Perturbation analysis of the matrix equation X - \sum_{i=1}^m A_i^* X^{p_i} A_i = Q

Abstract

Consider the nonlinear matrix equation X-sum_{i=1}^{m}A_{i}^{*}X^{p_{i}}A_{i}=Q with p_{i}>0. Sufficient and necessary conditions for the existence of positive definite solutions to the equation with p_{i}>0 are derived. Two perturbation bounds for the unique solution to the equation with 0<p_{i}<1 are evaluated. The backward error of an approximate solution for the unique solution to the equation with 0<p_{i}<1 is given. Explicit expressions of the condition number for the equation with 0<p_{i}<1 are obtained. The theoretical results are illustrated by numerical examples.