# A projected preconditioned conjugate gradient algorithm for computing many extreme eigenpairs of a Hermitian matrix [A projected preconditioned conjugate gradient algorithm for computing a large eigenspace of a Hermitian matrix]

## Abstract

Here, 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:

- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1325863

- Report Number(s):
- LLNL-JRNL-695289

Journal ID: ISSN 0021-9991

- Grant/Contract Number:
- AC52-07NA27344

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 290; Journal Issue: C; Journal ID: ISSN 0021-9991

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; 97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; preconditioned eigenvalue solvers; density functional theory based electronic structure calculations

### Citation Formats

```
Vecharynski, Eugene, Yang, Chao, and Pask, John E.
```*A projected preconditioned conjugate gradient algorithm for computing many extreme eigenpairs of a Hermitian matrix [A projected preconditioned conjugate gradient algorithm for computing a large eigenspace of a Hermitian matrix]*. United States: N. p., 2015.
Web. doi:10.1016/j.jcp.2015.02.030.

```
Vecharynski, Eugene, Yang, Chao, & Pask, John E.
```*A projected preconditioned conjugate gradient algorithm for computing many extreme eigenpairs of a Hermitian matrix [A projected preconditioned conjugate gradient algorithm for computing a large eigenspace of a Hermitian matrix]*. United States. doi:10.1016/j.jcp.2015.02.030.

```
Vecharynski, Eugene, Yang, Chao, and Pask, John E. Wed .
"A projected preconditioned conjugate gradient algorithm for computing many extreme eigenpairs of a Hermitian matrix [A projected preconditioned conjugate gradient algorithm for computing a large eigenspace of a Hermitian matrix]". United States.
doi:10.1016/j.jcp.2015.02.030. https://www.osti.gov/servlets/purl/1325863.
```

```
@article{osti_1325863,
```

title = {A projected preconditioned conjugate gradient algorithm for computing many extreme eigenpairs of a Hermitian matrix [A projected preconditioned conjugate gradient algorithm for computing a large eigenspace of a Hermitian matrix]},

author = {Vecharynski, Eugene and Yang, Chao and Pask, John E.},

abstractNote = {Here, 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.},

doi = {10.1016/j.jcp.2015.02.030},

journal = {Journal of Computational Physics},

number = C,

volume = 290,

place = {United States},

year = {Wed Feb 25 00:00:00 EST 2015},

month = {Wed Feb 25 00:00:00 EST 2015}

}

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