Research On The Fast Algorithms And Properties Of The Kind Of Toeplitz Type Matrices

Toeplitz type matrices and Vandermonde type matrices, Loewner type matrices, circulant partitioned matrices and generalized central symmetric matrices,which are nearly correlative with Toeplitz type matrices, belong to one of widely applied special matrices kinds. The research on the kind of matrices is very active because the kind of matrices is applied in many science and technology fields. Aiming at Toeplitz type matrices and Vandermonde-Loewner type matrices , the paper gives several new fast algorithms of solving linear systems, computing the Moore-Penrose inverse and the inverse. Finally, we research on the problems about positive stable matrices and give the necessary and sufficient conditions for special. symmetric circulant partitioned matrices and special generalized central symmetric matrices being positive stable matrices. The organization of the paper is as follows:In Chapter one, we introduce the background and research actuality of the problems.In Chapter two, we give the definitions and basic formula .In Chapter three, by constructing a special partitioned matrix and studying the fast triangular factorization of its inverse matrix, we respectively give fast algorithms for the minimal norm least squares solutions of linear equations which coefficient matrices are n×m Toeplitz type matrices and Vandermonde-Loewner type matrices with full column rank. Further we compare the new algorithms with the method by structuring normal equations and the orthogonalization method by numerical examplesIn Chapter four, by constructing a special partitioned matrix and studying the fastalgorithms of computing the inverse, the algorithms of computing the Moore-Penrose inverse of Toeplitz type matrices are given. At the same time we give numerical examples.In Chapter five, using the special structure of Toeplitz type matrices , we study the fast algorithms of computing the inverse of Toeplitz type matrices.Last we give numerical examples.In Chapter six, some necessary and sufficient conditions for special symmetric circulant partitioned matrices and special generalized central symmetric matrices being positive stable matrices are given.