The Lanczos Algorithm with Selective Orthogonalization

Parlett, Beresford Ν.; Scott, David

doi:10.2307/2006037

Cited by 102 publications

(113 citation statements)

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“…In the literature there are actually three different approaches to reduce the cost of reorthogonalization, namely periodic reorthogonalization [7], selective reorthogonalization [22], and partial reorthogonalization [27]. See also [28, chapter 5.3] for an introduction.…”

Section: Algorithmmentioning

confidence: 99%

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A Filtered Lanczos Procedure for Extreme and Interior Eigenvalue Problems

Fang

Saad²

2012

SIAM J. Sci. Comput.

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Abstract.When combined with Krylov projection methods, polynomial filtering can provide a powerful method for extracting extreme or interior eigenvalues of large sparse matrices. This general approach can be quite efficient in the situation when a large number of eigenvalues is sought. However, its competitiveness depends critically on a good implementation. This paper presents a technique based on such a combination to compute a group of extreme or interior eigenvalues of a real symmetric (or complex Hermitian) matrix. The technique harnesses the effectiveness of the Lanczos algorithm with partial reorthogonalization and the power of polynomial filtering. Numerical experiments indicate that the method can be far superior to competing algorithms when a large number of eigenvalues and eigenvectors is to be computed.Key words. Lanczos algorithm; polynomial filtering; partial reorthogonalization; interior eigenvalue problems.1. Introduction. The problem addressed in this paper is to compute eigenvalues located in a specified interval of a large real symmetric or complex Hermitian matrix, along with their associated eigenvectors. The interval, which we will also refer to as a 'window', can consist of a subset of the largest or smallest eigenvalue, in which case the eigenvalues requested are in one of the two ends of the spectrum. When the window is well inside the interval containing the spectrum, this is often referred to as an 'interior eigenvalue problem'. Eigenvalues in the inner portion of the spectrum are called 'interior eigenvalues', though this is clearly a loose definition.Computing a large number of interior eigenvalues of a large symmetric matrix remains one of the most difficult problems in computational linear algebra today. The classical approach to the problem is to use a form of shift-and-invert technique. If we are interested in eigenvalues around a certain shift σ, shift-andinvert consists of using a projection-type method (subspace iteration, Lanczos) to compute the eigenvalues of the matrix (A − σI) −1 . The eigenvalues (λ i − σ) −1 of this matrix become the dominant eigenvalues for those λ i 's close to σ and as a result they are easy to compute with the projection method. This approach has been the most common in structural analysis codes [1]. Computational codes based on this approach select a shift dynamically and perform a factorization of the matrix A − σI (or A − σB for the generalized case).There are a number of situations when shift-and-invert will be either inapplicable, or too slow to be of practical interest. For example, it is known that problems based on a 3-D physical mesh will tend to give matrices that are very expensive to factor due to both computational and memory requirements. There are also situations when the matrix A is not available explicitly but only through subroutines to perform matrix-vector products. Finally, in the situation, common in electronic structure calculations, when a very large number of eigenvalues is to be computed, the number of factorizations to be p...

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Section: Algorithmmentioning

confidence: 99%

“…A key feature of the Lanczos algorithm is to exploit partial reorthogonalization [27]. This is favored over the alternatives of 'selective reorthogonalization' [22] and 'full reorthogonalization' [25], due to its compromise between accuracy, cost, and ease of implementation.…”

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confidence: 99%

A Filtered Lanczos Procedure for Extreme and Interior Eigenvalue Problems

Fang

Saad²

2012

SIAM J. Sci. Comput.

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“…The following results on partial reorthogonalization of bi-Lanczos are obtained by investigating and generalizing theory known for the symmetric Lanczos method on the subject of partial [13] and selective [9] reorthogonalization.…”

Section: Partial Reorthogonalizatlon Of Bi-lanczosmentioning

confidence: 99%

Bi-Lanczos with partial orthogonalization

Veen

Vuik

1995

Computers & Structures

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“…matrix size > 5000 × 5000). To overcome this problem, various projection methods have been developed since the 1950s [Wu and Simon, 2000;Parlett et al, 1985;Freund et al, 1993;Parlett and cott, 1979;Bathe, 1971Bathe, , 2013Morgan, 2000]. Among these, the Arnoldi-type method [Morgan, 2000] is the one that can be used to solve large sparse asymmetric (non-Hermitian) eigenvalue problems.…”

Section: Introductionmentioning

confidence: 99%

An Arnoldi-Frontal Approach for the Stability Analysis of Flows in a Collapsible Channel

Hao

Cai

Roper

et al. 2016

Int. J. Appl. Mechanics

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In this paper, we have developed a new approach based on a combination of the Arnoldi and frontal methods, which is suitable for solving large sparse asymmetric and generalized complex eigenvalue problems. The new eigensolver seeks the most unstable eigen-solution in the Krylov subspace and makes use of the efficiency of the frontal solver developed for the finite element methods. The approach is used for a stability analysis of flows in a collapsible channel and is found to significantly improve the computational efficiency compared to the traditionally used QZ solver or a standard Arnoldi method. With the new approach, we are able to validate the previous results obtained either on a much coarser mesh or estimated from unsteady simulations. New neutral stability solutions of the system are also obtained which are beyond the limit of previously used methods.

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The Lanczos Algorithm with Selective Orthogonalization

Cited by 102 publications

References 2 publications

A Filtered Lanczos Procedure for Extreme and Interior Eigenvalue Problems

A Filtered Lanczos Procedure for Extreme and Interior Eigenvalue Problems

Bi-Lanczos with partial orthogonalization

An Arnoldi-Frontal Approach for the Stability Analysis of Flows in a Collapsible Channel

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