Solutions and improved perturbation analysis for the matrix equation X-A^{*}X^{-p}A=Q (p>0)
Provides theoretical advances for solving a specific nonlinear matrix equation, which is incremental for researchers in matrix theory and numerical linear algebra.
The paper derives new sufficient conditions for the existence of a unique positive definite solution to the matrix equation X - A*X^{-p}A = Q for p>0, and provides improved perturbation bounds and condition numbers for both cases p>1 and 0<p<1. Numerical examples validate the results.
In this paper the nonlinear matrix equation X-A^{*}X^{-p}A=Q with p>0 is investigated. We consider two cases of this equation: the case p>1 and the case 0<p<1. In the case p>1, a new sufficient condition for the existence of a unique positive definite solution for the matrix equation is obtained. A perturbation estimate for the positive definite solution is derived. Explicit expressions of the condition number for the positive definite solution are given. In the case 0<p<1, a new sharper perturbation bound for the unique positive definite solution is evaluated. A new backward error of an approximate solution to the unique positive definite solution is obtained. The theoretical results are illustrated by numerical examples.