Psd Methods For Solving Saddle Point Problems, And A Special Class Of Nonlinear Equations Of The N-psd Method Convergence

There are two methods to solve the linear equations Ax=b, the direct solving method and the iterative method. The direct solving method is convenient for the linear system with not very high order. The completely accurate answer will be produced through several limited operation if there is no rounding errors. The solutions of many problems are summed up to the solutions of one or some large sparse linear systems. The large linear equations are usually solved by iterative method.In this paper, they are discussed that preconditioned simultaneous displace-ment(PSD) method for the problem of large scale sparse saddle point and the con-vergence theories of N-PSD iteration methods.In chapter 2, the large and sparse saddle-point problems are solved by the PSD iteration method proposed by Evans in 1980. The large and sparse saddle-point problems are appeared in many fields, such as the constraint optimization problems, the fluid mechanics problems, the least squares problems. Therefore, constructing efficient iteration methods and establishing their convergence theories for these specially structured linear systems of the large and sparse saddle-point problems are theoretically important and practically valuable. The PSD iteration method is proposed to solve large linear equations. Later. the PSD method is used to solve the least squares problems by some scholars, the PSD method is proposed to solve the large and sparse saddle-point problems inspired by them. This new method is based on the splitting form of the coefficient matrix, the introduction of preconditioning matrix Q, and the application of the PSD iterative patten. Firstly, the function equation among the eigenvalues of the interaction matrix of the PSD method and the matrix Q-1BT A-1 B is established. Secondly, the necessary and sufficient condition for the convergence of the PSD method is derived by giving the restrictions imposed on the parameters,such as r=1orω=1. Thirdly. the optimum parameter and the most superior spectrum radius are obtained under certain conditions. Lastly, some numerical examples are given for illustrating, esting and verfing the theorem results.In chapter 3, they are discussed that the convergence theories of N-PSD it-eration methods. It is a useful method for the nonlinear systems. The two step iteration method is proposed by combining PSD method with Newton method. The large range convergence theorems are obtained under certain conditions. Lastly, some numerical examples are given for illustrating, easting and verifying the theo-rem results.