| In this paper,we study the problems relative to the stochastic algebraic riccati equation.First,we introduce the history and background of the stochastic algebraic riccati equation.Also,we give some methods to solve the classic algebraic riccati equation which some other researchers have studied.In the second part,we introduce some basic knowledge which would be used when we solve the stochastic algebraic riccati equation.Then we present the process of Newton's method for the stochastic algebraic riccati equation in detail and study the convergence of the Newton's method.The third part is the most important part of this paper.We find that some mixed-type Lyapunov equations need to be solved when we solve the stochastic algebraic riccati equation using the Newton's method.At present,usually we use iteration method to solve the mixed-type Lyapunov equation and will get the inexact solution.So,this would impact the solution of the stochastic algebraic riccati equation and the Newton's method becomes the inexact Newton's method.Then in the third part,we study the convergence of the inexact Newton's method for the stochastic algebraic riccati equation.We find that if the error of the solution of mixed-type Lyapunov equation satisfies some conditions,then the inexact Newton's method for the stochastic algebraic riccati equation still converge.Also,given more qualification,the convergence is quadratic.In the following,we introduce several iteration methods for the mixed-type Lyapunov equation.In the fourth part,we make up a new iteration to solve the stochastic algebraic riccati equation,and we prove this new method converges to the solution of this equation.The advantage of this new method is that we need to solve a regular Lyapunov equation which is easier that the mixed-type Lyapunov equation.However,the rate of the new method's convergence is smaller than the other methods.In the end of this paper,we give some numerical examples and develop the arithmetic compare of these three methods.
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