The QR iteration method for Hermitian quasiseparable matrices of an arbitrary order

Eidelman, Yuli; Gohberg, Israel; Olshevsky, Vadim

doi:10.1016/j.laa.2005.02.037

Cited by 48 publications

(53 citation statements)

References 12 publications

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“…This coincides with computing eigenvalues of a Hermitian quasiseparable matrix. Fast O(n 2 )-algorithms for computing these eigenvalues can be found, e.g., in [17,18,41].…”

Section: Rational Krylov Sequencesmentioning

confidence: 99%

On the Convergence of Rational Ritz Values

Beckermann¹,

Güttel²,

Vandebril³

2010

SIAM J. Matrix Anal. & Appl.

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Abstract. Ruhe's rational Krylov method is a popular tool for approximating eigenvalues of a given matrix, though its convergence behavior is far from being fully understood. Under fairly general assumptions we characterize in an asymptotic sense which eigenvalues of a Hermitian matrix are approximated by rational Ritz values and how fast this approximation takes place. Our main tool is a constrained extremal problem from logarithmic potential theory, where an additional external field is required for taking into account the poles of the underlying rational Krylov space. Several examples illustrate our analytic results.

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“…This coincides with computing eigenvalues of a Hermitian quasiseparable matrix. Fast O(n 2 )-algorithms for computing these eigenvalues can be found, e.g., in [17,18,41].…”

Section: Rational Krylov Sequencesmentioning

confidence: 99%

On the Convergence of Rational Ritz Values

Beckermann¹,

Güttel²,

Vandebril³

2010

SIAM J. Matrix Anal. & Appl.

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“…As such matrices will be used often, we will create a special notation for them: (5) where the arguments of are called the 'generator' matrices of . This type of data-sparse structured matrix has recently been studied with respect to LTV systems theory and inversion [38], scattering theory [37], and for their own sake [39], [40]. The facts in which we are interested are that SSS matrices can be stored using only memory, there exist algorithms of only computational complexity for SSS matrix-matrix addition and multiplication, inversion, LU, and QR factorization, and further, that the class of SSS matrices is closed under these operations, that is, they are structure preserving.…”

Section: Subsystem Model/interconnection Structurementioning

confidence: 99%

Distributed Control: A Sequentially Semi-Separable Approach for Spatially Heterogeneous Linear Systems

Rice

Verhaegen

2009

IEEE Trans. Automat. Contr.

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Abstract-We consider the problem of designing controllers for spatially-varying interconnected systems distributed in one spatial dimension. The matrix structure of such systems can be exploited to allow fast analysis and design of centralized controllers with simple distributed implementations. Iterative algorithms are provided for stability analysis, analysis and sub-optimal controller synthesis. For practical implementation of the algorithms, approximations can be used, and the computational efficiency and accuracy are demonstrated on an example.

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“…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%

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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The QR iteration method for Hermitian quasiseparable matrices of an arbitrary order

Cited by 48 publications

References 12 publications

On the Convergence of Rational Ritz Values

On the Convergence of Rational Ritz Values

Distributed Control: A Sequentially Semi-Separable Approach for Spatially Heterogeneous Linear Systems

Rational QR-iteration without inversion

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