A map of contour integral-based eigensolvers for solving generalized eigenvalue problems

TL;DR

This paper analyzes five contour integral-based eigensolvers for generalized eigenvalue problems, revealing they are all projection methods and categorizing them based on their subspace, projection type, and application.

Contribution

It provides a unified framework by mapping the relationships among these eigensolvers, clarifying their underlying structure and classification.

Findings

All five eigensolvers are projection methods.

They can be categorized by subspace, projection type, and problem.

The analysis offers a unified understanding of contour integral eigensolvers.

Abstract

Recently, contour integral-based methods have been actively studied for solving interior eigenvalue problems that find all eigenvalues located in a certain region and their corresponding eigenvectors. In this paper, we reconsider the algorithms of the five typical contour integral-based eigensolvers from the viewpoint of projection methods, and then map the relationships among these methods. From the analysis, we conclude that all contour integral-based eigensolvers can be regarded as projection methods and can be categorized based on their subspace used, the type of projection and the problem to which they are applied implicitly.