Year: 2003
Author: Zhen-Yue Zhang, Tiang-Wei Ouyang
Journal of Computational Mathematics, Vol. 21 (2003), Iss. 5 : pp. 657–670
Abstract
It is well-known that if we have an approximate eigenvalue $\widehat{\lambda}$ of a normal matrix $A$ of order $n$, a good approximation to the corresponding eigenvector $u$ can be computed by one inverse iteration provided the position, say $k_{max}$, of the largest component of $u$ is known. In this paper we give a detailed theoretical analysis to show relations between the eigenvector $u$ and vector $x_k,k=1,\cdots,n$, obtained by simple inverse iteration, i.e., the solution to the system $(A-\widehat{\lambda}I)x=e_k$ with $e_k$ the $k$th column of the identity matrix $I$. We prove that under some weak conditions, the index $k_{max}$ is of some optimal properties related to the smallest residual and smallest approximation error to $u$ in spectral norm and Froenius norm. We also prove that the normalized absolute vector $v=|u|/\|u\|_\infty$ of $u$ can be approximated by the normalized vector of $(\|x_1\|_2,\cdots,\|x_n\|_2)^T$. We also give some upper bounds of $|u(k)|$ for those "optimal" indexes such as Fernando's heuristic for $k_{max}$ without any assumptions. A stable double orthogonal factorization method and a simpler but may less stable approach are proposed for locating the largest component of $u$.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2003-JCM-10244
Journal of Computational Mathematics, Vol. 21 (2003), Iss. 5 : pp. 657–670
Published online: 2003-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 14
Keywords: Eigenvector Inverse iteration Accuracy Error estimation.