Efficient eigenvalue computation for quasiseparable Hermitian matrices under low rank perturbations

Yuli Eidelman, Luca Gemignani*, Israel Gohberg

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


In this paper we address the problem of efficiently computing all the eigenvalues of a large N×N Hermitian matrix modified by a possibly non Hermitian perturbation of low rank. Previously proposed fast adaptations of the QR algorithm are considerably simplified by performing a preliminary transformation of the matrix by similarity into an upper Hessenberg form. The transformed matrix can be specified by a small set of parameters which are easily updated during the QR process. The resulting structured QR iteration can be carried out in linear time using linear memory storage. Moreover, it is proved to be backward stable. Numerical experiments show that the novel algorithm outperforms available implementations of the Hessenberg QR algorithm already for small values of N.

Original languageEnglish
Pages (from-to)253-273
Number of pages21
JournalNumerical Algorithms
Issue number3
StatePublished - Mar 2008


  • Complexity
  • Hessenberg reduction
  • QR eigenvalue algorithm
  • Quasiseparable matrices


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