Solving two generalized nonlinear matrix equations

Abstract

In this paper, we consider the numerical solutions of two generalized nonlinear matrix equations. Newton’s method is applied to compute one of the generalized nonlinear matrix equations and a generalized Stein equation is obtained, then we adapt the generalized Smith method to find the maximal Hermitian positive definite solution. Furthermore, we consider the properties of the solution for the generalized nonlinear matrix equation. Newton’s method is also applied to the other generalized nonlinear matrix equation to find the minimal Hermitian positive definite solution. Finally, two numerical examples are presented to illustrate the effectiveness of the theoretical results and the convergence behaviour of the considered methods for two generalized nonlinear matrix equations, respectively.