A Fast Algorithm for the Inversion of Quasiseparable Vandermonde-like Matrices

The results on Vandermonde-like matrices were introduced as a generalization of polynomial Vandermonde matrices, and the displacement structure of these matrices was used to derive an inversion formula. In this paper we first present a fast Gaussian elimination algorithm for the polynomial Vandermonde-like matrices. Later we use the said algorithm to derive fast inversion algorithms for quasiseparable, semiseparable and well-free Vandermonde-like matrices having $\mathcal{O}(n^2)$ complexity. To do so we identify structures of displacement operators in terms of generators and the recurrence relations(2-term and 3-term) between the columns of the basis transformation matrices for quasiseparable, semiseparable and well-free polynomials. Finally we present an $\mathcal{O}(n^2)$ algorithm to compute the inversion of quasiseparable Vandermonde-like matrices

Eigenstructure of order-one-quasiseparable matrices. Three-term and two-term recurrence relations

AbstractThis paper presents explicit formulas and algorithms to compute the eigenvalues and eigenvectors of order-one-quasiseparable matrices. Various recursive relations for characteristic polynomials of their principal submatrices are derived. The cost of evaluating the characteristic polynomial of an N×N matrix and its derivative is only O(N). This leads immediately to several versions of a fast quasiseparable Newton iteration algorithm. In the Hermitian case we extend the Sturm property to the characteristic polynomials of order-one-quasiseparable matrices which yields to several versions of a fast quasiseparable bisection algorithm.Conditions guaranteeing that an eigenvalue of a order-one-quasiseparable matrix is simple are obtained, and an explicit formula for the corresponding eigenvector is derived. The method is further extended to the case when these conditions are not fulfilled. Several particular examples with tridiagonal, (almost) unitary Hessenberg, and Toeplitz matrices are considered.The algorithms are based on new three-term and two-term recurrence relations for the characteristic polynomials of principal submatrices of order-one-quasiseparable matrices R. It turns out that the latter new class of polynomials generalizes and includes two classical families: (i) polynomials orthogonal on the real line (that play a crucial role in a number of classical algorithms in numerical linear algebra), and (ii) the Szegö polynomials (that play a significant role in signal processing). Moreover, new formulas can be seen as generalizations of the classical three-term recurrence relations for the real orthogonal polynomials and of the two-term recurrence relations for the Szegö polynomials

Stable factorization of Hankel and Hankel-like matrices

This paper gives displacement structure algorithms for the factorization positive defjnite and indefinite Hankel and Hankel-like matrices. The positive definite algorithm uses orthogonal symplectic transformations in place of the E-orthogonal transformations used in Toeplitz algorithms. The indefinite algorithm uses a look-ahead step and is based on the observation that displacement structure algorithms for Hankel factorization have a natural and simple block generalization. Both algorithms can be applied to Hankel-like matrices of arbitrary displacement rank

Eigenvector Computation for Almost Unitary Hessenberg Matrices and Inversion of Szegö-Vandermonde Matrices via Discrete Transmission Lines

In this paper we use a discrete transmission line model (known to geophysicists as a layered earth model) to derive several computationally ecient solutions for the following three problems. (i) As is well-known, a Hessenberg matrix capturing recurrence relations for Szegö polynomials differs from unitary only by its last column. Hence, the first problem is how to rapidly evaluate the eigenvectors of this almost unitary Hessenberg matrix. (ii) The second problem is to design a fast O(n ) algorithm for inversion of Szegö-Vandermonde matrices (generalizing the well-known Traub algorithm for inversion of the usual Vandermonde matrices). (iii) Finally, the third problem is to extend the well-known Horner rule to evaluate a polynomial represented in the basis of Szegö polynomials. As we shall see, all three problems are closely related, and their solutions can be computed by the same family of fast algorithms

Fast algorithms for structured matrices : theory and applications : AMS-IMS-SIAM Joint Summer Research Conference on Fast Algorithms in Mathematics, Computer Science, and Engineering, August 5-9, 2001, Mount Holyoke College, South Hadley, Massachusetts

viii, 433 p. : ill. ; 21x30 cm