrix 2-norms $\parallel \xb7\parallel $. Perturbation properties are related to the condition number $\kappa (A,P)=(\frac{1}{2}-\parallel A\parallel \parallel {P}^{-1}{\parallel )}^{-1}$.

The results are illustrated by numerical examples carried out using MATLAB on a PC Pentium III/500 computer, with machine epsilon $\epsilon =2\xb72\times {10}^{-16}$.

##### MSC:

15A24 | Matrix equations and identities |