| Quasi-Toeplitz matrix plays an important role in signal processing and control theory.In this paper,we mainly study the splitting iterative solution of Toeplitz linear equations,and discuss the properties of quasi-Toeplitz matrix obtained by discretization of Laplace operator by difference scheme.Based on the fact that a Toeplitz matrix admits a circulant and skew circulant splitting(CSCS),we propose the extrapolated CSCS method to solve Hermitian definite Toeplitz systems and discuss the strategy to select the optimal two-parameters α,β,and the extrapolated parameter ω.The effectiveness of our method is verified by numerical experiments.Then we consider the eigenvalue problem of the quasi-Toeplitz matrix obtained by the fourth-order central difference discrete Laplace operator.We discuss the properties of the quasi-Toeplitz matrix,and prove that the eigenvalues of the quasi-Toeplitz matrix are all real numbers,and the first three eigenvalues have good approximation.In order to study the properties of the eigenvalues of the quasi-Toeplitz matrix,we introduce an auxiliary Toeplitz matrix.The Toeplitz matrix and quasi Toeplitz matrix are transformed into a block diagonal matrix of 2 × 2 under the same orthogonal transformation,and the difference between the corresponding sub-blocks is a rank-one correction,so as to obtain the characteristic polynomial of the quasi-Toeplitz matrix and the distribution of the zeros of the characteristic polynomial.This thesis consists of five chapters::The first chapter is the introduction,mainly introduces the research background of Toeplitz and quasi-Toeplitz matrices,the research status at home and abroad,and the innovation of this paper.The second chapter is the preliminary knowledge,mainly introduces the mathematical symbols,basic definitions,related lemmas and important properties;In chapter 3,we study the splitting iterative method and convergence analysis for solving Toeplitz linear equations.In chapter 4,we study the extrapolation method,convergence analysis and numerical experiments for solving Toeplitz linear equations.In chapter 5,we discuss the properties of quasi-Toeplitz matrix obtained by fourth-order central difference discretization of Laplace operator. |
