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On the positive definite solutions of nonlinear matrix equation $X+A^*x^{-\delta}A=Q$. (English) Zbl 1083.15018
The authors consider the positive definite solutions of the nonlinear matrix equation $X+A^*X^{-\delta}A=Q$ where $\delta \in (0,1]$. (For $\delta =1$ this equation arises in the analysis of ladder networks, in dynamic programming, control theory, stochastic filtering and statistics.) For $\delta \in (0,1]$ the equation has been first considered in {\it S. M. El-Sayed} and {\it A. C. M. Ran} [SIAM J. Matrix Anal. Appl. 23, 632--645 (2001; Zbl 1002.65061)]. In the present paper necessary and sufficient conditions for the existence of a solution are derived. An iterative algorithm for obtaining the positive definite solutions is proposed, and an error bound is found.

##### MSC:
 15A24 Matrix equations and identities 65F30 Other matrix algorithms 65H10 Systems of nonlinear equations (numerical methods)
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##### References:
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