Hermite positive definite solution of the matrix equation X+A^*（R+B^*XB）^-tA=Q

The nonlinear matrix equation X+ A ^*( R+ B ^* XB) ^-t A= Q( t≥ 1) is derived from the discrete-time algebraic Riccati equation. The sufficient conditions for the existence of Hermite positive definite solutions and the upper and lower bounds are given. The fixed point iteration and inverse-free iteration algorithms for solving the equation are constructed and the convergence of the algorithm is proved by using the monotone boundedness theorem. Finally, two numerical examples are given to illustrate the effectiveness and feasibility of the proposed algorithm for solving the matrix equation.