On the Iterative Method for the System of Nonlinear Matrix Equations

Abstract

The positive definite solutions for the system of nonlinear matrix equations $X+A^{\ast }{Y}^{-n}A=I,Y+B^{\ast }{X}^{-m}B=I$ are considered, where $n,\hspace{0.17em}m$ are two positive integers and A, B are nonsingular complex matrices. Some sufficient conditions for the existence of positive definite solutions for the system are derived. Under some conditions, an iterative algorithm for computing the positive definite solutions for the system is proposed. Also, the estimation of the error is obtained. Finally, some numerical examples are given to show the efficiency of the proposed iterative algorithm.