A new inversion free iteration for solving the equation X+A☠X-1A=Q

In this paper, we introduce a new inversion free variant of the basic fixed point iteration method for obtaining a maximal positive definite solution of the nonlinear matrix equation X+A★X-1A=Q. It is more accurate than Zhan's algorithm (J. Sci. Comput. 17 (1996) 1167) and has less number of operations than the algorithm of Guo and Lancaster (Math. Comput. 68 (1999) 1589). We derive convergence conditions of the iteration and existence conditions of a solution to the problem. Finally, we give some numerical results to illustrate the behavior of the considered algorithm.