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arXiv · 1208.3518

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

Jing Li
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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.

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Jing Li. 2012-08-17. Perturbation analysis of the matrix equation X - \sum_{i=1}^m A_i^* X^{p_i} A_i = Q. https://arxiv.org/abs/1208.3518

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