Newton’s Iteration Method for Solving the Nonlinear Matrix Equation X+∑i=1mAi*X−1Ai=Q

In this paper, we study the nonlinear matrix equation (NME) X+∑i=1mAi*X−1Ai=Q. We transform this equation into an equivalent zero-point equation, then we use Newton’s iteration method to solve the equivalent equation. Under some mild conditions, we obtain the domain of approximation solutions and prove that the sequence of approximation solutions generated by Newton’s iteration method converges to the unique solution of this equation. In addition, the error estimation of the approximation solution is given. Finally, the comparison of two well-known approaches with Newton’s iteration method by some numerical examples demonstrates the superiority of Newton’s iteration method in the convergence speed.