Convergence of the shifted $QR$ algorithm  for unitary Hessenberg matrices

Wang, Tai-Lin; Gragg, William B.

doi:10.1090/s0025-5718-01-01387-4

Cited by 30 publications

(30 citation statements)

References 16 publications

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“…The proof of the convergence of the basic QR algorithm can be found in numerous papers [20,21,22], many of them with focus on the various shift strategies or the different types of matrices like unitary Hessenberg or real symmetric tridiagonal [5,23,24]. Whereas Parlett [21] and…”

Section: Proof Of Convergencementioning

confidence: 99%

An extended QR‐solver for large profiled matrices

Ruess

Pahl

2009

Numerical Meth Engineering

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A new method for the solution of the standard eigenvalue problem with large symmetric profile matrices is presented. The method is based on the well-known QR-method for dense matrices. A new, flexible and reliable extension of the method is developed that is highly suited for the independent computation of any set of eigenvalues. In order to analyze the weak convergence of the method in the presence of clustered eigenvalues, the QR-method is studied. Two effective, stable and numerically cheap extensions are introduced to overcome the troublesome stagnation of the convergence. A repeated preconditioning process in combination with Jacobi rotations in the parts of the matrix with the strongest convergence is developed to significantly improve both local and global convergence. The extensions preserve the profile structure of the matrix. The efficiency of the new method is demonstrated with several examples.

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Section: Proof Of Convergencementioning

confidence: 99%

An extended QR‐solver for large profiled matrices

Ruess

Pahl

2009

Numerical Meth Engineering

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“…The Schur parameter representation for unitary Hessenberg matrices is a widespread tool for dealing with these matrices (see [31,32,33,34]). Suppose a unitary Hessenberg matrix H ∈ C n×n is given.…”

Section: Unitary Hessenberg Matricesmentioning

confidence: 99%

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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“…To make it work (efficiently), it is important to take the fact that the underlying matrix is orthogonal into account. A careful choice of shifts can lead to cubic convergence or even ensure global convergence [46,126,127]. Even better, an orthogonal (or unitary) Hessenberg matrix can be represented by O(n) so called Schur parameters [52,27].…”

Section: Orthogonal Matricesmentioning

confidence: 99%

Structured Eigenvalue Problems

Faßbender

Kreßner

2006

GAMM-Mitteilungen

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Most eigenvalue problems arising in practice are known to be structured. Structure is often introduced by discretization and linearization techniques but may also be a consequence of properties induced by the original problem. Preserving this structure can help preserve physically relevant symmetries in the eigenvalues of the matrix and may improve the accuracy and efficiency of an eigenvalue computation. The purpose of this brief survey is to highlight these facts for some common matrix structures. This includes a treatment of rather general concepts such as structured condition numbers and backward errors as well as an overview of algorithms and applications for several matrix classes including symmetric, skew‐symmetric, persymmetric, block cyclic, Hamiltonian, symplectic and orthogonal matrices. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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Convergence of the shifted $QR$ algorithm for unitary Hessenberg matrices

Cited by 30 publications

References 16 publications

An extended QR‐solver for large profiled matrices

An extended QR‐solver for large profiled matrices

Rational QR-iteration without inversion

Structured Eigenvalue Problems

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