The nonlinear matrix equation is considered where , are complex matrices with Hermitian positive definite and denoting the conjugate transpose of a matrix . Such equations arise in control theory, ladder networks, dynamic programming, stochastic filtering, statistics etc. Hermitian positive definite solutions are of particular interest. The inequality means that is positive semidefinite. One defines the maximal and minimal solutions and such that for any Hermitian positive definite solution .
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 . Perturbation properties are related to the condition number .
The results are illustrated by numerical examples carried out using MATLAB on a PC Pentium III/500 computer, with machine epsilon .