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Keywords = clustered eigenpairs

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29 pages, 911 KB  
Article
A Modified Inverse Iteration Method for Computing the Symmetric Tridiagonal Eigenvectors
by Wei Chu, Yao Zhao and Hua Yuan
Mathematics 2022, 10(19), 3636; https://doi.org/10.3390/math10193636 - 5 Oct 2022
Cited by 2 | Viewed by 2012
Abstract
This paper presents a novel method for computing the symmetric tridiagonal eigenvectors, which is the modification of the widely used Inverse Iteration method. We construct the corresponding algorithm by a new one-step iteration method, a new reorthogonalization method with the general Q iteration [...] Read more.
This paper presents a novel method for computing the symmetric tridiagonal eigenvectors, which is the modification of the widely used Inverse Iteration method. We construct the corresponding algorithm by a new one-step iteration method, a new reorthogonalization method with the general Q iteration and a significant modification when calculating severely clustered eigenvectors. The numerical results show that this method is competitive with other existing methods, especially when computing part eigenvectors or severely clustered ones. Full article
(This article belongs to the Special Issue Computational Methods and Applications for Numerical Analysis)
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13 pages, 1131 KB  
Article
Subspace Reduction for Stochastic Planar Elasticity
by Harri Hakula and Mikael Laaksonen
Appl. Mech. 2022, 3(1), 1-13; https://doi.org/10.3390/applmech3010001 - 22 Dec 2021
Viewed by 2768
Abstract
Stochastic eigenvalue problems are nonlinear and multiparametric. They require their own solution methods and remain one of the challenge problems in computational mechanics. For the simplest possible reference problems, the key is to have a cluster of at the low end of the [...] Read more.
Stochastic eigenvalue problems are nonlinear and multiparametric. They require their own solution methods and remain one of the challenge problems in computational mechanics. For the simplest possible reference problems, the key is to have a cluster of at the low end of the spectrum. If the inputs, domain or material, are perturbed, the cluster breaks and tracing of the eigenpairs become difficult due to possible crossing of the modes. In this paper we have shown that the eigenvalue crossing can occur within clusters not only by perturbations of the domain, but also of material parameters. What is new is that in this setting, the crossing can be controlled; that is, the effect of the perturbations can actually be predicted. Moreover, the basis of the subspace is shown to be a well-defined concept and can be used for instance in low-rank approximation of solutions of problems with static loading. In our industrial model problem, the reduction in solution times is significant. Full article
(This article belongs to the Special Issue Mechanics and Control using Fractional Calculus)
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