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

Journal Article · Mon Jun 01 04:00:00 EDT 2015 · Journal of Computational Physics

DOI:https://doi.org/10.1016/J.JCP.2015.02.030· OSTI ID:22465628

Vecharynski, Eugene ^[1]; Yang, Chao ^[1]; Pask, John E. ^[2]

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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)

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.

• OSTI ID: 22465628
• Journal Information: Journal of Computational Physics, Journal Name: Journal of Computational Physics Vol. 290; ISSN JCTPAH; ISSN 0021-9991
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ALGORITHMS
COMPLEXES
DENSITY FUNCTIONAL METHOD
EIGENVALUES
ELECTRONIC STRUCTURE
HERMITIAN MATRIX
ITERATIVE METHODS