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Notes on two perturbation estimates of the extreme solutions to the equations $X±{A}^{*}{X}^{-1}A=Q$. (English) Zbl 1222.15020
Summary: Two perturbation estimates of the maximal positive definite solutions to the matrix equations $X+{A}^{*}{X}^{-1}A=Q$ and $X-{A}^{*}{X}^{-1}A=Q$ are considered. These estimates are like to the estimates discussed by V. I. Hasanov and I. G. Ivanov [Linear Algebra Appl. 413, No. 1, 81–92 (2006; Zbl 1087.15016)]. The conditions $\parallel {X}_{L}^{-1}{A\parallel }_{2}<1$ and $\parallel {X}_{+}^{-1}{A\parallel }_{2}<1$ in [loc. cit.] are not always satisfied. We replace this conditions by $\parallel P{X}_{L}^{-1}A{P}^{-1}{\parallel }_{2}<1$ and $\parallel P{X}_{+}^{-1}A{P}^{-1}{\parallel }_{2}<1$ respective, where $P$ is positive definite matrix. The theoretical results are illustrated by numerical examples.
##### MSC:
 15A24 Matrix equations and identities
##### Keywords:
nonlinear matrix equation; perturbation estimates
##### References:
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