A further study on a nonlinear matrix equation (Q2227322)

The authors consider the nonlinear matrix equation \(X^p = R + M^T(X^{-1}+B)^{-1}M,\) where \(p\) is a positive integer, \(M\) is an \(n\times n\) real matrix and \(R\) and \(B\) are symmetric positive semidefinite matrices. For \(p=1\), this is a simplified symmetric form of the discrete-time algebraic Riccati equation and may be solved by an iterative method given by [\textit{J. Meng} and \textit{H.-M. Kim}, J. Comput. Appl. Math. 322, 139--147 (2017; Zbl 1364.15010)]. The authors study the convergence rate of this method. For \(p\geq 2\), the authors define a structured condition number for the equation and provide numerical examples to show the sharpness of the structured condition number; this extends existing results in the literature.