A note on the fixed-point iteration for the matrix equations $X\pm A^*X^{-1}A=I$

Abstract

The fixed-point iteration is a simple method for finding the maximal Hermitian positive definite solutions of the matrix equations X±A∗X−1A=I (the plus/minus equations). The convergence of this method may be very slow if the initial matrix is not chosen carefully. A strategy for choosing better initial matrices has been recently proposed by Ivanov, Hasanov and Uhlig. They proved that this strategy can improve the convergence in general and observed from numerical experiments that dramatic improvement happens for the plus equation with some matrices A. It turns out that the matrices A are normal for those examples. In this note we prove a result that explains the dramatic improvement in convergence for normal (and thus nearly normal) matrices for the plus equation. A similar result is also proved for the minus equation.