A unification of unitary similarity transforms to compressed representations

Abstract

In this paper a new framework for transforming arbitrary matrices to compressed representations is presented. The framework provides a generic way of transforming a matrix via unitary similarity transformations to, e.g., Hessenberg, Hessenberg-like form and combinations of both. The new algorithms are deduced, based on the Q R-factorization of the original matrix. Relying on manipulations with rotations, all the algorithms consist of eliminating the correct set of rotations, resulting in a matrix obeying the desired structural constraints. Based on this new reduction procedure we investigate further correspondences such as irreducibility, uniqueness of the reduction procedure and the link with (rational) Krylov methods. The unitary similarity transform to Hessenberg-like form as presented here, differs significantly from the one presented in earlier work. Not only does it use less rotations to obtain the desired structure, also the convergence to rational Ritz-values is not observed in the conventional approach. Mathematics Subject Classification (2000) 15A21 · 65F99R. Vandebril has a grant as "Postdoctoraal Onderzoeker" from the Fund for Scientific Research-Flanders (Belgium).

Block tridiagonal reduction of perturbed normal and rank structured matrices

“…It is worth noticing that the matrix C plays an important role for the design of fast structured variants of the QR iteration applied to perturbed Hermitian matrices. Specifically, in [4,20] it is shown that the sequence of perturbations C k : = Q H k C k−1 Q k yields a description of the upper rank structure of the matrices A k : = Q H k A k−1 Q k generated under the QR process applied to A 0 = A. The case where A = U + xy H is a rank-one correction of a unitary matrix U is particularly interesting for applications to polynomial root-finding.…”

“…It is worth noticing that the matrix C plays an important role for the design of fast structured variants of the QR iteration applied to perturbed Hermitian matrices. Specifically, in [4,20] it is shown that the sequence of perturbations…”

“…Thus the problem of simultaneously reducing both A and C to symmetric band structure is theoretically interesting but it also might be beneficial for the design of fast effective eigenvalue algorithms for these matrices. Furthermore, condensed representations expressed in terms of block matrices [5] or the product of simpler matrices [3,20] tends to become inefficient as the length of the perturbation increases [10]. The exploitation of condensed representations in banded form can circumvent these difficulties.…”