Positive definite solutions of the nonlinear matrix equation X + A∗XqA = Q (q > 0)

In this paper, the nonlinear matrix equation X + A∗XqA = Q (q > 0) is investigated. Some necessary and sufficient conditions for existence of Hermitian positive definite solutions of the nonlinear matrix equations are derived. An effective iterative method to obtain the positive definite solution is presented. Some numerical results are given to illustrate the effectiveness of the iterative methods.