Finding the Maximal Eigenpair for a Large, Dense, Symmetric Matrix Based on Mufa Chen's Algorithm (Communications in Mathematical Research, 2020)
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 somerequire only the maximal one. In a series of papers, efficient algorithms havebeen 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 employa 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 formby using similarity transformations. We then handle the cases that prevent usfrom applying Chen's algorithm directly, e.g., the cases with zero or negativesuper- or sub-diagonal elements. Serval numerical experiments are carried outto demonstrate the efficiency of the proposed hybrid method.