Toeplitz Matrix Calculation

This thesis presents a systematic research on Toeplitz matrices such as computing the inversion of Toeplitz matrices and solving the Toeplitz linear systems. The thesis consists three parts with four chapers.In part one (chapter one), we give the definitions of the Toeplitz matrix and the special Toeplitz matrix. The simple properties of the Toeplitz matrix are presented.The algorithms and the expressions for the inversion of Toeplitz matrix and special Toeplitz matrix are given in part two (chapter two and three). In chapter two, we introduce the methods computing the inversion of Toeplitz matrix and the factorization for the inversion of Toeplitz matrix. A new conclusion is obtained: the inversion of a Toeplitz matrix can be denoted as a sum of products of circulant matrices and upper triangular Toeplitz matrices. In chapter three, we give a new fast algorithm to computing the inversion of the five-diagonal Toeplitz matrix. The inversion of the tridiagonal Toeplitz matrix is also considered.In the last part (chapter four), the algorithms for solving the Toeplitz linear systems, circulant Toeplitz linear systems and the band Toeplitz linear systems are presented. At first, we introduce some classical methods for solving the Toeplitz linear equations. Then, two algorithms for solving the five-diagonal Toeplitz matrix linear equations are given.