Elegant Iterative Methods for Solving a Nonlinear Matrix Equation X-A^* e^X A=I

Abstract

The nonlinear matrix equation   was solved by Gao (2016) via standard fixed point method. In this paper, three more elegant iterative methods are proposed to find the approximate solution of the nonlinear matrix equation  namely: Newton’s method; modified fixed point method and a combination of Newton’s method and fixed point method. The convergence of Newton’s method and modified fixed point method are derived. Comparative numerical experimental results indicate that the new developed algorithms have both less computational time and good convergence properties when compared to their respective standard algorithms.

Keywords: Hermitian positive definite solution, nonlinear matrix equation, modified fixed point method, iterative method.