CIRR: a Rayleigh-Ritz type method with contour integral for generalized eigenvalue problems

Sakurai, Tetsuya; Tadano, Hiroshi

doi:10.14492/hokmj/1272848031

Cited by 107 publications

(107 citation statements)

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“…Later on Tetsuya Sakurai joined us in our study and co-authored some papers [13,17,12]. To solve eigenvalue problems, Sakurai and his co-authors applied the idea of the generalized eigenvalue problem involving the Hankel and shifted Hankel matrix using moments based on the resolvent function [18,10,15,9,19,25,1,2]. Eric Polizzi and co-authors also used contour integrals based on the resolvent function resulting in the FEAST algorithm [16,22,7].…”

Section: Introductionmentioning

confidence: 99%

Designing rational filter functions for solving eigenvalue problems by contour integration

Barel

2016

Linear Algebra and its Applications

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Solving (nonlinear) eigenvalue problems by contour integration, requires an e↵ective discretization for the corresponding contour integrals. In this paper it is shown that good rational filter functions can be computed using (nonlinear least squares) optimization techniques as opposed to designing those functions based on a thorough understanding of complex analysis. The conditions that such an effective filter function should satisfy, are derived and translated in a nonlinear least squares optimization problem solved by optimization algorithms from Tensorlab. Numerical experiments illustrate the validity of this approach.Keywords : linear, polynomial, nonlinear eigenvalue problems; contour integration; filter function; rational approximation; nonlinear least squares; resolvent. MSC : Primary : 65-F15, Secondary : 47-J10, 30-E05. Designing rational filter functions for solving eigenvalue problems by contour integrationMarc Van Barel ⇤ January 20, 2015Abstract Solving (nonlinear) eigenvalue problems by contour integration, requires an e↵ective discretization for the corresponding contour integrals. In this paper it is shown that good rational filter functions can be computed using (nonlinear least squares) optimization techniques as opposed to designing those functions based on a thorough understanding of complex analysis. The conditions that such an e↵ective filter function should satisfy, are derived and translated in a nonlinear least squares optimization problem solved by optimization algorithms from Tensorlab. Numerical experiments illustrate the validity of this approach.

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Section: Introductionmentioning

confidence: 99%

Designing rational filter functions for solving eigenvalue problems by contour integration

Barel

2016

Linear Algebra and its Applications

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“…In the moment based method [3] On the other hand, in a Rayleigh-Ritz type approach [6], by constructing an orthonormal basis Q ∈ C n×m via the orthogonalization of S, approximate eigenvalues are given by the Ritz values of a projected matrix pencil…”

Section: The Ss Methodsmentioning

confidence: 99%

A quadrature-based eigensolver with a Krylov subspace method for shifted linear systems for Hermitian eigenproblems in lattice QCD

et al. 2010

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We consider a quadrature-based eigensolver to find eigenpairs of Hermitian matrices arising in lattice quantum chromodynamics. To reduce the computational cost for finding eigenpairs of such Hermitian matrices, we propose a new technique for solving shifted linear systems with complex shifts by means of the shifted CG method. Furthermore, by using integration paths along horizontal lines corresponding to the real axis of the complex plane, the number of iterations for the shifted CG method is also reduced. Some numerical experiments illustrate the accuracy and efficiency of the proposed method by comparison with a conventional method.

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“…In this section, we describe an eigensolver using a contour integration presented in [8]. This method finds eigenpairs inside a given circle.…”

Section: An Eigensolver Using a Contour Integrationmentioning

confidence: 99%

“…Moreover, the values which are used in a numerical integration can be evaluated independently on each integration node, it provides a variety of parallel programming models [7]. We recently proposed a Rayleigh-Ritz type method [8] in order to improve numerical stability. The block method [2] performs well if there are multiple eigenvalues in the interested spectrum.…”

Section: Introductionmentioning

confidence: 99%

A Parallel Eigensolver Using Contour Integration for Generalized Eigenvalue Problems in Molecular Simulation

Sakurai¹,

Tadano²,

Ikegami³

et al. 2010

Taiwanese J. Math.

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In this paper, we consider a parallel method for computing interior eigenvalues and corresponding eigenvectors of generalized eigenvalue problems which arise from molecular orbital computation of biochemistry applications. Matrices in such applications are sparse but often have relatively large number of nonzero elements, and we may require some eigenpairs in a specific part of the spectrum. We use a contour integration to construct a desired subspace. Some properties of the subspace obtained by numerical integration are discussed, and then a parallel implementation is presented. We report the numerical aspects and parallel performance of our method with matrices derived from molecular orbital computation.

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CIRR: a Rayleigh-Ritz type method with contour integral for generalized eigenvalue problems

Cited by 107 publications

References 10 publications

Designing rational filter functions for solving eigenvalue problems by contour integration

Designing rational filter functions for solving eigenvalue problems by contour integration

A quadrature-based eigensolver with a Krylov subspace method for shifted linear systems for Hermitian eigenproblems in lattice QCD

A Parallel Eigensolver Using Contour Integration for Generalized Eigenvalue Problems in Molecular Simulation

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