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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 $\varepsilon =2.2\times 10^{-16}$.

15A24Matrix equations and identities
Full Text: DOI
[1] Anderson, W. N.; Morley, T. D.; Trapp, G. E.: Positive solutions to X=A-BX-1B\ast. Linear algebra appl. 134, 53-62 (1990) · Zbl 0702.15009
[2] Engwerda, J. C.: On the existence of a positive definite solution of the matrix equation X+ATX-1A=I. Linear algebra appl. 194, 91-108 (1993) · Zbl 0798.15013
[3] Engwerda, J. C.; Ran, A. C. M.; Rijkeboer, A. L.: Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation X+A\astX-1A=I. Linear algebra appl. 186, 255-275 (1993) · Zbl 0778.15008
[4] Ran, A. C. M.; Rodman, L.: Stable Hermitian solutions of discrete algebraic Riccati equations. Math. control signals systems 5, 165-193 (1992) · Zbl 0771.93059
[5] Xu, S. F.: On the maximal solution of the matrix equation X+ATX-1A=I. Acta sci. Natur. univ. Pekinensis 36, 29-38 (2000)
[6] Zhan, X.: On the matrix equation X+ATX-1A=I. Linear algebra appl. 247, 337-345 (1996)
[7] Zhan, X.: Computing the extremal positive definite solutions of a matrix equation. SIAM J. Sci. comput. 17, 1167-1174 (1996) · Zbl 0856.65044