The maximal positive definite solution of the nonlinear matrix equation X + A*X-1 A + B*X-1 B = I

In the current study, we present a new inversion free variant of the fixed point iteration method to find a maximal positive definite solution for the matrix equation X + A*X-1 A + B*X-1 B = I. The existence conditions of the nonlinear matrix equation are derived. Some numerical examples are presented to show the behavior and efficiency of the considered iterative method. The comparison of the results shows that the new algorithm is more accurate and has fewer operations than the other algorithms. Some conclusions and open problems on possible future research direction close the paper.