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 solu-tions of the nonlinear matrix equation X C A?X−1A D Q, where Q is positive definite, has been studied. In this paper, a two-sided iterative process for the same equation is investi-gated. 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, Xk X Yk 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