ON POSITIVE DEFINITE SOLUTIONS OF THE MATRIX EQUATION $X+A∗X^{-q}A=Q(0 < q ≤ 1)$
Xiaoyan Yin, Sanyang Liu, Tiexiang Li · Taiwanese Journal of Mathematics · 2012
Consider the nonlinear matrix equation $X+A^{*}X^{-q}A=Q$ where $0<q\leq 1$. A new sufficient condition for this equation to have positive definite solution is provided and two iterative methods for the maximal positive definite solution are proposed. Applying the theory of condition number developed by Rice, an explicit expression of the condition number of the maximal positive definite solution is obtained. The theoretical results are illustrated by numerical examples.