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Volume 17, Issue 3
Arnoldi Type Algorithms for Large Unsymmetric Multiple Eigenvalue Problems

Zhong-Xiao Jia

J. Comp. Math., 17 (1999), pp. 257-274.

Published online: 1999-06

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  • Abstract

As is well known, solving matrix multiple eigenvalue problems is a very difficult topic. In this paper, Arnoldi type algorithms are proposed for large unsymmetric multiple eigenvalue problems when the matrix $A$ involved is diagonalizable. The theoretical background is established, in which lower and upper error bounds for eigenvectors are new for both Arnoldi's method and a general perturbation problem, and furthermore, these bounds are shown to be optimal and they generalize a classical perturbation bound due to W. Kahan in 1967 for $A$ symmetric. The algorithms can adaptively determine the multiplicity of an eigenvalue and a basis of the associated eigenspace. Numerical experiments show reliability of the algorithms.

  • Keywords

Arnoldi's process, Large unsymmetric matrix, Multiple eigenvalue, Diagonalizable, Error bounds.

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COPYRIGHT: © Global Science Press

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@Article{JCM-17-257, author = {Jia , Zhong-Xiao}, title = {Arnoldi Type Algorithms for Large Unsymmetric Multiple Eigenvalue Problems}, journal = {Journal of Computational Mathematics}, year = {1999}, volume = {17}, number = {3}, pages = {257--274}, abstract = {

As is well known, solving matrix multiple eigenvalue problems is a very difficult topic. In this paper, Arnoldi type algorithms are proposed for large unsymmetric multiple eigenvalue problems when the matrix $A$ involved is diagonalizable. The theoretical background is established, in which lower and upper error bounds for eigenvectors are new for both Arnoldi's method and a general perturbation problem, and furthermore, these bounds are shown to be optimal and they generalize a classical perturbation bound due to W. Kahan in 1967 for $A$ symmetric. The algorithms can adaptively determine the multiplicity of an eigenvalue and a basis of the associated eigenspace. Numerical experiments show reliability of the algorithms.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9100.html} }
TY - JOUR T1 - Arnoldi Type Algorithms for Large Unsymmetric Multiple Eigenvalue Problems AU - Jia , Zhong-Xiao JO - Journal of Computational Mathematics VL - 3 SP - 257 EP - 274 PY - 1999 DA - 1999/06 SN - 17 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9100.html KW - Arnoldi's process, Large unsymmetric matrix, Multiple eigenvalue, Diagonalizable, Error bounds. AB -

As is well known, solving matrix multiple eigenvalue problems is a very difficult topic. In this paper, Arnoldi type algorithms are proposed for large unsymmetric multiple eigenvalue problems when the matrix $A$ involved is diagonalizable. The theoretical background is established, in which lower and upper error bounds for eigenvectors are new for both Arnoldi's method and a general perturbation problem, and furthermore, these bounds are shown to be optimal and they generalize a classical perturbation bound due to W. Kahan in 1967 for $A$ symmetric. The algorithms can adaptively determine the multiplicity of an eigenvalue and a basis of the associated eigenspace. Numerical experiments show reliability of the algorithms.

Jia , Zhong-Xiao. (1999). Arnoldi Type Algorithms for Large Unsymmetric Multiple Eigenvalue Problems. Journal of Computational Mathematics. 17 (3). 257-274. doi:
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