Inversion free Iterative Method for Finding Symmetric Solution of the Nonlinear Matrix Equation ? − ?∗??? = ? (? ≥ ?)

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