On the matrix equation X+ATX-12mA=I

In this paper we study iterative methods for finding the extremal positive definite solutions of the matrix equation X+ATX-1MA=I. First, a condition on the existence of a positive definite solution of this matrix equation is given. Then, the existence as well as the rate of convergence of some proposed algorithms to obtain the extremal positive definite solutions of this equation is presented. Moreover, a generalization of computationally simple and efficient known algorithm is applied for obtaining the extremal positive definite solutions. In addition, both the necessary and sufficient conditions for this matrix equation to have positive definite solution are presented. Numerical examples are given to illustrate the performance and the effectiveness of the algorithms.