A new framework for implicit restarting of the Krylov–Schur algorithm

Bujanović, Zvonimir; Drmač, Zlatko

doi:10.1002/nla.1944

Cited by 3 publications

(5 citation statements)

References 21 publications

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“…Another variant of implicit restarting using the Schur factorization was proposed by Stewart in [23], and a new implementation was recently proposed by Bujanović and Drmač in [5].…”

Section: Implicitly Restartingmentioning

confidence: 99%

A restarted Induced Dimension Reduction method to approximate eigenpairs of large unsymmetric matrices

Astudillo

Gijzen

2016

Journal of Computational and Applied Mathematics

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A new algorithm to compute eigenpairs of large unsymmetric matrices is presented. Using the Induced Dimension Reduction method (IDR), which was originally proposed for solving linear systems, we obtain a Hessenberg decomposition from which we approximate the eigenvalues and eigenvectors of a matrix. This decomposition has two main advantages. First, the computational efficiency since IDR is a short recurrence method. Second, the IDR polynomial used to create this Hessenberg decomposition is also used as a filter to discard the unwanted eigenvalues. Additionally, we incorporate the implicitly restarting technique proposed by D.C. Sorensen, in order to approximate specific portions of the spectrum and improve the convergence.

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“…Another variant of implicit restarting using the Schur factorization was proposed by Stewart in [23], and a new implementation was recently proposed by Bujanović and Drmač in [5].…”

Section: Implicitly Restartingmentioning

confidence: 99%

A restarted Induced Dimension Reduction method to approximate eigenpairs of large unsymmetric matrices

Astudillo

Gijzen

2016

Journal of Computational and Applied Mathematics

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“…Several possible continuations of this result appear feasible. There are several variants of the Arnoldi method that might be extendible, for example, a block Krylov–Schur or advanced filtering techniques . The understanding of the algorithm can also certainly be improved by further adapting results known for the standard Arnoldi method (for matrixes) (e.g., ).…”

Section: Discussionmentioning

confidence: 99%

A rank‐exploiting infinite Arnoldi algorithm for nonlinear eigenvalue problems

Beeumen

Jarlebring

Michiels

2016

Numerical Linear Algebra App

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SUMMARYWe consider the nonlinear eigenvalue problem M. /x D 0, where M. / is a large parameter-dependent matrix. In several applications, M. / has a structure where the higher-order terms of its Taylor expansion have a particular low-rank structure. We propose a new Arnoldi-based algorithm that can exploit this structure. More precisely, the proposed algorithm is equivalent to Arnoldi's method applied to an operator whose reciprocal eigenvalues are solutions to the nonlinear eigenvalue problem. The iterates in the algorithm are functions represented in a particular structured vector-valued polynomial basis similar to the construction in the infinite Arnoldi method [Jarlebring, Michiels, and Meerbergen, Numer. Math., 122 (2012), pp. . In this paper, the low-rank structure is exploited by applying an additional operator and by using a more compact representation of the functions. This reduces the computational cost associated with orthogonalization, as well as the required memory resources. The structure exploitation also provides a natural way in carrying out implicit restarting and locking without the need to impose structure in every restart. The efficiency and properties of the algorithm are illustrated with two large-scale problems.

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“…In applications this technique is usually used to deal with large memory requirements or orthogonalization costs for V m+1 , or to purge unwanted or spurious eigenvalues (see, e.g., [5,8,9] and the references therein). Implicit filtering for RADs was first introduced in [9] and further studied in [8].…”

Section: Implicit Filters In the Rational Krylov Method Implicit Filmentioning

confidence: 99%

“…This is, however, not done here. Pertinent ideas for polynomial Krylov methods have recently appeared in [5], where the authors relate implicit filtering in the KrylovSchur algorithm [43,45] with partial eigenvalue assignment.…”

Section: Implicit Filters In the Rational Krylov Method Implicit Filmentioning

confidence: 99%

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Generalized Rational Krylov Decompositions with an Application to Rational Approximation

Berljafa¹,

Güttel

2015

SIAM J. Matrix Anal. & Appl.

102

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Abstract. Generalized rational Krylov decompositions are matrix relations which, under certain conditions, are associated with rational Krylov spaces. We study the algebraic properties of such decompositions and present an implicit Q theorem for rational Krylov spaces. Transformations on rational Krylov decompositions allow for changing the poles of a rational Krylov space without recomputation, and two algorithms are presented for this task. Using such transformations we develop a rational Krylov method for rational least squares fitting. Numerical experiments indicate that the proposed method converges fast and robustly. A MATLAB toolbox with implementations of the presented algorithms and experiments is provided.Key words. rational Krylov decomposition, inverse eigenvalue problem, rational approximation AMS subject classifications. 15A22, 65F15, 65F18, 30E10 DOI. 10.1137/140998081 1. Introduction. Numerical methods based on rational Krylov spaces have become an indispensable tool of scientific computing. Rational Krylov spaces were initially proposed by Ruhe in the 1980s for the purpose of solving large sparse eigenvalue problems [37,39,40]. Since then many more applications have been found in model order reduction [22,17], large-scale matrix functions and matrix equations [13,15,1,26,27], and nonlinear eigenvalue problems [41,30,47,28], to name a few.In this paper we study various algebraic properties of rational Krylov spaces, using as a starting point a generalized rational Krylov decomposition

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A new framework for implicit restarting of the Krylov–Schur algorithm

Cited by 3 publications

References 21 publications

A restarted Induced Dimension Reduction method to approximate eigenpairs of large unsymmetric matrices

A restarted Induced Dimension Reduction method to approximate eigenpairs of large unsymmetric matrices

A rank‐exploiting infinite Arnoldi algorithm for nonlinear eigenvalue problems

Generalized Rational Krylov Decompositions with an Application to Rational Approximation

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