Rational Krylov: A Practical Algorithm for Large Sparse Nonsymmetric Matrix Pencils

Ruhe, Axel

doi:10.1137/s1064827595285597

Cited by 120 publications

(125 citation statements)

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“…The rational Arnoldi procedure [32][33][34] is an algorithm for constructing orthonormal bases of the union of Krylov subspaces. Let V m , W m ∈ C n×m be the bases of such subspaces and let P m be a projector defined as…”

Section: The Rational Arnoldi Methodsmentioning

confidence: 99%

“…To ameliorate this problem rational Arnoldi and Lanczos algorithms [17,21,[32][33][34] have been developed which produce reduced models that match the moments of G(s) at different frequencies. Notwithstanding the greatly improved approximation offered by the rational Arnoldi and Lanczos techniques, there are some outstanding issues that need to be addressed and are summarized below.…”

Section: Introductionmentioning

confidence: 99%

“…No simple Arnoldi-and Lanczos-like equations have been derived in the literature in the rational case. The authors in [17,18,28,34] derive equations that describe the rational algorithms; however these are not in the standard Arnoldi and Lanczos form.…”

Section: Introductionmentioning

confidence: 99%

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Adaptive Rational Interpolation: Arnoldi and Lanczos-like Equations

Frangos

Jaimoukha

2008

European Journal of Control

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The Arnoldi and Lanczos algorithms, which belong to the class of Krylov subspace methods, are increasingly used for model reduction of large scale systems.The standard versions of the algorithms tend to create reduced order models that poorly approximate low frequency dynamics. Rational Arnoldi and Lanczos algorithms produce reduced models that approximate dynamics at various frequencies. This paper tackles the issue of developing simple Arnoldi and Lanczos equations for the rational case. This allows a simple error analysis to be carried out for both algorithms and permits the development of computationally efficient model reduction algorithms, where the frequencies at which the dynamics are to be matched can be updated adaptively.

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Section: The Rational Arnoldi Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Adaptive Rational Interpolation: Arnoldi and Lanczos-like Equations

Frangos

Jaimoukha

2008

European Journal of Control

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“…Thus, if we want an eigenvalue in the neighborhood of α, then we should not iterate with A, but with B = (A − αI) −1 . The rational Krylov method (RKS) of A. Ruhe [26,27,28] allows for a different shift α in every iteration step. Thus v k = (A − α k I) −1 v k−1 , or even more generally v k = (A − σ k I)(A − α k I) −1 v k−1 , where α k is used to enforce the influence of the eigenspaces of the eigenvalues in the neighborhood of α k , while σ k is used to suppress the influence of the eigenspaces of the eigenvalues in the neighborhood of σ k .…”

Section: Krylov Subspace Methodsmentioning

confidence: 99%

Orthogonal Rational Functions on the Unit Circle: from the Scalar to the Matrix Case

Bultheel

González-Vera

Hendriksen

et al. 2006

Lecture Notes in Mathematics

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The purpose of these lecture notes is to give a short introduction to the theory of orthogonal rational functions (ORF) on the unit circle. We start with the classical problem of linear prediction of a stochastic process to give a motivation for the study of Szegő's problem and to show that in this context it will turn out that not as much the ORF but rather the reproducing kernels will play a central role. Another example of rational Krylov iteration shows that it might also be interesting to consider ORF on the real line, which we shall not discuss in these lectures.In a second part we will show that most of the results of the scalar case thanslate easily to the case of matrix valued orthogonal rational functions (MORF).There are however many aspects that are intimately related to these ideas that we do not touch upon like continued fractions, Nevanlinna-Pick interpolation, moment problems, and many other aspects of what is generally known as Schur analysis.

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“…3. Construct V j ,W j ∈ C n×k j such that their columns are bases for the rational Krylov [85] spaces …”

Section: Improving Krylov Models By Using Dominant Polesmentioning

confidence: 99%

Model Order Reduction: Methods, Concepts and Properties

Antoulas

Ionutiu

Martins

et al. 2015

Mathematics in Industry

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Model order reduction : methods, concepts and propertiesAntoulas, A.C.; Ionutiu, R.; Martins, N.; ter Maten, E.J.W.; Mohaghegh, K.; Pulch, R.; Rommes, J.; Saadvandi, M.; Striebel, M.Published: 01/01/2015 Document VersionPublisher's PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication:• A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication Citation for published version (APA):Antoulas, A. C., Ionutiu, R., Martins, N., Maten, ter, E. J. W., Mohaghegh, K., Pulch, R., ... Striebel, M. (2015). Model order reduction : methods, concepts and properties. (CASA-report; Vol. 1507). Eindhoven: Technische Universiteit Eindhoven. General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.• You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ? Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Here eigenvalues are the starting point. The eigenvalue problems related to large-scale dynamical systems are usually too large to be solved completely. The algorithms described in this section are efficient and effective methods for the computation of a few specific dominant eigenvalues of these large-scale systems. It is shown how these algorithms can be used for computing reduced-order models with modal approximation and Krylov-based methods. Section 3.3, written by Maryam Saadvandi and Joost Rommes, concerns passivity preserving model order reduction using the spectral zero method. It detailedly discusses two algorithms, one by Antoulas and one by Sorenson. These two approaches are based on a projection method by selecting spectral zeros of the original transfer function to produce a reduced transfer function that has the specified roots as its spectral zeros. The reduced model preserves passivity. Section 3.4, written by Roxana Ionutiu, Joost Rommes and Athanasios C. Antoulas, refines the spectra...

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Rational Krylov: A Practical Algorithm for Large Sparse Nonsymmetric Matrix Pencils

Cited by 120 publications

References 12 publications

Adaptive Rational Interpolation: Arnoldi and Lanczos-like Equations

Adaptive Rational Interpolation: Arnoldi and Lanczos-like Equations

Orthogonal Rational Functions on the Unit Circle: from the Scalar to the Matrix Case

Model Order Reduction: Methods, Concepts and Properties

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