The Study Of Preconditioning Interative Methods For Large Sparse Linear Systems

The solution of large linear systems is central to many numerical simulations in science and engineering, such as the nuclear power industry, petroleum industry, the design and computer analysis of circuits, PDEs, and image processing, which is also often the most time-consuming part of a computation. Therefore, deriving the efficient iterative algorithms for solving large linear systems is an important project in scientific and engineering computation field.As well known, there are two kinds of methods for solving linear systems, i.e., direct methods and iterative methods. Direct methods, based on the factorization of the coefficient matrix into easily invertible matrices, are widely used when the coefficient matrix is small and density. However, when the coefficient matrix is large and sparse, iterative methods are preferred. Also, there is a common shortcoming for iterative methods, i.e., the convergence rate is slow. Hence, combining the preconditioning technique with iterative methods is very popular and used widely.In this dissertation, we studied the preconditioning iterative methods for solving large and sparse linear systems. Firstly, we presented a preconditioned matrix based discrete cosine transformation for symmetric Toeplitz linear systems. Secondly, a variable preconditioning SOR-BICR method has been proposed for solving linear systems. Then, when the coefficient matrix is non-Hermitian positive definite, a variable preconditioning HSS-generalized conjugate residual method is given. Finally, a fast inversion algorithm for lower triangular Toeplitz matrices has been derived, based on discrete sine transformation. This dissertation includes five chapters, which is organized as follows:Firstly, the research background and research status are given, as well as the preliminary knowledge. Furthermore, the main contents of this paper are briefed.In the second chapter, a preconditioned matrix based discrete cosine transformation has been studied for symmetric Toeplitz linear systems. Numerical experimental results show that the new preconditioned matrix is efficient.In the third chapter, a variable preconditioning SOR-BICR method has been proposed for solving large linear systems. Also, several numerical experiments have been given to verify the efficiency of the algorithm.In Chapter 4, when the coefficient matrix is non-Hermitian positive definite, a variable preconditioning HSS-generalized conjugate residual method has been proposed and the algorithm is tested on computer. The performance of the new algorithm is compared with the existing ones on several numerical examples. The faster convergence behavior is illustrated.Finally, the research work of this dissertation is summarized and the possible research lines are discussed.