On the nonlinear matrix equation $X+\sum_{i=1}^{m} A_{i}^{*}X^{-q}A_{i}=Q (0 < q \leq 1 )$
Bull. Korean Math. Soc. 2014 Vol. 51, No. 3, 739-763
Published online May 1, 2014
Xiaoyan Yin, Ruiping Wen, and Liang Fang
Xidian University, Taiyuan Normal University, Xidian University
Abstract : In this paper, the nonlinear matrix equation $$X+\sum_{i=1}^{m} A_{i}^{*} X^{-q} A_{i} = Q ~ (0 < q \leq 1)$$ is investigated. Some necessary conditions and sufficient conditions for the existence of positive definite solutions for the matrix equation are derived. Two iterative methods for the maximal positive definite solution are proposed. A perturbation estimate and an explicit expression for the condition number of the maximal positive definite solution are obtained. The theoretical results are illustrated by numerical examples.
Keywords : nonlinear matrix equation, positive definite solution, perturbation estimate, condition number
MSC numbers : 15A24, 15A45, 65H05
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