Solutions and perturbation analysis of the matrix equation X - \sum_{i=1}^m A_i^* X^{-1} A_i = Q
For researchers in numerical linear algebra, this provides theoretical guarantees and sensitivity analysis for a class of nonlinear matrix equations, though it is an incremental extension of existing work.
This paper proves the existence of a unique positive definite solution to the matrix equation X - ∑ A_i^* X^{-1} A_i = Q without restrictions on A_i, and provides perturbation bounds, backward error, and condition number for the solution.
Consider the nonlinear matrix equation X-sum_{i=1}^{m}A_{i}^{*}X^{-1}A_{i}=Q. This paper shows that there exists a unique positive definite solution to the equation without any restriction on A_{i}. Three perturbation bounds for the unique solution to the equation are evaluated. A backward error of an approximate solution for the unique solution to the equation is derived. Explicit expressions of the condition number for the unique solution to the equation are obtained. The theoretical results are illustrated by numerical examples.