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Perturbation analysis of the maximal solution of the matrix equation X+A * X -1 A=P. (English) Zbl 0992.15013

The nonlinear matrix equation X+A * X -1 A=P is considered where A, P are n×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 XY means that X-Y is positive semidefinite. One defines the maximal and minimal solutions X L and X S such that X S XX 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 ·. Perturbation properties are related to the condition number κ(A,P)=(1 2-AP -1 ) -1 .

The results are illustrated by numerical examples carried out using MATLAB on a PC Pentium III/500 computer, with machine epsilon ε=2·2×10 -16 .

15A24Matrix equations and identities