A refined Arnoldi type method for large scale eigenvalue problems

Xiang Wang; Qiang Niu; Lin Zhang Lu

doi:10.1007/s13160-012-0090-0

A refined Arnoldi type method for large scale eigenvalue problems

Xiang Wang^*, Qiang Niu, Lin Zhang Lu

^*Corresponding author for this work

Department of Applied Mathematics

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

Abstract

We present a refined Arnoldi-type method for extracting partial eigenpairs of large matrices. The approximate eigenvalues are the Ritz values of (A-τ I)^-1 with respect to a shifted Krylov subspace. The approximate eigenvectors are derived by satisfying certain optimal properties, and they can be computed cheaply by a small sized singular value problem. Theoretical analysis show that the approximate eigenpairs computed by the new method converges as the approximate subspace expands. Finally, numerical results are reported to show the efficiency of the new method.

Original language	English
Pages (from-to)	129-143
Number of pages	15
Journal	Japan Journal of Industrial and Applied Mathematics
Volume	30
Issue number	1
DOIs	https://doi.org/10.1007/s13160-012-0090-0
Publication status	Published - Feb 2013

Keywords

Arnoldiprocess
Eigenvalue problem
Harmonic Ritz values
Rayleigh-Ritz
Ritz values

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10.1007/s13160-012-0090-0

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title = "A refined Arnoldi type method for large scale eigenvalue problems",

abstract = "We present a refined Arnoldi-type method for extracting partial eigenpairs of large matrices. The approximate eigenvalues are the Ritz values of (A-τ I)-1 with respect to a shifted Krylov subspace. The approximate eigenvectors are derived by satisfying certain optimal properties, and they can be computed cheaply by a small sized singular value problem. Theoretical analysis show that the approximate eigenpairs computed by the new method converges as the approximate subspace expands. Finally, numerical results are reported to show the efficiency of the new method.",

keywords = "Arnoldiprocess, Eigenvalue problem, Harmonic Ritz values, Rayleigh-Ritz, Ritz values",

author = "Xiang Wang and Qiang Niu and Lu, {Lin Zhang}",

note = "Funding Information: This work is supported by NNSF of China No. 11101204, supported by Young Scientists of Jiangxi province, NSF of Jiangxi, China No. 20114BAB201004, Science Funds of The Education Department of Jiangxi Province No.GJJ12011, and the Scientific Research Foundation of Graduate School of Nanchang University with No.YC2011-S007.",

year = "2013",

month = feb,

doi = "10.1007/s13160-012-0090-0",

language = "English",

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TY - JOUR

T1 - A refined Arnoldi type method for large scale eigenvalue problems

AU - Wang, Xiang

AU - Niu, Qiang

AU - Lu, Lin Zhang

N1 - Funding Information: This work is supported by NNSF of China No. 11101204, supported by Young Scientists of Jiangxi province, NSF of Jiangxi, China No. 20114BAB201004, Science Funds of The Education Department of Jiangxi Province No.GJJ12011, and the Scientific Research Foundation of Graduate School of Nanchang University with No.YC2011-S007.

PY - 2013/2

Y1 - 2013/2

N2 - We present a refined Arnoldi-type method for extracting partial eigenpairs of large matrices. The approximate eigenvalues are the Ritz values of (A-τ I)-1 with respect to a shifted Krylov subspace. The approximate eigenvectors are derived by satisfying certain optimal properties, and they can be computed cheaply by a small sized singular value problem. Theoretical analysis show that the approximate eigenpairs computed by the new method converges as the approximate subspace expands. Finally, numerical results are reported to show the efficiency of the new method.

AB - We present a refined Arnoldi-type method for extracting partial eigenpairs of large matrices. The approximate eigenvalues are the Ritz values of (A-τ I)-1 with respect to a shifted Krylov subspace. The approximate eigenvectors are derived by satisfying certain optimal properties, and they can be computed cheaply by a small sized singular value problem. Theoretical analysis show that the approximate eigenpairs computed by the new method converges as the approximate subspace expands. Finally, numerical results are reported to show the efficiency of the new method.

KW - Arnoldiprocess

KW - Eigenvalue problem

KW - Harmonic Ritz values

KW - Rayleigh-Ritz

KW - Ritz values

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U2 - 10.1007/s13160-012-0090-0

DO - 10.1007/s13160-012-0090-0

M3 - Article

AN - SCOPUS:84874324820

SN - 0916-7005

VL - 30

SP - 129

EP - 143

JO - Japan Journal of Industrial and Applied Mathematics

JF - Japan Journal of Industrial and Applied Mathematics

IS - 1

ER -

A refined Arnoldi type method for large scale eigenvalue problems

Abstract

Keywords

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