Quasi gradient-based inversion-free iterative algorithm for solving a class of the nonlinear matrix equations

Inspired by the gradient-based and inversion-free iterations, a new quasi gradient-based inversion-free iterative algorithm is proposed for solving the nonlinear matrix equation $X+A^{\mathrm{T}}{X}^{?n}A=I$. The convergence proof of the suggested algorithm is given. Several matrix norm inequalities are established to depict the convergence properties of this algorithm. Three numerical examples are given to illustrate the effectiveness of the suggested algorithms.