Computation of a Few Small Eigenvalues of a Large Matrix with Application to Liquid Crystal Modeling

Baglama, James; Calvetti, Daniela; Reichel, Lothar; Ruttan, Arden

doi:10.1006/jcph.1998.6064

Cited by 16 publications

(13 citation statements)

References 26 publications

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“…Let the residual error (2.5) be larger than a specified tolerance for the Ritz values of interest. We then apply recursion formulas derived in [4] to compute the matrix…”

Section: The Irbl Method Let {Vmentioning

confidence: 99%

“…However, we found that when there are multiple or very close eigenvalues a block-version of the code performs better. Therefore, an IRBL method was developed and described in [4], where an application to liquid crystal modeling was also discussed. This application gives rise to large-scale path-following problems.…”

mentioning

confidence: 99%

“…The limited amount of fast computer memory available on a workstation and the large sizes of the Jacobian matrices that arise in this application made it necessary to develop a code that does not demand the factorization of the Jacobian matrices and requires only the storage of very few n-vectors, in addition to the computed eigenvectors. Further discussion on numerical methods for large-scale bifurcation problems based on the IRBL method can be found in [4,7,8].…”

mentioning

confidence: 99%

“…The IRBL algorithm discussed in [4] is designed for the computation of a few extreme eigenvalues and associated eigenvectors but cannot be applied to determine eigenvalues in the vicinity of an arbitrary point on the real axis. The code discussed in the present paper removes this restriction by applying a judiciously chosen acceleration polynomial.…”

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confidence: 99%

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IRBL: An Implicitly Restarted Block-Lanczos Method for Large-Scale Hermitian Eigenproblems

Baglama¹,

Calvetti²,

Reichel³

2003

SIAM J. Sci. Comput.

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Abstract. The irbleigs code is an implementation of an implicitly restarted block-Lanczos method for computing a few selected nearby eigenvalues and associated eigenvectors of a large, possibly sparse, Hermitian matrix A. The code requires only the evaluation of matrix-vector products with A; in particular, factorization of A is not demanded, nor is the solution of linear systems of equations with the matrix A. This, together with a fairly small storage requirement, makes the irbleigs code well suited for large-scale problems. Applications of the irbleigs code to certain generalized eigenvalue problems and to the computation of a few singular values and associated singular vectors are also discussed. Numerous computed examples illustrate the performance of the method and provide comparisons with other available codes.

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“…Let the residual error (2.5) be larger than a specified tolerance for the Ritz values of interest. We then apply recursion formulas derived in [4] to compute the matrix…”

Section: The Irbl Method Let {Vmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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IRBL: An Implicitly Restarted Block-Lanczos Method for Large-Scale Hermitian Eigenproblems

Baglama¹,

Calvetti²,

Reichel³

2003

SIAM J. Sci. Comput.

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“…For large scale structural problems, in order to save computational cost, only a small number of the eigenvalues and their associated eigenvectors around the eigenvalue concerned are extracted by the existing technique, such as subspace iterative method and Lanzcos method [22,23] . Meanwhile, the linear combination of these eigenvectors can provide a good approximation of the derivative ∂ φ i / ∂ b j .…”

Section: The Quadratic Programming Modelmentioning

confidence: 99%

Eigenvalue analysis of structures with interval parameters using the second-order Taylor series expansion and the DCA for QB

Qiu

Zhang³

2017

Applied Mathematical Modelling

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a b s t r a c tWhen the uncertainties in interval parameters are fairly large, the current analysis methods, which are usually based on the information of the first-order partial derivatives of eigenvalues, may not work well for the structural eigenvalue problem with interval parameters. To overcome this drawback, in this work, the structural eigenvalue problem with interval parameters is modeled as a series of QB (quadratic programming with box constrains) problems by taking advantage of the information of the second-order partial derivatives of eigenvalues. Then the series of QB problems would be solved by using the DCA (difference of convex functions algorithm) which is turn out be very effective for the QB problem. The specific examples, a concrete frame with sixty bars and a plate discretized with 300 finite elements, are given to show the effectiveness and feasibility of the proposed method compared with other methods.

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An implicitly restarted block Lanczos bidiagonalization method using Leja shifts

Baglama

Reichel

2012

Bit Numer Math

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Computation of a Few Small Eigenvalues of a Large Matrix with Application to Liquid Crystal Modeling

Cited by 16 publications

References 26 publications

IRBL: An Implicitly Restarted Block-Lanczos Method for Large-Scale Hermitian Eigenproblems

IRBL: An Implicitly Restarted Block-Lanczos Method for Large-Scale Hermitian Eigenproblems

Eigenvalue analysis of structures with interval parameters using the second-order Taylor series expansion and the DCA for QB

An implicitly restarted block Lanczos bidiagonalization method using Leja shifts

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