A greedy algorithm for computing eigenvalues of a symmetric matrix
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 manybody calculations and manybody localization studies of quantum spin chains.
 Publication:

arXiv eprints
 Pub Date:
 November 2019
 arXiv:
 arXiv:1911.10041
 Bibcode:
 2019arXiv191110041H
 Keywords:

 Physics  Computational Physics
 EPrint:
 14 pages, 6 figures, 1 table