A greedy algorithm for computing eigenvalues of a symmetric matrix

TL;DR

This paper introduces a greedy algorithm that efficiently computes localized eigenpairs of large sparse matrices, improving eigenvalue calculations in quantum physics and network analysis.

Contribution

The paper presents a novel greedy method for identifying eigenvector components and approximating eigenvalues, enhancing computational efficiency for large sparse matrices.

Findings

Effective in localizing eigenvectors in quantum many-body problems

Accelerates convergence of iterative eigensolvers using approximate eigenvectors

Demonstrated success on real-world large sparse matrices

Abstract

We present a greedy algorithm for computing selected eigenpairs of a large sparse matrix $H$ that can exploit localization features of the eigenvector. When the eigenvector to be computed is localized, meaning only a small number of its components have large magnitudes, the proposed algorithm identifies the location of these components in a greedy manner, and obtains approximations to the desired eigenpairs of $H$ by computing eigenpairs of a submatrix extracted from the corresponding rows and columns of $H$. Even when the eigenvector is not completely localized, the approximate eigenvectors obtained by the greedy algorithm can be used as good starting guesses to accelerate the convergence of an iterative eigensolver applied to $H$. We discuss a few possibilities for selecting important rows and columns of $H$ and techniques for constructing good initial guesses for an iterative eigensolver using the approximate eigenvectors returned from the greedy algorithm. We demonstrate the effectiveness of this approach with examples from nuclear quantum many-body calculations, many-body localization studies of quantum spin chains and road network analysis.