Krylov-Schur-Type restarts for the two-sided arnoldi method

I.N. Zwaan, M.E. Hochstenbach

Research output: Contribution to journalArticleAcademicpeer-review

9 Citations (Scopus)


We consider the two-sided Arnoldi method and propose a two-sided Krylov-Schurtype restarting method. We discuss the restart for standard Rayleigh-Ritz extraction as well as harmonic Rayleigh-Ritz extraction. Additionally, we provide error bounds for Ritz values and Ritz vectors in the context of oblique projections and present generalizations of, e.g., the Bauer-Fike theorem and Saad's theorem. Applications of the two-sided Krylov-Schur method include the simultaneous computation of left and right eigenvectors and the computation of eigenvalue condition numbers. We demonstrate how the method can be used to -nd the least sensitive eigenvalues of a nonnormal matrix and how to approximate pseudospectra by using left and right shift-invariant subspaces. The results demonstrate that significant improvements in quality can be obtained over approximations with the (one-sided) Krylov-Schur method.

Original languageEnglish
Pages (from-to)297-321
Number of pages25
JournalSIAM Journal on Matrix Analysis and Applications
Issue number2
Publication statusPublished - 2017


  • Dual Arnoldi
  • Eigenvalue condition number
  • Harmonic two-sided extraction
  • Implicit restart
  • Krylov-Schur
  • Least sensitive eigenvalues
  • Pseudospectra
  • Two-sided Arnoldi
  • Two-sided Krylov-Schur


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