On deflations in extended QR algorithms

Thomas MacH, Raf Vandebril

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)


In this paper we discuss the deflation criterion used in the extended QR algorithm based on the chasing of rotations. We provide absolute and relative perturbation bounds for this deflation criterion. Further, we present a generalization of aggressive early deflation to the extended QR algorithms. Aggressive early deflation is the key technique for the identification and deflation of already converged, but hidden, eigenvalues. Often these possibilities for deflation are not detected by the standard technique. We present numerical results underpinning the power of aggressive early deflation also in the context of extended QR algorithms. We further generalize these ideas by the transcription of middle deflations.

Original languageEnglish
Pages (from-to)559-579
Number of pages21
JournalSIAM Journal on Matrix Analysis and Applications
Issue number2
Publication statusPublished - 2014
Externally publishedYes


  • Aggressive early deflations
  • Deflation
  • Extended Hessenberg matrices
  • Extended QR algorithms
  • Middle deflations
  • Perturbation bounds
  • Rotations

ASJC Scopus subject areas

  • Analysis

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