The Stationary Iterative Methods For Toeplitz Systems

Toeplitz matrices and Toeplitz linear system of equations have an extensive appli-cations in mathematics and engineering,including information and image processing,queuing theory and cybernetics,differential equations and numerical solution of in-tegral equations.So it is of great scientific value to systemically study the effective algorithms of Toeplitz linear equations.Direct method and iterative method are the two classes of methods to solve the linear system of equations.In general,the numer-ical solution of the direct method is unstable,and the iterative method is more stable than the direct method.And an effective iterative scheme can provide an efficient precondit,ioner for the Krylov subspace method for solving Toeplitz system of linear equations,which greatly encourages the scientists to study fast and efficient algorithms to solve Toeplitz systems.In this thesis,we mainly consider the solution to the Hermitian positive definite Toeplitz systems Tx = b by iterative method.It is known that every Toeplitz matrix T enjoys a circulant and skew circulant splitting(denoted by CSCS),i.e.,T = C-S with C a circulant matrix and S a skew circulant matrix.Based on the splitting,we first develop classical CSCS splitting iterative methods and then introduce shift-ed classical CSCS splitting iterative methods for solving hermitian positive dcefinite Toeplitz systems in this paper.The convergence of each method is analyzed.To verify the effectiveness of our method,we made a lot of numerical experiments.Numerical experiments show that:(1)the spectrum of the coefficient matrix are clustered around 1,if T enjoys a P-regular circulant and skew circulant splitting,the original linear system converges to the exact solution,in addition,the classical CSCS iterative methods work slightly better than the Gauss-Seidel(GS)iterative methods;(2)There is always a constant ? such that the shifted classical CSCS iteration converges much faster than the Gauss-Seidel iteration and classical CSCS iteration,no matter whether the CSCS itself is convergent or not.The structure of this thesis is devided into six chapters as follows:The first chapter is an introduction which give a brief introduction for research backgrounds,current situation of Toeplitz linear system of equations,the research content as well as innovation of this paper.The second chapter is the preliminaries which review some of the commonly used definitions,lemma and its basic properties involved in this paper.In the third chapter,we introduced several kinds of basic iterative methods which include the classical iteration methods,Gauss-Seidel iterative method,Jacobi iterative method,SOR methods and SSOR method.In the fourth chapter,the first part,we propose the circulant and skew-circulant splitting method for hermitian positive define Toeplitz systems Tx = b.The second part,we introduce the classical Gauss-Seidel iterative method for hermitian positive define Toeplitz systems Tx = b.At last,We compare the algorithms of the two methods.In the fifth chapter,we introduce a new method-the shifted classical CSCS iter-ative methods for solving hermitian positive definite Toeplitz systems and its conver-gence properties are analyzed.In the last chapter,the numerical experiments are given,for different forms of splitting matrix,we analyze the data from numerical examples and compare with each other.In the end,we give concluding remarks.