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 • Previous Articles

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

Abstract

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

CLC Number:

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.

TrendMD

[1]	Wenhui Guan, Xuenian Cao. A Numerical Algorithm for the Caputo Tempered Fractional Advection-Diffusion Equation [J]. Communications on Applied Mathematics and Computation, 2021, 3(1): 41-59.
[2]	Mehdi Samiee, Ehsan Kharazmi, Mark M. Meerschaert, Mohsen Zayernouri. A Unified Petrov–Galerkin Spectral Method and Fast Solver for Distributed-Order Partial Differential Equations [J]. Communications on Applied Mathematics and Computation, 2021, 3(1): 61-90.
[3]	Xiao, Xia Guo, Hong, Xiao Wu. Two Structure-Preserving-Doubling Like Algorithms to Solve the Positive Definite Solution of the Equation X - A^HX^-1A = Q [J]. Communications on Applied Mathematics and Computation, 2021, 3(1): 123-135.
[4]	Somayeh Yeganeh, Reza Mokhtari, Jan S. Hesthaven. A Local Discontinuous Galerkin Method for Two-Dimensional Time Fractional Difusion Equations [J]. Communications on Applied Mathematics and Computation, 2020, 2(4): 689-709.
[5]	Daniel J. Frean, Jennifer K. Ryan. Superconvergence and the Numerical Flux: a Study Using the Upwind-Biased Flux in Discontinuous Galerkin Methods [J]. Communications on Applied Mathematics and Computation, 2020, 2(3): 461-486.
[6]	Xue-lei Lin, Pin Lyu, Michael K. Ng, Hai-Wei Sun, Seakweng Vong. An Efcient Second-Order Convergent Scheme for One-Side Space Fractional Difusion Equations with Variable Coefcients [J]. Communications on Applied Mathematics and Computation, 2020, 2(2): 215-239.
[7]	Yaoqiang Yan, Weihua Deng, Daxin Nie. A Finite-Diference Approximation for the One- and Two-Dimensional Tempered Fractional Laplacian [J]. Communications on Applied Mathematics and Computation, 2020, 2(1): 129-145.
[8]	Xiangcheng Zheng, V. J. Ervin, Hong Wang. An Indirect Finite Element Method for Variable-Coefcient Space-Fractional Difusion Equations and Its Optimal-Order Error Estimates [J]. Communications on Applied Mathematics and Computation, 2020, 2(1): 147-162.
[9]	Paola Antonietti, Claudio Canuto, Marco Verani. An Adaptive hp-DG-FE Method for Elliptic Problems: Convergence and Optimality in the 1D Case [J]. Communications on Applied Mathematics and Computation, 2019, 1(3): 309-331.
[10]	Wen-Ting Wu. On Minimization of Upper Bound for the Convergence Rate of the QHSS Iteration Method [J]. Communications on Applied Mathematics and Computation, 2019, 1(2): 263-282.
[11]	Yong Liu, Chi-Wang Shu, Mengping Zhang. Superconvergence of Energy-Conserving Discontinuous Galerkin Methods for Linear Hyperbolic Equations [J]. Communications on Applied Mathematics and Computation, 2019, 1(1): 101-116.

On Convergence of MRQI and IMRQI Methods for Hermitian Eigenvalue Problems

PDF

Knowledge

Abstract

Cite this article

share this article

References

Related Articles 11

Recommended Articles

Metrics

Comments