Mixed Type Division And Gaussian Type Preconditioned Iteration

With the development of science and technology, when we numerical simulate the Practical problems of the natural and social sciences, partial differential equations are often make use of as mathematical models, To get the numerical solution of differential functions, we usually use difference method to get the linear system, then solve it. In according to the needs of practical problems, this linear system is usually a large sparse system. Therefore, this article mainly discusses how to increase the convergence speed of large sparse linear systems. For such systems, it’s difficult to use the direct method to get exact solutions. Therefore we usually use the iterative method to solve, and then it is worth to discuss the convergence and the convergence speed of the iterative method. Non-convergence or slow convergence of the iterative method is of no practical value.In recent years, the preconditioned method is widely studied, it can speed up the convergence rate of the iterative process greatly. This article main discusses this method, how to choose the good preconditioner makes the pre-condition method to speed up the convergence rate. Because of the convergence speed is related with the spectrum radius of the iteration matrix, therefore this article is mainly comparing the spectrum radius.This work is organized as follows six chapters. The first chapter is the introduction part, which first export the linear system from the differential equation, then introduce iterative method of linear systems, and give the forms of several common iterative methods, at last introduce the precondition method; The second chapter is preliminary, it mainly lists the definition and the lemma which will be used in this article; The third chapter is relevant knowledge, it mainly introduce several preconditioners, and describe briefly the development of precondition method. From Chapter four, it is main body. Below we give the detailed explanation.The fourth chapter is divided into two parts, The first part talk about when A is the L-matrix the auxiliary matrix D and L of the mixed-type split iteration method with the different selection influence on the convergent speed, and the theoretical proof are given. The second section discusses when A is the non-singular irreducible M-the matrix left Gauss preconditioners iterative method, we extend the left Gauss preconditioners by giving family of preconditioners based on the elimination of elements at the strictly upper triangular part, and the theoretical analysis is presented which theoretically conform the results in [5], and improved Gauss preconditioning algorithms associated with inbuilt technique are given.The fifth chapter basics of the fourth chapter given the Gaussian type preconditions iteration and mixed-split type iteration talk about the iterative convergence of comparison with the original iteration method, then we give the numerical example to confirmation theorem the accuracy and give the research prospect briefly. The sixth part is Summary and Prospect. This part mainly introduces the methods and main conclusions of this paper to do one sum up, then on the pre-conditions for methods to do the prospects for the future.