Inversion–free Iterative Method for Finding Symmetric Solution of the Nonlinear Matrix Equation X-A^* X^q A=I (q≥2)

Abstract

In this paper, we propose the inversion free iterative method to find symmetric solution of the nonlinear matrix equation  where  is an unknown symmetric solution,  is a given Hermitian matrix and  is a positive integer. The convergence of the proposed method is derived. Numerical examples demonstrate that the proposed iterative method is quite efficient and converges well when the initial guess is sufficiently close to the approximate solution.

Keywords: Symmetric solution, nonlinear matrix equation, inversion free, iterative method.