Iterative methods for the extremal positive definite solution of the matrix equation

In this paper, the inversion free variant of the basic fixed point iteration methods for obtaining the maximal positive definite solution of the nonlinear matrix equation X + A * X - α A = Q with the case 0 < α ⩽ 1 and the minimal positive definite solution of the same matrix equation with the case α ⩾ 1 are proposed. Some necessary conditions and sufficient conditions for the existence of positive definite solutions for the matrix equation are derived. Numerical examples to illustrate the behavior of the considered algorithms are also given.