The positive definite solution of the nonlinear matrix equation Xp=A+M(B+X−1)−1M∗

We consider the nonlinear matrix equation Xp=A+M(B+X−1)−1M∗, where p≥1 is a positive integer, M is an arbitrary n×n matrix, and A and B are Hermitian positive semidefinite matrices. An elegant estimate of the Hermitian positive definite (HPD) solution is derived. A fixed-point iteration and an inversion-free variant iteration for obtaining the HPD solution are proposed. Some numerical examples are presented to show the efficiency of the proposed two iterative methods.