Some Researches On Numerical Methods Of The Nonsymmetric Algebraic Riccati Equations

Solving the minimal non-negative solution of the nonsymmetric algebraic Ric-cati equations is often needed in scientific computing and engineering applications. When the scale of the matrix gets bigger, the numerical iteration methods are more effective than the derect methods. So far, many experts and scholars have put for-ward many numerical methods with the good properties about the equations. But we still can get some better numerical methods through some creative ideas and skills. In this thesis, three new numerical methods are proposed to solve the min-imal non-negative solution of the nonsymmetric algebraic Riccati equations. The main contents are as follows:In the Introduction, the development and current situation of the nonsymmetric algebraic Riccati equations are summarized. Then some existing numerical methods and fundamental lemma for solving the nonsymmetric algebraic Riccati equations are introduced briefly.In Chapter 1, the linear implicit iterative algorithm (LI) is proposed to solve the minimal non-negative solution of the nonsymmetric algebraic Riccati equations. Then the modified linear implicit iterative method (MLI) is proposed by using the Sa Manski technic. Under the proper conditions, we prove the monotone convergence of the LI and MLI iterative methods. Numerical experiments show that the LI and MLI iteration methods are feasible and effective.In Chapter 2, a new alternately linearied implicit iteration method is proposed to solve the minimal non-negative solution of the nonsymmetric algebraic Riccati equations based on the alternately linearied implicit iteration method. The conver-gence speed of the method is further improved. Under the proper conditions, the monotone convergence of the method is proved. The optimal parameters and the the asymptotic convergence factor are estimated by using the theory of generalized Carlisle transform. Finally, the theoretical knowledge and numerical experiments show that the new alternately linearied implicit iteration method is effective.In Chapter 3, we know that the Newton method is an effective method for solving the minimal non-negative solution of the nonsymmetric algebraic Riccati equations. The generalized parameter iteration method is proposed to solve the Sylvester equation in each Newton step, then we get the generalized inexact Newton iteration method by controlling the solution in each Newton iteration steps. Then we prove the convergence of the method and get the optimal parameters in the method through the theory of generalized Carlisle transform. Numerical experiments show that the proposed method is feasible and effective.In the last chapter, we make a conclusion of the work in this thesis and point out how to further our research work on the nonsymmetric algebraic Riccati equations.