A greedy algorithm for computing eigenvalues of a symmetric matrix with localized eigenvectors

Hernandez, Taylor M.; Van Beeumen, Roel; Caprio, Mark A.; Yang, Chao

doi:10.1002/nla.2341

A greedy algorithm for computing eigenvalues of a symmetric matrix with localized eigenvectors

Journal Article · Fri Oct 09 00:00:00 EDT 2020 · Numerical Linear Algebra with Applications

DOI:https://doi.org/10.1002/nla.2341· OSTI ID:1762164

Hernandez, Taylor M. ^[1]; ^[2]; Caprio, Mark A. ^[1]; ^[2]

Univ. of Notre Dame, IN (United States)
Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

Here, 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 and many-body localization studies of quantum spin chains.

View Accepted Manuscript (DOE)

Research Organization:: Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States); Univ. of Notre Dame, IN (United States)

Sponsoring Organization:: USDOE; USDOE Office of Science (SC) Office of Nuclear Physics (NP); USDOE Office of Science (SC), Nuclear Physics (NP)

Grant/Contract Number:: AC02-05CH11231; FG02-95ER40934

OSTI ID:: 1762164

Alternate ID(s):: OSTI ID: 1804777
OSTI ID: 1864858

Journal Information:: Numerical Linear Algebra with Applications, Journal Name: Numerical Linear Algebra with Applications Journal Issue: 2 Vol. 28; ISSN 1070-5325

Publisher:: WileyCopyright Statement

Country of Publication:: United States

Language:: English

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