On the nonlinear matrix equation Xp = A + MT(X#B)M

The nonlinear matrix equation X-p = A + M-T (X#B)M, where p >= 1 is a positive integer, M is an n x n nonsingular matrix, A is a positive semidefinite matrix and B is a positive definite matrix, is considered. We denote by C#D the geometric mean of positive definite matrices C and D. Based on the properties of the Thompson metric, we prove that this nonlinear matrix equation always has a unique positive definite solution and that the fixed-point iteration method can be efficiently employed to compute it. In addition, estimates of the positive definite solution and perturbation analysis are investigated. Numerical experiments are given to confirm the theoretical analysis. (C) 2019 Elsevier B.V. All rights reserved.