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 - <p style="text-align: justify;">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&#39;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.</p>