Perturbation analysis for the positive definite solution of the nonlinear matrix equation $X-\sum_{i=1}^{m} A_{i}^{*}X^{-1}A_{i}=Q$

Abstract

Consider the perturbation analysis for positive definite solution of the nonlinear matrix equation \(X-\sum_{i=1}^{m}A_{i}^{*}X^{-1}A_{i}=Q\) which arises in an optimal interpolation problem. Two perturbation bounds for the unique positive definite solution are obtained, and an explicit expression of the condition number for the unique positive definite solution is derived. The theoretical results are illustrated by several numerical examples.