An improved inversion-free method for solving the matrix equation X+A∗X−αA=QX+A∗X−αA=Q

In this paper, based on the damped Newton method, we present an inversion-free iterative method with a stepsize parameter for solving the nonlinear matrix equation X+A∗X−αA=QX+A∗X−αA=Q, where QQ is a Hermitian positive definite matrix and α≥1α≥1. The proposed method is an extension of Algorithm 2.2 in [Z. Peng, S.M. El-Sayed, X. Zhang, Iterative methods for the extremal positive definite solution of the matrix equation X+A∗X−αA=QX+A∗X−αA=Q, J. Comput. Appl. Math. 200 (2007) 520–527]. Preliminary numerical results show that the proposed method can improve numerical effect.