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