The Research On The Computing Problems And The Properties About Special Matrices

The paper is concerned with some special matrices that arise in science and engineering, for example,periodic tridiagonal matrix,crossed tridiagonal matrix, Hankel matrix and Vandermonde matrix. About these special matrices,we obtain several algorithms of solving linear systems,computing the inverses and the triangular factorization, among which some are new ones, some are improvements for old ones , some are extensions of old ones . Finally we research on properties of Kronecker product on diagonally dominant matrices. The paper is composed of the following main parts:In Chapter one, we introduce the definitions and the significance of some special matrix.In Chapter two, we first give the inverse matrix of periodic tridiagonal matrix and the upper bounds of the elements of the inverse matrix . Then two new algorithms for solving respective linear systems are given. At last, numerical examples are presented to check the validity of the algorithms.In Chapter three , we mainly research on the crossed tridiagonal matrix. We introduce the fast algorithms of inverting and LU factorizating it. Following this , we give three fast algorithms to solve respective linear systems. As before, we enclose some numerial examples which verify the correctness of the algorithms.In Chapter four, we present the improvements for the fast triangular factorization of Hankel matrix, its inverse matrix and the inverse matrix of Vandermonde matrix . The numerial examples in the end of the chapter show that the improved algorithms are less in computing time and more higher in prescion than the classical algorithms.In Chapter five, we give some new properties of Kronecker product on diagonally dominant matrices.