Iterative positive definite solutions of the two nonlinear matrix equations X ± AT X−2A = I

In this paper iterative methods for obtaining positive definite solutions of the two matrix equations X ± AT X−2A = I are proposed. We show that under some conditions on the real square matrix A the constructed iterative methods converge to positive definite solutions for the two equations. Numerical examples are given to test and illustrate the performance of the algorithms in terms of convergence, accuracy as well as the efficiency.