On global convergence of subspace projection methods for Hermitian eigenvalue problems

Abstract

We provide a concise overview of some iterative approaches, by which large and sparse eigenvalue problems are successfully dealt with. Our main goal in this paper is to prove the global convergence of a few state-of-the art iteration methods, such as the Jacobi–Davidson, the rational Krylov sequence and the Lanczos methods. We also derive some conditions to ensure global convergence of the corresponding inexact variants of the above-mentioned methods. These global convergence analyses result either in a depth understanding or in an efficient improvement to the original methods.

Keywords:

• Global convergence
• iteration method
• Jacobi–Davidson method
• rational Krylov sequence method
• Lanczos method

AMS(MOS) Subject Classifications:

• 65F15
• 65N25

Acknowledgments

The authors are very much indebted to the referees for their constructive comments and valuable suggestions, which greatly improved the original manuscript of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Cun-Qiang Miao was supported by the National Natural Science Foundation of China (No. 11901361) and Natural Science Foundation of Shandong Province (No. ZR2018BG002), People's Republic of China. Xue-Yuan Tan was supported by the National Natural Science Foundation of China (No. 11871280) and the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province (No. 17KJB110008).