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Volume 21, Issue 5
Computing Eigenvectors of Normal Matrices with Simple Inverse Iteration

Zhen-Yue Zhang & Tiang-Wei Ouyang

J. Comp. Math., 21 (2003), pp. 657-670.

Published online: 2003-10

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  • 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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@Article{JCM-21-657, author = {Zhang , Zhen-Yue and Ouyang , Tiang-Wei}, title = {Computing Eigenvectors of Normal Matrices with Simple Inverse Iteration}, journal = {Journal of Computational Mathematics}, year = {2003}, volume = {21}, number = {5}, pages = {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 $u$.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/10244.html} }
TY - JOUR T1 - Computing Eigenvectors of Normal Matrices with Simple Inverse Iteration AU - Zhang , Zhen-Yue AU - Ouyang , Tiang-Wei JO - Journal of Computational Mathematics VL - 5 SP - 657 EP - 670 PY - 2003 DA - 2003/10 SN - 21 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/10244.html KW - Eigenvector, Inverse iteration, Accuracy, Error estimation. AB -

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 $u$.

Zhang , Zhen-Yue and Ouyang , Tiang-Wei. (2003). Computing Eigenvectors of Normal Matrices with Simple Inverse Iteration. Journal of Computational Mathematics. 21 (5). 657-670. doi:
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