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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.

15A24Matrix equations and identities
65F30Other matrix algorithms
65H10Systems of nonlinear equations (numerical methods)
Full Text: DOI
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