An implicit <i>QR</i> algorithm for symmetric semiseparable matrices

Vandebril, Raf; Barel, Marc Van; Mastronardi, Nicola

doi:10.1002/nla.425

Cited by 29 publications

(27 citation statements)

References 32 publications

(41 reference statements)

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“…We know that the semiseparable structure will be preserved under a QR and RQ-iterations [17] and hence also under the rational method presented here. Nevertheless, it is worth trying to prove the preservation of the structure directly by investigating in detail the structure of the intermediate matrices.…”

Section: Preservation Of the Semiseparable Structurementioning

confidence: 92%

“…Any kind of structure can be considered such as Hessenberg [15,16], tridiagonal [2], band, Hessenberg-like [17], semiseparable, quasiseparable [18], unitary plus low rank [19,20,21], etc. 2 Similar to the QR-case one can prove that a step of the RQ-algorithm preserves the structure of the matrix A.…”

Section: Preservation Of the Structurementioning

confidence: 99%

“…These 2n − 2 transformations are effectively needed when performing a step of the QR-method onto semiseparable matrices (see, e.g., [17]). …”

Section: 11mentioning

confidence: 99%

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Rational QR-iteration without inversion

2008

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In this manuscript a new method will be presented for performing a QR-iteration with (A − σI)(A − κI) −1 = QR without explicit inversion of the factor (A − κI) −1 . A QR-method driven by a rational function is attractive since convergence can occur at both sides of the matrix.Each step of this new iteration consists of two substeps. In the explicit version, first an RQ-factorization of the initial matrix A − κI = RQ will be computed, followed by a QR-factorization of the matrix (A − σI)Q H . The factorization of (A − σI)Q H can be computed in an intelligent manner, exploiting properties of the already known RQ-factorization of A − κI. The similarity transformation yielding the QR-step is defined by the unitary factor Q in the QR-factorization of the transformed matrix (A − σI)Q H . Examples will be given, illustrating how to efficiently compute the factorization for some specific classes of matrices. The novelties of this approach with respect to these matrix classes will be discussed. Article information• Vandebril, Raf; Van Barel, Marc; Mastronardi, Nicola. Rational QR-iteration without inversion, Numerische Mathematik, volume 110, issue 4, pages 561-575, 2008.• The content of this article is identical to the content of the published paper, but without the final typesetting by the publisher.• Journal's homepage: http://link.springer.com/journal/211• Abstract In this manuscript a new method will be presented for performing a QR-iteration with (A − σI)(A − κI) −1 = QR without explicit inversion of the factor (A − κI) −1 . A QR-method driven by a rational function is attractive since convergence can occur at both sides of the matrix. Each step of this new iteration consists of two substeps. In the explicit version, first an RQfactorization of the initial matrix A − κI = RQ will be computed, followed by a QR-factorization of the matrix (A − σI)Q H . The factorization of (A − σI)Q H can be computed in an intelligent manner, exploiting properties of the already known RQ-factorization of A − κI. The similarity transformation yielding the QR-step is defined by the unitary factor Q in the QR-factorization of the transformed matrix (A − σI)Q H .Examples will be given, illustrating how to efficiently compute the factorization for some specific classes of matrices. The novelties of this approach with respect to these matrix classes will be discussed.

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Section: Preservation Of the Semiseparable Structurementioning

confidence: 92%

Section: Preservation Of the Structurementioning

confidence: 99%

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Rational QR-iteration without inversion

2008

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“…Recently, many authors have been intrigued by hidden rank properties in dense matrices. Basic theoretical results on rank structured matrices [3,4], quickly evolved to system solvers for these matrices [5,6] and to eigenvalue methods based on these matrix structures [7,8,9]. 1 A companion matrix, which is Hessenberg, can be written as the sum of a unitary and a rank 1 matrix.…”

Section: Definition 1 Given a Monic Polynomial P(z)mentioning

confidence: 99%

“…The authors showed that the matrix U k can be written as a product of three sequences of 2 × 2 unitary transformations as represented in (7). The presented method is an explicit QR-algorithm, in which special compression is needed to maintain the unitary structure.…”

Section: Definition 1 Given a Monic Polynomial P(z)mentioning

confidence: 99%

Implicit double shift QR-algorithm for companion matrices

et al. 2010

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In this paper an implicit (double) shifted QR-method for computing the eigenvalues of companion and fellow matrices will be presented. Companion and fellow matrices are Hessenberg matrices, that can be decomposed into the sum of a unitary and a rank 1 matrix. The Hessenberg, the unitary as well as the rank 1 structures are preserved under a step of the QR-method. This makes these matrices suitable for the design of a fast QR-method.Several techniques already exist for performing a QR-step. The implementation of these methods is highly dependent on the representation used. Unfortunately for most of the methods compression is needed since one is not able to maintain all three, unitary, Hessenberg and rank 1 structures.In this manuscript an implicit algorithm will be designed for performing a step of the QR-method on the companion or fellow matrix based on a new representation consisting of Givens transformations. Moreover, no compression is needed as the specific representation of the involved matrices is maintained. Finally, also a double shift version of the implicit method is presented. Abstract In this paper an implicit (double) shifted QR-method for computing the eigenvalues of companion and fellow matrices will be presented. Companion and fellow matrices are Hessenberg matrices, that can be decomposed into the sum of a unitary and a rank 1 matrix. The Hessenberg, the unitary as well as the rank 1 structures are preserved under a step of the QR-method. This makes these matrices suitable for the design of a fast QR-method. Several techniques already exist for performing a QR-step. The implementation of these methods is highly dependent on the representation used. Unfortunately for most of the methods compression is needed since one is not able to maintain all three, unitary, Hessenberg and rank 1 structures.In this manuscript an implicit algorithm will be designed for performing a step of the QRmethod on the companion or fellow matrix based on a new representation consisting of Givens transformations. Moreover, no compression is needed as the specific representation of the involved matrices is maintained. Finally, also a double shift version of the implicit method is presented.

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A Matrix and its Inverse: Revisiting Minimal Rank Completions

Woerdeman

Operator Theory: Advances and Applications

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An implicit QR algorithm for symmetric semiseparable matrices

Cited by 29 publications

References 32 publications

Rational QR-iteration without inversion

Rational QR-iteration without inversion

Implicit double shift QR-algorithm for companion matrices

A Matrix and its Inverse: Revisiting Minimal Rank Completions

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