## Convergence Analysis of a Locally Accelerated Preconditioned Steepest Descent Method for Hermitian-Definite Generalized Eigenvalue Problems

## Abstract

By extending the classical analysis techniques due to Samokish, Faddeev and Faddeeva, and Longsine and McCormick among others, we prove the convergence of preconditioned steepest descent with implicit deflation (PSD-id) method for solving Hermitian-definite generalized eigenvalue problems. Moreover, we derive a nonasymptotic estimate of the rate of convergence of the PSD-id method. We show that with a proper choice of the shift, the indefinite shift-and-invert preconditioner is a locally accelerated preconditioner, and is asymptotically optimal that leads to superlinear convergence. Numerical examples are presented to verify the theoretical results on the convergence behavior of the PSDid method for solving ill-conditioned Hermitian-definite generalized eigenvalue problems arising from electronic structure calculations. While rigorous and full-scale convergence proofs of preconditioned block steepest descent methods in practical use still largely eludes us, we postulate that the theoretical results introduced in this paper sheds light on an improved understanding of the convergence behavior of these block methods.

- Authors:

- Peking Univ., Beijing (China)
- Univ. of California, Davis, 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 National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1526870

- Report Number(s):
- LLNL-JRNL-776239

Journal ID: ISSN 0254-9409; 968787

- Grant/Contract Number:
- AC52-07NA27344; DMS-1522697; CCF-1527091

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Computational Mathematics

- Additional Journal Information:
- Journal Volume: 36; Journal Issue: 5; Journal ID: ISSN 0254-9409

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Eigenvalue problem; Steepest descent method; Preconditioning; Superlinear convergence

### Citation Formats

```
Cai, Yunfeng, Bai, Zhaojun, Pask, John E., and Sukumar, N. Convergence Analysis of a Locally Accelerated Preconditioned Steepest Descent Method for Hermitian-Definite Generalized Eigenvalue Problems. United States: N. p., 2018.
Web. doi:10.4208/jcm.1703-m2016-0580.
```

```
Cai, Yunfeng, Bai, Zhaojun, Pask, John E., & Sukumar, N. Convergence Analysis of a Locally Accelerated Preconditioned Steepest Descent Method for Hermitian-Definite Generalized Eigenvalue Problems. United States. doi:10.4208/jcm.1703-m2016-0580.
```

```
Cai, Yunfeng, Bai, Zhaojun, Pask, John E., and Sukumar, N. Fri .
"Convergence Analysis of a Locally Accelerated Preconditioned Steepest Descent Method for Hermitian-Definite Generalized Eigenvalue Problems". United States. doi:10.4208/jcm.1703-m2016-0580. https://www.osti.gov/servlets/purl/1526870.
```

```
@article{osti_1526870,
```

title = {Convergence Analysis of a Locally Accelerated Preconditioned Steepest Descent Method for Hermitian-Definite Generalized Eigenvalue Problems},

author = {Cai, Yunfeng and Bai, Zhaojun and Pask, John E. and Sukumar, N.},

abstractNote = {By extending the classical analysis techniques due to Samokish, Faddeev and Faddeeva, and Longsine and McCormick among others, we prove the convergence of preconditioned steepest descent with implicit deflation (PSD-id) method for solving Hermitian-definite generalized eigenvalue problems. Moreover, we derive a nonasymptotic estimate of the rate of convergence of the PSD-id method. We show that with a proper choice of the shift, the indefinite shift-and-invert preconditioner is a locally accelerated preconditioner, and is asymptotically optimal that leads to superlinear convergence. Numerical examples are presented to verify the theoretical results on the convergence behavior of the PSDid method for solving ill-conditioned Hermitian-definite generalized eigenvalue problems arising from electronic structure calculations. While rigorous and full-scale convergence proofs of preconditioned block steepest descent methods in practical use still largely eludes us, we postulate that the theoretical results introduced in this paper sheds light on an improved understanding of the convergence behavior of these block methods.},

doi = {10.4208/jcm.1703-m2016-0580},

journal = {Journal of Computational Mathematics},

number = 5,

volume = 36,

place = {United States},

year = {2018},

month = {6}

}