Volume 36, Issue 1
Finding the Maximal Eigenpair for a Large, Dense, Symmetric Matrix Based on Mufa Chen's Algorithm

Commun. Math. Res., 36 (2020), pp. 93-112.

Published online: 2020-03

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• Abstract

A hybrid method is presented for determining maximal eigenvalue and its eigenvector (called eigenpair) of a large, dense, symmetric matrix. Many problems require finding only a small part of the eigenpairs, and some require only the maximal one. In a series of papers, efficient algorithms have been developed by Mufa Chen for computing the maximal eigenpairs of tridiagonal matrices with positive off-diagonal elements. The key idea is to explicitly construct effective initial guess of the maximal eigenpair and then to employ a self-closed iterative algorithm. In this paper, we will extend Mufa Chen's algorithm to find maximal eigenpair for a large scale, dense, symmetric matrix. Our strategy is to first convert the underlying matrix into the tridiagonal form by using similarity transformations. We then handle the cases that prevent us from applying Chen's algorithm directly, e.g., the cases with zero or negative super- or sub-diagonal elements. Serval numerical experiments are carried out to demonstrate the efficiency of the proposed hybrid method.

60J60, 34L15

tangt@sustc.edu.cn (Tao Tang)

yangj7@sustech.edu.cn (Jiang Yang)

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@Article{CMR-36-93, author = {Tang , Tao and Yang , Jiang}, title = {Finding the Maximal Eigenpair for a Large, Dense, Symmetric Matrix Based on Mufa Chen's Algorithm}, journal = {Communications in Mathematical Research }, year = {2020}, volume = {36}, number = {1}, pages = {93--112}, abstract = {

A hybrid method is presented for determining maximal eigenvalue and its eigenvector (called eigenpair) of a large, dense, symmetric matrix. Many problems require finding only a small part of the eigenpairs, and some require only the maximal one. In a series of papers, efficient algorithms have been developed by Mufa Chen for computing the maximal eigenpairs of tridiagonal matrices with positive off-diagonal elements. The key idea is to explicitly construct effective initial guess of the maximal eigenpair and then to employ a self-closed iterative algorithm. In this paper, we will extend Mufa Chen's algorithm to find maximal eigenpair for a large scale, dense, symmetric matrix. Our strategy is to first convert the underlying matrix into the tridiagonal form by using similarity transformations. We then handle the cases that prevent us from applying Chen's algorithm directly, e.g., the cases with zero or negative super- or sub-diagonal elements. Serval numerical experiments are carried out to demonstrate the efficiency of the proposed hybrid method.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2020-0005}, url = {http://global-sci.org/intro/article_detail/cmr/15791.html} }
TY - JOUR T1 - Finding the Maximal Eigenpair for a Large, Dense, Symmetric Matrix Based on Mufa Chen's Algorithm AU - Tang , Tao AU - Yang , Jiang JO - Communications in Mathematical Research VL - 1 SP - 93 EP - 112 PY - 2020 DA - 2020/03 SN - 36 DO - http://doi.org/10.4208/cmr.2020-0005 UR - https://global-sci.org/intro/article_detail/cmr/15791.html KW - Maximal eigenpair, symmetric matrix, dense matrix, tridiagonal matrix, Householder transformation, complexity, iteration. AB -

A hybrid method is presented for determining maximal eigenvalue and its eigenvector (called eigenpair) of a large, dense, symmetric matrix. Many problems require finding only a small part of the eigenpairs, and some require only the maximal one. In a series of papers, efficient algorithms have been developed by Mufa Chen for computing the maximal eigenpairs of tridiagonal matrices with positive off-diagonal elements. The key idea is to explicitly construct effective initial guess of the maximal eigenpair and then to employ a self-closed iterative algorithm. In this paper, we will extend Mufa Chen's algorithm to find maximal eigenpair for a large scale, dense, symmetric matrix. Our strategy is to first convert the underlying matrix into the tridiagonal form by using similarity transformations. We then handle the cases that prevent us from applying Chen's algorithm directly, e.g., the cases with zero or negative super- or sub-diagonal elements. Serval numerical experiments are carried out to demonstrate the efficiency of the proposed hybrid method.

Tao Tang & Jiang Yang. (2020). Finding the Maximal Eigenpair for a Large, Dense, Symmetric Matrix Based on Mufa Chen's Algorithm. Communications in Mathematical Research . 36 (1). 93-112. doi:10.4208/cmr.2020-0005
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