Trilogy on Computing Maximal Eigenpair

TL;DR

This paper presents a three-step study on computing the maximal eigenpair, introducing efficient initializations and global algorithms applicable to a broad class of matrices, including complex ones.

Contribution

It develops new initialization methods for dangerous algorithms and introduces two global algorithms effective for various matrices.

Findings

Efficient initializations improve convergence for known algorithms.

Global algorithms work well for large and complex matrices.

Applicable to matrices with nonnegative off-diagonal elements.

Abstract

The eigenpair here means the twins consist of eigenvalue and its eigenvector. This paper introduces the three steps of our study on computing the maximal eigenpair. In the first two steps, we construct efficient initials for a known but dangerous algorithm, first for tridiagonal matrices and then for irreducible matrices, having nonnegative off-diagonal elements. In the third step, we present two global algorithms which are still efficient and work well for a quite large class of matrices, even complex for instance.