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Title: A projected preconditioned conjugate gradient algorithm for computing many extreme eigenpairs of a Hermitian matrix

We present an iterative algorithm for computing an invariant subspace associated with the algebraically smallest eigenvalues of a large sparse or structured Hermitian matrix A. We are interested in the case in which the dimension of the invariant subspace is large (e.g., over several hundreds or thousands) even though it may still be small relative to the dimension of A. These problems arise from, for example, density functional theory (DFT) based electronic structure calculations for complex materials. The key feature of our algorithm is that it performs fewer Rayleigh–Ritz calculations compared to existing algorithms such as the locally optimal block preconditioned conjugate gradient or the Davidson algorithm. It is a block algorithm, and hence can take advantage of efficient BLAS3 operations and be implemented with multiple levels of concurrency. We discuss a number of practical issues that must be addressed in order to implement the algorithm efficiently on a high performance computer.
Authors:
 [1] ;  [1] ;  [2]
  1. Computational Research Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720 (United States)
  2. Physics Division, Lawrence Livermore National Laboratory, 7000 East Avenue, Livermore, CA 94550 (United States)
Publication Date:
OSTI Identifier:
22465628
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Computational Physics; Journal Volume: 290; Other Information: Copyright (c) 2015 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGORITHMS; COMPLEXES; DENSITY FUNCTIONAL METHOD; EIGENVALUES; ELECTRONIC STRUCTURE; HERMITIAN MATRIX; ITERATIVE METHODS