A New Algorithm for Solving the Equation X + A*X-1A = I

In this paper, we investigate the existence of positive definite solutions for the matrix equation X+A*X^-1 A=I. A new inversion free variant of the basic fixed point iteration method for obtaining the maximal positive definite solution is established. Moreover, some necessary conditions and sufficient conditions for the existence of positive definite solutions of the matrix equation are obtained. In the end, one numerical example is given to illustrate the effectiveness of our results.