Iteration Method for Solving Nonlinear Matrix Equation X -- A* 2m (square root X-1) A = I

Matrix equation problem is one of the topics of active research in the context of computational mathematics. The Hermitian positive definite solutions of a matrix equation play an important role in real applications. In this paper, we present the sufficient conditions for the existence of the positive definite solution to the nonlinear matrix equation X - A* (X^-1)^1/2 A = I and propose a natural and stable iteration algorithm for obtaining a positive definite solution of this matrix equation. Finally, two numerical examples for the convergence behavior of the proposed algorithm are conducted to demonstrate the effectiveness.