A new inversion-free method for a rational matrix equation

Motivated by the classical Newton-Schulz method for finding the inverse of a nonsingular matrix, we develop a new inversion-free method for obtaining the minimal Hermitian positive definite solution of the matrix rational equation X+A^∗X^-1A=I, where I is the identity matrix and A is a given nonsingular matrix. We present convergence results and discuss stability properties when the method starts from the available matrix AA^∗. We also present numerical results to compare our proposal with some previously developed inversion-free techniques for solving the same rational matrix equation.