On Convergence of MRQI and IMRQI Methods for Hermitian Eigenvalue Problems

doi:10.1007/s42967-020-00079-1

Communications on Applied Mathematics and Computation ›› 2021, Vol. 3 ›› Issue (1): 189-197.doi: 10.1007/s42967-020-00079-1

• ORIGINAL PAPER • 上一篇

On Convergence of MRQI and IMRQI Methods for Hermitian Eigenvalue Problems

Fang Chen¹, Cun², Qiang Miao³

1 School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China;
2 School of Mathematics and Statistics, Central South University, Changsha 410083, China;
3 Laboratory of Computational Mechanics, I.I. Vorovich Institute of Mathematics, Mechanics and Computer Science, Southern Federal University, Rostov-on-Don 344090, Russia

收稿日期:2020-03-03 修回日期:2020-05-21 发布日期:2021-03-15
通讯作者: Fang Chen, chenfreesky@126.com;Cun-Qiang Miao, cqmiao@csu.edu.cn;Galina V. Muratova, muratova@sfedu.ru E-mail:chenfreesky@126.com;cqmiao@csu.edu.cn;muratova@sfedu.ru

On Convergence of MRQI and IMRQI Methods for Hermitian Eigenvalue Problems

Fang Chen¹, Cun², Qiang Miao³

1 School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China;
2 School of Mathematics and Statistics, Central South University, Changsha 410083, China;
3 Laboratory of Computational Mechanics, I.I. Vorovich Institute of Mathematics, Mechanics and Computer Science, Southern Federal University, Rostov-on-Don 344090, Russia

Received:2020-03-03 Revised:2020-05-21 Published:2021-03-15
Contact: Fang Chen, chenfreesky@126.com;Cun-Qiang Miao, cqmiao@csu.edu.cn;Galina V. Muratova, muratova@sfedu.ru E-mail:chenfreesky@126.com;cqmiao@csu.edu.cn;muratova@sfedu.ru

摘要/Abstract

摘要： Bai et al. proposed the multistep Rayleigh quotient iteration (MRQI) as well as its inexact variant (IMRQI) in a recent work (Comput. Math. Appl. 77: 2396–2406, 2019). These methods can be used to effectively compute an eigenpair of a Hermitian matrix. The convergence theorems of these methods were established under two conditions imposed on the initial guesses for the target eigenvalue and eigenvector. In this paper, we show that these two conditions can be merged into a relaxed one, so the convergence conditions in these theorems can be weakened, and the resulting convergence theorems are applicable to a broad class of matrices. In addition, we give detailed discussions about the new convergence condition and the corresponding estimates of the convergence errors, leading to rigorous convergence theories for both the MRQI and the IMRQI.

关键词: Hermitian eigenvalue problem, MRQI, IMRQI, Convergence

Abstract: Bai et al. proposed the multistep Rayleigh quotient iteration (MRQI) as well as its inexact variant (IMRQI) in a recent work (Comput. Math. Appl. 77: 2396–2406, 2019). These methods can be used to effectively compute an eigenpair of a Hermitian matrix. The convergence theorems of these methods were established under two conditions imposed on the initial guesses for the target eigenvalue and eigenvector. In this paper, we show that these two conditions can be merged into a relaxed one, so the convergence conditions in these theorems can be weakened, and the resulting convergence theorems are applicable to a broad class of matrices. In addition, we give detailed discussions about the new convergence condition and the corresponding estimates of the convergence errors, leading to rigorous convergence theories for both the MRQI and the IMRQI.

Key words: Hermitian eigenvalue problem, MRQI, IMRQI, Convergence

中图分类号:

65F10
65F15

Fang Chen, Cun, Qiang Miao, Galina V. Muratova. On Convergence of MRQI and IMRQI Methods for Hermitian Eigenvalue Problems[J]. Communications on Applied Mathematics and Computation, 2021, 3(1): 189-197.

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On Convergence of MRQI and IMRQI Methods for Hermitian Eigenvalue Problems

On Convergence of MRQI and IMRQI Methods for Hermitian Eigenvalue Problems

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