On positive definite solutions of the matrix equation \(X + A^* X^{-q} A = Q(0 < q \leq 1)\) (Q1936069)

This paper gives a new sufficient condition for the nonlinear matrix equation \(X+A^{\ast }X^{-q}A=Q\) where \(0<q\leq 1\) to have positive definite solution. The author presents two iterative methods to find the maximal positive definite solution of this equation. Applying the theory of condition number developed by \textit{J. R. Rice} [SIAM J. Numer. Anal. 3, 287--310 (1966; Zbl 0143.37101)], the author gives an explicit expression of the condition number of the maximal positive definite solution. The theoretical results are also illustrated by numerical examples.