In this paper, we propose the inversion free iterative method to find symmetric solution of the nonlinear matrix equation X - A*XqA = I (q ≥ 2), where X is an unknown symmetric solution, A is a given Hermitian matrix and q 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. [ABSTRACT FROM AUTHOR]
In this manuscript, for approximation of solutions to equations that are nonlinear, a new class of two-point iterative structure that is based on a weight function involving two converging power series, is developed. For any method constructed from the developed class of methods, it requires three separate functions evaluation in a complete iteration cycle that is of order four convergence. Also, some well-known existing methods are typical members of the new class of methods. The numerical test on some concrete methods derived from the class of methods indicates that they are effective and competitive when employed in solving a nonlinear equation. [ABSTRACT FROM AUTHOR]