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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 $\delta \in \left(0,1\right]$. (For $\delta =1$ this equation arises in the analysis of ladder networks, in dynamic programming, control theory, stochastic filtering and statistics.) For $\delta \in \left(0,1\right]$ the equation has been first considered in S. M. El-Sayed and 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)