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Some iterative algorithms for positive definite solution to nonlinear matrix equations
Keywords:Nonlinear matrix equation; Structure-preserving algorithm; Cyclic reduction algorithm; Positive definite solution; Convergence theory.
      This paper is concerned with the unique positive definite solutions of the system of nonlinear matrix equations $X-A^*\bar{Y}^{-1}A=I_n$ and $Y-B^*\bar{X}^{-1}B=I_n$, where $A,B\in\mathbf{C}^{n\times n}$ are given matrices. Based on the special structure of the system of nonlinear matrix equations, the system can be equivalently reformulated as $V-C^*\bar{V}^{-1}C=I_{2n}$. Moreover, by means of Sherman-Moorison-Woodbury formula, we derive the relationships between the solutions of $V-C^*\bar{V}^{-1}C =I_{2n}$ and the well studied standard nonlinear matrix equation $Z+D^*Z^{-1}D=Q$, where $D$, $Q$ are uniquely determined by $C$. Then we present a structure-preserving doubling algorithm and two modified structure-preserving doubling algorithms to compute the positive definite solution of the system. Furthermore, cyclic reduction algorithm and two modified cyclic reduction algorithms for the positive definite solution of the system are proposed. Finally, some numerical examples are presented to illustrate the effectiveness of the theoretical results and the behavior of the considered algorithms.