Convergence of the unitary $QR$ algorithm  with a unimodular Wilkinson shift

Wang, Tai-Lin; Gragg, William B.

doi:10.1090/s0025-5718-02-01444-8

Cited by 17 publications

(14 citation statements)

References 11 publications

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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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Section: Orthogonal Matricesmentioning

confidence: 99%

Structured Eigenvalue Problems

Faßbender

Kreßner

2006

GAMM-Mitteilungen

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“…For the special case of a unitary Hessenberg matrix, William B. Gragg used the so-called Schur parametrization for solving the corresponding eigenvalue problem [24]. For related implicit QR-algorithms, we refer the reader to [26,12,31,32]. This Schur parametrization can also be used to derive efficient algorithms in the context of orthogonal polynomials on the unit circle (Szegö polynomials), least squares approximation using trigonometric polynomials, the construction of Gaussian quadrature on the unit circle, frequency estimation, .…”

Section: Introductionmentioning

confidence: 99%

Unitary rank structured matrices

Delvaux

Barel

2008

Journal of Computational and Applied Mathematics

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In this paper we describe how one can represent a unitary rank structured matrix in an efficient way as a product of elementary unitary or Givens transformations. We also provide some basic operations for manipulating the representation, such as the transition to zero-creating form, the transition to a unitary/Givens-weight representation, as well as an internal pull-through process of the two branches of the representation. Finally, we characterize how to determine the 'shift' correction term to the rank structure, and we provide some applications to this result.

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“…In [23], Rutishauser designed an LR-iteration. Implicit QR-algorithms for unitary Hessenberg matrices were described and analysed in [18,24,10,29,30]. In [19,5,6,20], divide and conquer algorithms were constructed.…”

Section: Introductionmentioning

confidence: 99%

Eigenvalue computation for unitary rank structured matrices

Delvaux

Barel

2008

Journal of Computational and Applied Mathematics

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In this paper we describe how to compute the eigenvalues of a unitary rank structured matrix in two steps. First we perform a reduction of the given matrix into Hessenberg form, next we compute the eigenvalues of this resulting Hessenberg matrix via an implicit QR-algorithm. Along the way, we explain how the knowledge of a certain 'shift' correction term to the structure can be used to speed up the QR-algorithm for unitary Hessenberg matrices, and how this observation was implicitly used in a paper due to William B. Gragg. We also treat an analogue of this observation in the Hermitian tridiagonal case.

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Convergence of the unitary $QR$ algorithm with a unimodular Wilkinson shift

Cited by 17 publications

References 11 publications

Structured Eigenvalue Problems

Structured Eigenvalue Problems

Unitary rank structured matrices

Eigenvalue computation for unitary rank structured matrices

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