The Multishift <i>QR</i> Algorithm. Part II: Aggressive Early Deflation

Braman, Karen; Byers, Ralph; Mathias, Roy

doi:10.1137/s0895479801384585

Cited by 88 publications

(107 citation statements)

References 28 publications

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“…The aggressive early deflation strategy, introduced in [4] for the nonsymmetric Hessenberg QR algorithm, is known to greatly speed up the algorithm for computing the eigenvalues of a nonsymmetric matrix by deflating converged eigenvalues long before a conventional deflation strategy does. Here we consider the simpler symmetric tridiagonal case.…”

Section: Aggressive Early Deflation Applied To Symmetric Tridiagonal Qrmentioning

confidence: 99%

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Eigenvalue perturbation bounds for Hermitian block tridiagonal matrices

Nakatsukasa¹

2012

Applied Numerical Mathematics

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We derive new perturbation bounds for eigenvalues of Hermitian matrices with block tridiagonal structure. The main message of this paper is that an eigenvalue is insensitive to blockwise perturbation, if it is well-separated from the spectrum of the diagonal blocks nearby the perturbed blocks. Our bound is particularly effective when the matrix is block-diagonally dominant and graded. Our approach is to obtain eigenvalue bounds via bounding eigenvector components, which is based on the observation that an eigenvalue is insensitive to componentwise perturbation if the corresponding eigenvector components are small. We use the same idea to explain two well-known phenomena, one concerning aggressive early deflation used in the symmetric tridiagonal QR algorithm and the other concerning the extremal eigenvalues of Wilkinson matrices.

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Section: Aggressive Early Deflation Applied To Symmetric Tridiagonal Qrmentioning

confidence: 99%

“…However when A, E are both allowed to be arbitrary Hermitian matrices 4 we cannot use this argument, which can be seen by a simple counterexample A = E = I, for which A + tE has a multiple eigenvalue for all 0 ≤ t ≤ 1. Hence in a general setting we need a different approach.…”

Section: Appendix a Multiple Eigenvaluesmentioning

confidence: 99%

Eigenvalue perturbation bounds for Hermitian block tridiagonal matrices

Nakatsukasa¹

2012

Applied Numerical Mathematics

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“…Recent improvements of the QR algorithm (e.g., aggressive early deflation [22]) may be extended to structured algorithms, but little work has been done in this direction so far. A commonly underappreciated aspect is the development of publicly available software for structured eigenvalue problems.…”

Section: Discussionmentioning

confidence: 99%

“…For other structures, such as skew-Hamiltonian and Hamiltonian matrices, this figure can be less dramatic [9]. Moreover, in view of recent progress made in improving the performance of general-purpose algorithms [21,22], it may require considerable implementation efforts to turn this reduction of flops into an actual reduction of computational time.…”

Section: Efficiencymentioning

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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“…Braman, Byers, and Mathias proposed in their SIAM Linear Algebra Prize winning work [27,28] an up to 10x faster Hessenberg QR-algorithm for the nonsymmetric EVD. This is the bottleneck of the overall nonsymmetric EVD, for which significant speedups should be expected.…”

Section: Algorithmic Improvements For the Solution Of Eigenvalue Probmentioning

confidence: 99%