Iterative Computation For Special Solutions Of Several Matrix Equations

Nonlinear matrix equation is one of the most important topics in numerical algebra.It is widely used in the optimal control theory,ladder network,dynamic programming,stochastic filtering and so on.The iterative method is usually used to solve the nonlinear matrix equation.However,the convergence rate of the solution is slow and the computation is large when the iterative method is used to solve the nonlinear matrix equation.In recent years,we use the fixed point iterative method and the inverse-free iterative method to solve the nonlinear matrix equation,where the inverse-free iterative method greatly simplifies the computational complexity.Firstly,based on the properties of the Kronecker product,the necessary and sufficient conditions for the existence of Hermitian positive definite solutions of nonlinear matrix equation X + A*?Im???X-C?^-t A = Q?t>0?are obtained.And then the fixed point iterative method and inverse-free iterative method for solving the equation are constructed by the principle of the bounded sequence.Finally,the efficiency of the two iterative methods are verified by some numerical examples.In addition,we also consider the nonlinear matrix equation Xs +A*X^-t₁xA+B*X^-t₂B=I?s,t₁,t₂>0?.Firstly,we obtain some new conditions for the existence of Hermitian positive definite solutions and the sufficient conditions for the existence of the unique Hermitian positive definite solutions.By discussing the range of s,t₁ and t₂,the existence interval of the solution for the equation is given.Secondly,the fixed point iterative method for solving the equation is constructed.Finally,the efficiency of the iterative method is verified by numerical examples.Furthermore,we study the nonlinear matrix equations???,Firstof all,we obtain the condition for the existence of positive definite solution for the equations.Secondly,we propose a fixed point iterative method for solving the equations.Finally,a numerical example is given to demonstrate the efficiency of the iterative method.Finally,we further study the nonlinear matrix equations???.Wepropose the steepest descent method and the Newton method for solving the least square solution of the nonlinear matrix equations.Finally,we verify the efficiency of the Newton method by a numerical example.