Iterative Algorithm for Solving a System of Nonlinear Matrix Equations

Abstract

We discuss the positive definite solutions for the system of nonlinear matrix equations $X-A^{\ast }{Y}^{-n}A=I$ and $Y-B^{\ast }{X}^{-m}B=I$, where $n$, $m$ are two positive integers. Some properties of solutions are studied, and the necessary and sufficient conditions for the existence of positive definite solutions are given. An iterative algorithm for obtaining positive definite solutions of the system is proposed. Moreover, the error estimations are found. Finally, some numerical examples are given to show the efficiency of the proposed iterative algorithm.