Roberto Bevilacqua

ON BAND MATRICES AND THEIR INVERSES

Abstract Structural properties of the inverses of band matrices are discussed. The definition of semiseparable matrices is given, and the theorem is proved that the inverse of a strict band matrix is a semiseparable matrix and vice versa. Finally, a recurrence algorithm is recommended for computing the blocks of the inverses of strict band matrices.

Parallel solution of block tridiagonal linear systems

Abstract The explicit structure of the inverse of block tridiagonal matrices is presented in terms of blocks defined by linear recurrence relations. Parallel algorithms are shown which solve block second order linear recurrences without using commutativity. Moreover we investigate the parallel solution of the associated block tridiagonal linear system. Using this theoretical background, the implementation of the algorithms is analyzed both on a small number of processors and on a hypercube. The resulting complexity is given in terms of parallel steps, each consisting of block operations, and the cost due to interprocessor communications is taken into account, too.

On the Inverse of Block Tridiagonal Matrices with Applications to the Inverses of Band Matrices and Block Band Matrices

In the present paper the authors make an attempt to give a uniform description of the main properties of tridiagonal, band, block tridiagonal and block band matrices and their inverses. Some basic concepts are recalled and also some new results are presented.

Qd-type methods for quasiseparable matrices

In the last few years many numerical techniques for computing eigenvalues of structured rank matrices have been proposed. Most of them are based on QR iterations since, in the symmetric case, the rank structure is preserved and high accuracy is guaranteed. In the unsymmetric case, however, the QR algorithm destroys the rank structure, which is instead preserved if LR iterations are used. We consider a wide class of quasiseparable matrices which can be represented in terms of the same parameters involved in their Neville factorization. This class, if assumptions are made to prevent possible breakdowns, is closed under LR steps. Moreover, we propose an implicit shifted LR method with a linear cost per step, which resembles the qd method for tridiagonal matrices. We show that for totally nonnegative quasiseparable matrices the algorithm is stable and breakdowns cannot occur if the Laguerre shift, or other shift strategy preserving nonnegativity, is used. Computational evidence shows that good accuracy is obt...

On Algebras of Toeplitz Plus Hankel Matrices

We characterize a wide class of maximal algebras of Toeplitz plus Hankel matrices by exploiting properties of displacement operators.

A CMV-Based Eigensolver for Companion Matrices

In this paper we present a novel matrix method for polynomial rootfinding. We approximate the roots by computing the eigenvalues of a permuted version of the companion matrix associated with the polynomial. This form, referred to as a lower staircase form of the companion matrix in the literature, has a block upper Hessenberg shape with possibly nonsquare subdiagonal blocks. It is shown that this form is well suited to the application of the QR eigenvalue algorithm. In particular, each matrix generated under this iteration is block upper Hessenberg and, moreover, all its submatrices located in a specified upper triangular portion are of rank two at most, with entries represented by means of four given vectors. By exploiting these properties we design a fast and computationally simple structured QR iteration which computes the eigenvalues of a companion matrix of size

Closure, commutativity and minimal complexity of some spaces of matrices

n

Structure of algebras of commutative matrices

in lower staircase form using

h-Space structure in matrix displacement formulas

O(n^2)

On algebras of symmetric Loewner matrices

flops and