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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 $n\times n$ 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 $X\geq Y$ means that $X-Y$ is positive semidefinite. One defines the maximal and minimal solutions $X_L$ and $X_S$ such that $X_S\leq X\leq X_L$ 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 $\|.\|$. Perturbation properties are related to the condition number $\kappa (A,P)=(\frac{1}{2}-\|A\|\|P^{-1}\|)^{-1}$. The results are illustrated by numerical examples carried out using MATLAB on a PC Pentium III/500 computer, with machine epsilon $\varepsilon =2.2\times 10^{-16}$.

##### MSC:
 15A24 Matrix equations and identities
Matlab
Full Text:
##### References:
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