A semiorthogonal generalized Arnoldi method and its variations for quadratic eigenvalue problems

Wei Qiang Huang, Tiexiang Li*, Yung Ta Li, Wen-Wei Lin

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


In this paper, we are concerned with the computation of a few eigenpairs with smallest eigenvalues in absolute value of quadratic eigenvalue problems. We first develop a semiorthogonal generalized Arnoldi method where the name comes from the application of a pseudo inner product in the construction of a generalized Arnoldi reduction for a generalized eigenvalue problem. The method applies the Rayleigh-Ritz orthogonal projection technique on the quadratic eigenvalue problem. Consequently, it preserves the spectral properties of the original quadratic eigenvalue problem. Furthermore, we propose a refinement scheme to improve the accuracy of the Ritz vectors for the quadratic eigenvalue problem. Given shifts, we also show how to restart the method by implicitly updating the starting vector and constructing better projection subspace. We combine the ideas of the refinement and the restart by selecting shifts upon the information of refined Ritz vectors. Finally, an implicitly restarted refined semiorthogonal generalized Arnoldi method is developed. Numerical examples demonstrate that the implicitly restarted semiorthogonal generalized Arnoldi method with or without refinement has superior convergence behaviors than the implicitly restarted Arnoldi method applied to the linearized quadratic eigenvalue problem.

Original languageEnglish
Pages (from-to)259-280
Number of pages22
JournalNumerical Linear Algebra with Applications
Issue number2
StatePublished - 1 Mar 2013


  • Implicit restart
  • Orthogonal projection
  • Quadratic eigenvalue problem
  • Refined shifts
  • Refinement
  • Semiorthogonal generalized Arnoldi method

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