On the existence of extremal positive definite solutions of the nonlinear matrix equation (Q604124)

The authors consider the matrix equation \(X^r+\sum _{j=1}^mA_j^*X^{\delta _j}A_j=I\), \(-1<\delta _j<0\), where \(A_j\) are nonsingular \(n\times n\) matrices, \(I\) is the identity matrix, \(r\) and \(m\) are positive integers, \(A_j^*\) is the transpose conjugate of \(A_j\). The authors derive a necessary condition for the existence of a positive definite solution and, based on the Banach fixed point theorem, a sufficient condition for the existence of a unique such solution. Iterative methods for obtaining the extremal (maximal-minimal) positive definite solutions of this equation are proposed. The rate of convergence of some proposed algorithms is proved and numerical examples are given to illustrate their performance and effectiveness.