Pre-conditions For Solution Of Linear Systems

The thesis mainly deals with how to accelerate the rate of convergence for the large-scale linear equations. As is well known, many problems are reduced to one or more of large (usually sparse) linear equations as a matter of fact. The iterative method is the usual way to solve the equations ,its convergence and its rate of convergence are great significa- nce , an iterative method doesn't work, while that has a low rate of convergence is no practical value. An iterative format that has a better rate of convergence must be found, the parameters in the format be determined, and the autologous improvements on the iterative format, and one of these ways is to seek better iterative method which has a fast rate of convergence. Concretely, the preconditioner P_α= I+C_αis applied to SOR iterative method( MSOR iterative method).This text includes five chapters. Chapter one is the introduction of the paper, mainly to discuss some common iterative methods and to introduce preconditioned theory and cited the preconditioned matrix P_α= I+C_αto propose by Hadjidmos. In chapter two , the writer provides the basic knowledge and the basic lemma required in this paper. The third chapter is an outline of the conclusions about preconditioner for the development of the theory in recent years . From the beginning of chapter four is the essence of the thesis and here we give details for chapter four and five .In the fourth chapter, the author discusses the convergence of the modified SOR ite- rative method which the preconditioner P_αis applied to SOR iterative method, when the matrix A is strictly diagonally dominant L -matrix or H -matrix. And a brief description of the parameters _αfor the best selection from the preconditioner P_α= I+C_α. At last, numerical examples are also given, which show the effectiveness of our method.Chapter five, we discuss the preconditioner P_α= I+C_αwhich accelerates the rate of convergence for iterative method. First, the author discusses the convergence of the modified SOR iterative method to accelerate SOR iterative method, when the matrix A is the nonsingular M - matrix, and gives the best parametersωof the MSOR iterative method and then we discuss that the rate of convergence of MSOR iterative method is faster than the AOR iterative method where reflects the advantage of the practical value of P_αand then we change a way to structure a series of preconditioner P_αi that proves the same method to enhance the convergence rate of SOR iterative method, but the convergence rate of the method which the preconditioners P_αi are applied to SOR iterative method ( ISOR iterative method) is lower thanMSOR and then numerical examples are also given, which show the effectiveness of our method.Finally, the author describes the material train of thought and research foreground of his article.