The nonlinear matrix equation (NMEQ) \(X+A^{T}X^{-1}A=Q,\) arises in numerous applied fields, such as optimal control, filtering theory, mechanical systems, and network analysis. This equation plays a crucial role in solving discrete-time algebraic Riccati equations, stability analysis, and other problems in control and dynamical systems. In this paper, we first establish theoretical conditions for the existence of solutions to this equation. Next, we introduce a class of inversion-free iterative methods based on fixed-point iteration to compute the solution efficiently. We prove the convergence of the proposed methods to the exact solution under appropriate conditions. Numerical experiments illustrate the effectiveness and computational accuracy of the methods in different dimension. Finally, we demonstrate an application of the proposed approach in optimal control, highlighting its practical utility.
