Solutions and perturbation estimates for the matrix equation Xs+A∗X-tA=Q

In this paper, the nonlinear matrix equation Xs+A∗X-tA=Q is investigated. Necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions are derived. An effective iterative method to obtain the special solution XL (We proved that if there is a maximal Hermitian positive definite solution, then it must be XL) is established. Moreover, some new perturbation estimates for XL are obtained. Several numerical examples are given to illustrate the effectiveness of the algorithm and the perturbation estimates.