## Notes on two perturbation estimates of the extreme solutions to the equations $$X\pm A^{*}X^{-1}A=Q$$.(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 $$\| X_L^{-1}A\|_2 < 1$$ and $$\| X_+^{-1}A\|_2 < 1$$ in [loc. cit.] are not always satisfied. We replace this conditions by $$\| PX_L^{-1}AP^{-1}\|_2 < 1$$ and $$\| PX_+^{-1}AP^{-1}\|_2 < 1$$ 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

Zbl 1087.15016
Full Text:

### References:

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