The Fast Algorithms Of Special Matrices And Their Structure

The paper is concerned with some matrices that arise in engineering. We give the definations of Hankel-type matrix and symmetrical Hankel-type matrix. Explicit expressions of Toeplitz-type matrix, Loewner-type matrix, Hankel-type matrix, symmetrical Hankel-type matrix are obtained. We also give fast algorithms for solving linear equations, triangular factorization, inverse matrices and so on. The paper is composed of the following main parts:In §2, firstly, we give explicit expressions of Toeplitz-type matrix, Loewner-type matrix, Hankel-type matrix and symmetrical Hankel-type matrix, then we give a necessary and sufficient condition for Hankel matrix.In §3, a fast algorithm for solving symmetrical Hankel-type systems is obtained, also some numerical examples are given.In §4, we obtain a fast algorithm for solving linear equations that coefficient matrix is Loewner-type matrix plus another Loewner-type matrix, we give a numerical example.In §5, we research a new fast triangular factorization algorithm of Hankel matrix. Then we give a numerical example.In §6, necessary and sufficient condition for Hankel matrix with Hankel inverses revisited are given.In §7, we give inversion of generalized Vandermonde matrix.In §8, we give the exist conditions, explicit expressions and fast algorithms of left inverse and right inverse of m× n Vandermonde-type matrix and Loewner-type matrix.In §9, we introduce definition of M- matrix, then we give some conclusions about M- matrix.