A further study on a nonlinear matrix equation

Abstract

The nonlinear matrix equation $$X^p=R+M^T(X^{-1}+B)^{-1}M$$ , where p is a positive integer, M is an arbitrary $$n\times n$$ real matrix, R and B are symmetric positive semidefinite matrices, is considered. When $$p=1$$ , this matrix equation is the well-known discrete-time algebraic Riccati equation (DARE), we study the convergence rate of an iterative method which was proposed in Meng and Kim (J Comput Appl Math 322:139–147, 2017). For the generalized case $$p\ge 1$$ , a structured condition number based on the classic definition of condition number is defined and its explicit expression is obtained. Finally, we give some numerical examples to show the sharpness of the structured condition number.