On the convergence of Ritz pairs and refined Ritz vectors for quadratic eigenvalue problems

Tsung Ming Huang, Zhongxiao Jia, Wen Wei Lin

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

For a given subspace, the Rayleigh-Ritz method projects the large quadratic eigenvalue problem (QEP) onto it and produces a small sized dense QEP. Similar to the Rayleigh-Ritz method for the linear eigenvalue problem, the Rayleigh-Ritz method defines the Ritz values and the Ritz vectors of the QEP with respect to the projection subspace. We analyze the convergence of the method when the angle between the subspace and the desired eigenvector converges to zero. We prove that there is a Ritz value that converges to the desired eigenvalue unconditionally but the Ritz vector converges conditionally and may fail to converge. To remedy the drawback of possible non-convergence of the Ritz vector, we propose a refined Ritz vector that is mathematically different from the Ritz vector and is proved to converge unconditionally. We construct examples to illustrate our theory.

Original languageEnglish
Pages (from-to)941-958
Number of pages18
JournalBIT Numerical Mathematics
Volume53
Issue number4
DOIs
Publication statusPublished - 2013 Dec 1

Fingerprint

Quadratic Eigenvalue Problem
Rayleigh-Ritz Method
Converge
Ritz Values
Subspace
Eigenvalues and eigenfunctions
Eigenvector
Eigenvalue Problem
Projection
Eigenvalue
Angle
Zero

Keywords

  • Convergence
  • Rayleigh-Ritz method
  • Refined Ritz vector
  • Ritz value
  • Ritz vector

ASJC Scopus subject areas

  • Software
  • Computer Networks and Communications
  • Computational Mathematics
  • Applied Mathematics

Cite this

On the convergence of Ritz pairs and refined Ritz vectors for quadratic eigenvalue problems. / Huang, Tsung Ming; Jia, Zhongxiao; Lin, Wen Wei.

In: BIT Numerical Mathematics, Vol. 53, No. 4, 01.12.2013, p. 941-958.

Research output: Contribution to journalArticle

@article{975652b1819e40f9a9ca2ba73209d649,
title = "On the convergence of Ritz pairs and refined Ritz vectors for quadratic eigenvalue problems",
abstract = "For a given subspace, the Rayleigh-Ritz method projects the large quadratic eigenvalue problem (QEP) onto it and produces a small sized dense QEP. Similar to the Rayleigh-Ritz method for the linear eigenvalue problem, the Rayleigh-Ritz method defines the Ritz values and the Ritz vectors of the QEP with respect to the projection subspace. We analyze the convergence of the method when the angle between the subspace and the desired eigenvector converges to zero. We prove that there is a Ritz value that converges to the desired eigenvalue unconditionally but the Ritz vector converges conditionally and may fail to converge. To remedy the drawback of possible non-convergence of the Ritz vector, we propose a refined Ritz vector that is mathematically different from the Ritz vector and is proved to converge unconditionally. We construct examples to illustrate our theory.",
keywords = "Convergence, Rayleigh-Ritz method, Refined Ritz vector, Ritz value, Ritz vector",
author = "Huang, {Tsung Ming} and Zhongxiao Jia and Lin, {Wen Wei}",
year = "2013",
month = "12",
day = "1",
doi = "10.1007/s10543-013-0438-0",
language = "English",
volume = "53",
pages = "941--958",
journal = "BIT",
issn = "0006-3835",
publisher = "Springer Netherlands",
number = "4",

}

TY - JOUR

T1 - On the convergence of Ritz pairs and refined Ritz vectors for quadratic eigenvalue problems

AU - Huang, Tsung Ming

AU - Jia, Zhongxiao

AU - Lin, Wen Wei

PY - 2013/12/1

Y1 - 2013/12/1

N2 - For a given subspace, the Rayleigh-Ritz method projects the large quadratic eigenvalue problem (QEP) onto it and produces a small sized dense QEP. Similar to the Rayleigh-Ritz method for the linear eigenvalue problem, the Rayleigh-Ritz method defines the Ritz values and the Ritz vectors of the QEP with respect to the projection subspace. We analyze the convergence of the method when the angle between the subspace and the desired eigenvector converges to zero. We prove that there is a Ritz value that converges to the desired eigenvalue unconditionally but the Ritz vector converges conditionally and may fail to converge. To remedy the drawback of possible non-convergence of the Ritz vector, we propose a refined Ritz vector that is mathematically different from the Ritz vector and is proved to converge unconditionally. We construct examples to illustrate our theory.

AB - For a given subspace, the Rayleigh-Ritz method projects the large quadratic eigenvalue problem (QEP) onto it and produces a small sized dense QEP. Similar to the Rayleigh-Ritz method for the linear eigenvalue problem, the Rayleigh-Ritz method defines the Ritz values and the Ritz vectors of the QEP with respect to the projection subspace. We analyze the convergence of the method when the angle between the subspace and the desired eigenvector converges to zero. We prove that there is a Ritz value that converges to the desired eigenvalue unconditionally but the Ritz vector converges conditionally and may fail to converge. To remedy the drawback of possible non-convergence of the Ritz vector, we propose a refined Ritz vector that is mathematically different from the Ritz vector and is proved to converge unconditionally. We construct examples to illustrate our theory.

KW - Convergence

KW - Rayleigh-Ritz method

KW - Refined Ritz vector

KW - Ritz value

KW - Ritz vector

UR - http://www.scopus.com/inward/record.url?scp=84888127716&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84888127716&partnerID=8YFLogxK

U2 - 10.1007/s10543-013-0438-0

DO - 10.1007/s10543-013-0438-0

M3 - Article

AN - SCOPUS:84888127716

VL - 53

SP - 941

EP - 958

JO - BIT

JF - BIT

SN - 0006-3835

IS - 4

ER -