Preconditioned Successive Over-relaxation(SOR) Iterative Methods And Comparison Theorems

To solve large linear system, iterative methods have become one of the most important methods instead of direct method. The rule whether the iterative is good is usually described by convergence rate. Therefore, convergence rate of iterative method has become a very important problem. Thus, we should find an iterative method which has fast convergence rate. This owns practical value. In order to solve linear system faster and more better, we quote nonsingular preconditioned matrices. By preconditioned matrices, we accelerate the convergence rate of iterative method. Based on [4] to [11],this paper proposes the preconditioned SOR iterative methods which have the generalization in practice.The preconditioned matrix Pr concludes some preconditioned matrices which have known. The condition of the given preconditioned comparison theorems in this paper is weaker than before. The coefficient matrix A of the linear system entends nonsingular M— matrix from irreducibly diagonal dominant Z—matrix.The followings are the construction and main contents of this paper:In Chapter 1, firstly, we review all the proposed preconditioned matrices. Secondly, we illustrate the relations of the preconditioned matrices in Milaszewicz[12], Gunawar-dena et al.[4], Kohno et al.[7], Evans et al.[3], Hadjidimos et al.[5] and Niki et al.|[14]. Finally, we review different iterative methods and some preconditioned comparison theorems which have been disscussed under different peconditioned matrices of Gunawardena et al.[4], etc.In Chapter 2, preliminaries. This part mainly makes preparations for Chapter 3. Firstly, we point out the definitions and theorems which will be used in Chapter 3, such as M—matrix, regular splitting'definitions, etc. Secondly, we point out general methods which obtain comparison theorems. We divide into two cases: Two splittings of the same matrix and two splittings of the different matrices. Therefore we use different methods to obtain comparison theorems according to two cases. In some case, we may change two splittings of same matrix to compare. For example, the proof of comparison theorems in L.-y.Sun[15] changes the two splittings of same matrix.In Chapter 3, preconditioned SOR iterative methods and comparison theorems. Thispart is the main part of this paper. Based on preconditioner Pr in Niki et al.[14] and the preconditioned Gauss-Seidel iterative method proposed by Gunawardena et al. [4] .Where the preconditioner Pr concludes the preconditioner Ps in Gunawardena et al.[4] and the preconditioner Pi in Evans et al.[3]. In this paper, the author proposes the preconditioned SOR iterative method under the preconditioner Pr. Firstly, when the real parameter w = 1, the preconditioned SOR iterative method of this paper reduces to the preconditioned Gauss-Seidel iterative method in Niki et al.[14]. This generalizes the iterative method by Niki et al.[14]. Secondly, when the coefficient matrix A of linear system is a nonsingular M—matrix, we obtain some preconditioned comparison theorems, which generalizes the comparison theorems in Niki et al.[14] under the assumption that the coefficient matrix A is an irreducibly diagonal dominant Z—matrix.In Chapter 4, numerical examples. This part mainly verifies the results in Chapter 3. The coefficient matrix A of Example 4.1 and Example 4.2 are irreducibly diagonal dominant .Z—matrices. However, Example 4.3 and Example 4.4 are reducibly nondiag-onal dominant Z—matrices, they are nonsingular M—matrices.This doesn't satisfy the condition of comparison theroems in Niki et al.[14]. But it satifies the theorem assumption of this paper, so this shows the preconditioned comparison theorems of this paper have the generalization in practice.