Iterative methods for nonlinear matrix equations X + A⋆X−αA = I

The present paper treats iterative methods for a class of nonlinear matrix equations X + A⋆X−αA = I, with two cases: α ⩾ 1 and 0 < α < 1. Here, the matrix A is nonsingular. We will derive the necessary and sufficient conditions for the existence of positive definite solutions.