Computing the Extremal Positive Definite Solutions of a Matrix Equation

Abstract

An efficient and numerically stable implementation of a known algorithm is suggested for finding the extremal positive definite solutions of the matrix equation $X + A * X^{ - 1} A = I$, if such solutions exist. The convergence rate is analyzed. A new algorithm that avoids matrix inversion is presented. Numerical examples are given to illustrate the effectiveness of the algorithms.