Latest Inversion-Free Iterative Scheme for Solving a Pair of Nonlinear Matrix Equations

Abstract

In this work, the following system of nonlinear matrix equations is considered, X1+A∗X1−1A+B∗X2−1B=I and X2+C∗X2−1C+D∗X1−1D=I, where A,B,C, and D are arbitrary n×n matrices and I is the identity matrix of order n. Some conditions for the existence of a positive-definite solution as well as the convergence analysis of the newly developed algorithm for finding the maximal positive-definite solution and its convergence rate are discussed. Four examples are also provided herein to support our results.