Skew-Hamiltonian and Hamiltonian Eigenvalue Problems: Theory, Algorithms and Applications

Benner, Peter; Kreßner, Daniel; Mehrmann, Volker

doi:10.1007/1-4020-3197-1_1

Cited by 47 publications

(55 citation statements)

References 73 publications

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“…The notation H and W, specifically reserved for the Hamiltonian matrix and the skew-Hamiltonian matrix, respectively, should offer a clue on which structure we are referring to in the context. Matrices with Hamiltonian structure arise from a variety of applications, including systems and controls, algebraic Riccati equations, and quadratic eigenvalue problems (Benner, Kressner and Mehrmann 2005). Inherent in the Hamiltonian structure are many interesting properties.…”

Section: Hamiltonian Structurementioning

confidence: 99%

“…3 (Benner et al 2005) There exist orthogonal symplectic matrices U, V ∈ R 2n×2n such that H = U ⊤ HV is of the form, (8.45) where N has no particular structure, T is upper triangular and R is upper quasitriangular.…”

Section: Hamiltonian Structurementioning

confidence: 99%

“…Currently, stable procedure for computing eigenvalues of skew-Hamiltonian matrices is well developed (Benner et al 2005, Van Loan 1984. For Hamiltonian matrices, the task is much harder.…”

Section: Hamiltonian Structurementioning

confidence: 99%

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Linear algebra algorithms as dynamical systems

Chu

2008

Acta Numerica

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Any logical procedure that is used to reason or infer either deductively or inductively so as to draw conclusions or make decisions can be called, in a broad sense, a realization process. A realization process usually assumes the recursive form that one state gets developed into another state by following a certain specific rule. Such an action is qualified as what is generally known as a dynamical system. In mathematics, especially for existence questions, a realization process often appears in the form of an iterative procedure or a differential equation. For years researchers have taken great effort to describe, analyze, and modify realization processes for various applications.The thrust in this exposition is to exploit the notion of dynamical systems as a special realization process for problems arising from the field of linear algebra. Several differential equations whose solutions evolve in some submanifolds of matrices are cast in fairly general frameworks of which special cases have been found to afford unified and fundamental insights into the structure and behavior of existing discrete methods and, now and then, suggest new and improved numerical methods. In some cases, there are remarkable connections between smooth flows and discrete numerical algorithms. In other cases, the flow approach seems advantageous in tackling very difficult open problems. Various aspects of the recent development and application in this direction are discussed in this paper.

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Section: Hamiltonian Structurementioning

confidence: 99%

Section: Hamiltonian Structurementioning

confidence: 99%

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Linear algebra algorithms as dynamical systems

Chu

2008

Acta Numerica

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“…QR-like algorithms that achieve this goal have been developed in [10,28,120] while Krylov subspace methods tailored to Hamiltonian matrices can be found in [8,9,49,96,129]. An efficient strongly backward stable method for computing invariant subspaces of H has recently been proposed in [34].…”

Section: Hamiltonian 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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“…For example, the QR algorithm applied to a symmetric matrix requires roughly 10% of the floating point operations (flops) required by the same algorithm applied to a general matrix [51]. 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

Lecture Notes in Computational Science and Engineering

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Skew-Hamiltonian and Hamiltonian Eigenvalue Problems: Theory, Algorithms and Applications

Cited by 47 publications

References 73 publications

Linear algebra algorithms as dynamical systems

Linear algebra algorithms as dynamical systems

Structured Eigenvalue Problems

Structured Eigenvalue Problems

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