Iterative Methods For Solving Toeplitz Systems

The name Toeplitz originates from the work of Otto Toeplitz in the early 1900s on bilinear forms related to Laurent series. Toeplitz systems arise in a varity of applications in mathematics, scientific computing and engineering, for instance, image restoration problems in image processing, numerical differential equations and integral equations, time series and control theory, etc.In the thesis, we will mainly investigate iterative methods for solving Toeplitz systems, which is derived by a circulant-block diagonal splitting. We discuss also the inverse of band Toeplitz matrices. It contains two parts. They are·Iterative methods for solving Toeplitz systems;·An iterative method for the inverse of non-symmetric 5-band Toeplitz matrices.The arrangement of this paper is as follows.In Chapter 1, we first introduce some background knowledgement of Toeplitz systems. Then give a brief survey on the developments in using preconditioning techniques to solve Toeplitz systems.In Chapter 2, we introduce a circulant/block-diagonal splitting of Toeplitz matrices. Based on this splitting we then derive the iterative method for solving the Toeplitz systems. We then discuss the convergence condition and the determination of the optimal parameters. We also define the SOR iterative method. Furthermore, we give some simple numerical experiments.In Chapter 3, we give an entry by entry algorithm for computing the inverse of non-symmetric 5-band Toeplitz matrix. The efficiency of this algorithm was verified by a C++ program.