Structural properties of matrix unitary reduction to semiseparable form

Bevilacqua, Roberto; Corso, Gianna M. Del

doi:10.1007/s10092-004-0093-6

Cited by 6 publications

(14 citation statements)

References 13 publications

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“…The generalization of these results under weaker hypotheses, for example, with the help of structural properties of semiseparable matrices analysed in References [4,7], is also a possible direction of further research.…”

Section: Discussionmentioning

confidence: 94%

“…The special case D = O of the preceding theorem was also shown in Reference [4], by exploiting the low-rank structure of the triangular parts of a semiseparable matrix. Details on the implementation of QR steps on matrix classes including dpss matrices, also in the non-symmetric case, are given in References [6,22,23].…”

Section: Proofmentioning

confidence: 90%

“…In fact, Hermitian semiseparable matrices are but a minor generalization of the real symmetric case, since for any Hermitian S there exists a unitary diagonal matrix such that S H is real and symmetric. We refer to References [4][5][6][7] for recent accounts on structural and computational properties of symmetric semiseparable matrices. Remark that Hermitian rank-one matrices are semiseparable.…”

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confidence: 99%

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Rational Krylov matrices and QR steps on Hermitian diagonal‐plus‐semiseparable matrices

Fasino¹

2005

Numerical Linear Algebra App

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SUMMARYWe prove that the unitary factor appearing in the QR factorization of a suitably deÿned rational Krylov matrix transforms a Hermitian matrix having pairwise distinct eigenvalues into a diagonal-plussemiseparable form with prescribed diagonal term. This transformation is essentially uniquely deÿned by its ÿrst column. Furthermore, we prove that the set of Hermitian diagonal-plus-semiseparable matrices is invariant under QR iteration. These and other results are shown to be the rational counterpart of known facts involving structured matrices related to polynomial computations.

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Section: Discussionmentioning

confidence: 94%

Section: Proofmentioning

confidence: 90%

mentioning

confidence: 99%

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Rational Krylov matrices and QR steps on Hermitian diagonal‐plus‐semiseparable matrices

Fasino¹

2005

Numerical Linear Algebra App

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“…The first constructive similarity transformations to alternative matrices, admitting low-cost storage, such as semiseparable (plus diagonal) and Hessenberg-like were proposed in [3,23,28]. Though interesting convergence results were observed [27], the methods were not competitive due to a nonneglectable extra computational cost w.r.t.…”

Section: Introductionmentioning

confidence: 99%

“…Detailed studies of the outcome of these reduction algorithms reveal close connections with orthogonal rational functions [6,21,22] and relations with rational Krylov sequences [2,3,13,18]. Whereas standard Krylov methods typically have a Ritz-value convergence towards the extreme eigenvalues, rational Krylov techniques are able to shift convergence to other, more interesting points in the plane, by selecting wellchosen poles.…”

Section: Introductionmentioning

confidence: 99%

A unification of unitary similarity transforms to compressed representations

Vandebril

Corso

2011

Numer. Math.

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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).

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Fast QR iterations for unitary plus low rank matrices

2019

Self Cite

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Some fast algorithms for computing the eigenvalues of a (block) companion matrix have recently appeared in the literature. In this paper we generalize the approach to encompass unitary plus low rank matrices of the form A = U + XY H where U is a general unitary matrix. Three important cases for applications are U unitary diagonal, U unitary block Hessenberg and U unitary in block CMV form. Our extension exploits the properties of a larger matrixÂ obtained by a certain embedding of the Hessenberg reduction of A suitable to maintain its structural properties. We show thatÂ can be factored as product of lower and upper unitary Hessenberg matrices possibly perturbed in the first k rows, and, moreover, such a data-sparse representation is well suited for the design of fast eigensolvers based on the QR iteration. The resulting algorithm is fast and backward stable.We first recall some basic properties of unitary matrices which play an important role in the derivation of our methods.Lemma 1 Let U be a unitary matrix of size n. Then rank (U (α, β)) = rank (U (J\α, J\β)) + |α| + |β| − n where J = {1, 2, . . . , n} and α and β are subsets of J. If α = {1, . . . , h} and β = J\α, then we have rank (U (1 : h, h+1 : n)) = rank (U (h+1 : n, 1 : h)), for all h = 1, . . . , n−1.

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Structural properties of matrix unitary reduction to semiseparable form

Cited by 6 publications

References 13 publications

Rational Krylov matrices and QR steps on Hermitian diagonal‐plus‐semiseparable matrices

Rational Krylov matrices and QR steps on Hermitian diagonal‐plus‐semiseparable matrices

A unification of unitary similarity transforms to compressed representations

Fast QR iterations for unitary plus low rank matrices

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