Based On The Two Steps Of Some Form Of Division Iteration Method Of Convergence Analysis

In many fields of scientific applications and engineering computation, we always encounter initial value and boundary value problems of the differential equations, but very few simple equations whose analytical solution can be expressed. However since those practical problems are complicated differential equations, such as elliptic, parabolic and hyperbolic equations, we should find out the solution of these equations or the values of the function at the discrete points, and we want to get the numerical solution of the differential equations. When we seek the numerical solution of the elliptic boundary value by using the differential method for the solutions, it comes down to solving linear sparse linear system finally. As we know, it has two kinds of methods which are direct and iterative methods to solve linear system, and the proportional of non-zero elements of the coefficient matrix generated by the large sparse linear system is small. These elements have great regularity in the distribution. The iterative method is a very important method which is not only easy to be carried out, but also saving the computer storage. As for a large sparse matrix, the convergence rate of the selected iterative methods to solve the linear system is extremely important, and only the convergent iterative methods work for practical applications. The two-stage multisplitting method is discussed ardently at present, in this paper we study the convergent condition of this method using various types.The first part is introduction which introduces the background of two-stage iterative method. The second part of this paper gives necessary theoretical knowledge prepared for studying. In the third part we discuss the convergence of two-stage iterative method using ILU decomposition as outer splitting and an inner splitting, we get some theories and give corresponding theoretic proof. And then we discuss the convergence of two-stage iterative method using H-compatible splitting as outer splitting and SSOR splitting as inner splitting, give the convergent conditions of this method, and give numerical example to prove the theoretic results. Further more, this paper also considers the convergence of two-stage splitting for the preconditioned linear system, Aα=(I+Sα)A is the preconditioned matrix in this system. Sα is a matrix that the elements in sub-diagonal line under main diagonal line are non-zero, and the rest elements are zeros, it is defined in the fifth part of this paper. We get some different theorems and corollaries when A is a H-matrix or M-matrix and give two numerical examples to prove corresponding theoretic results, then we can find some amazing results.This paper discussed the convergence of two-stage iterative method using three types, and obtained some theoretical results. It is helpful for the improvement and development of existing conclusions about two-stage multisplitting which is discussed frequently. There is certain referential and practical value for the scholars or researchers who are engaged in the numerical calculations.