A two-sided iterative method for computing positive definite solutions of a nonlinear matrix equation

In several recent papers, a one-sided iterative process for computing positive definite solutions of the nonlinear matrix equation X + A * X-1 A = Q, where Q is positive definite, has been studied. In this paper, a two-sided iterative process for the same equation is investigated. The novel idea here is that the two sequences obtained by starting at two different values provide (a) an interval in which the solution is located, that is, X-k less than or equal to X less than or equal to Y-k for all k and (b) a better stopping criterion. Some properties of solutions are discussed. Sufficient solvability conditions on a matrix A are derived. Moreover, when the matrix A is normal and satisfies an additional condition, the matrix equation has smallest and largest positive definite solutions. Finally, some numerical examples are given to illustrate the effectiveness of the algorithm.