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

Tsung Ming Huang, Zhongxiao Jia*, Wen Wei Lin

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)


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
Issue number4
Publication statusPublished - 2013 Dec


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

ASJC Scopus subject areas

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


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