Barzilai-Borwein-like methods for the extreme eigenvalue problem

Huan Gao; Yu-Hong Dai; Xiao-Jiao Tong

doi:10.3934/jimo.2015.11.999

Article Contents

2015, Volume 11, Issue 3: 999-1019. Doi: 10.3934/jimo.2015.11.999

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Barzilai-Borwein-like methods for the extreme eigenvalue problem

1.
College of Applied Sciences, Beijing University of Technology, Beijing 100124
2.
State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS, Chinese Academy of Sciences, Beijing 100190
3.
Hunan Province Key Laboratory of Smart Grids Operation and Control, Changsha University of Science and Technology, Changsha 410004, Hunan Province

Received: November 2012

Revised: June 2014

Published: July 2015

Abstract / Introduction Related Papers Cited by

Abstract

We consider numerical methods for the extreme eigenvalue problem of large scale symmetric positive definite matrices. By the variational principle, the extreme eigenvalue can be obtained by minimizing some unconstrained optimization problem. Firstly, we propose two adaptive nonmonotone Barzilai-Borwein-like methods for the unconstrained optimization problem. Secondly, we prove the global convergence of the two algorithms under some conditions. Thirdly, we compare our methods with eigs and the power method for the standard test problems from the UF Sparse Matrix Collection. The primary numerical experiments indicate that the two algorithms are promising.

Keywords:

Mathematics Subject Classification: Primary: 90C25, 90C30; Secondary: 49M30.

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