A New Inversion-Free Iterative Method for Solving a Class of Nonlinear Matrix Equations

In this paper, we propose a new inversion-free iterative method for computation of positive definite solution of the nonlinear matrix equation where $p\geq 1$ is a positive integer, $A$ and $B$ are Hermitian positive semidefinite matrices and $M$ is an arbitrary square complex matrix. This matrix equation has been studied recently in (J. Compt Appl Math 322, 139-147, 2017), where the authors proposed an inversion-free algorithm for solving this equation with the hypothesis that the matrix $B$ is nonsingular. For our part, we propose a new algorithm that is applicable for all choices of the positive semidefinite matrix $B$ even if it is singular. In order to prove the convergence of the proposed algorithm, we prove a new matrix inequality. The efficiency of the proposed algorithm is confirmed by some numerical simulations.