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Restarted Q-Arnoldi-type methods exploiting symmetry in quadratic eigenvalue problems

Abstract

We investigate how to adapt the Q-Arnoldi method for the case of symmetric quadratic eigenvalue problems, that is, we are interested in computing a few eigenpairs \((\lambda ,x)\) of \((\lambda ^2M+\lambda C+K)x=0\) with MCK symmetric \(n\times n\) matrices. This problem has no particular structure, in the sense that eigenvalues can be complex or even defective. Still, symmetry of the matrices can be exploited to some extent. For this, we perform a symmetric linearization \(Ay=\lambda By\), where AB are symmetric \(2n\times 2n\) matrices but the pair (AB) is indefinite and hence standard Lanczos methods are not applicable. We implement a symmetric-indefinite Lanczos method and enrich it with a thick-restart technique. This method uses pseudo inner products induced by matrix B for the orthogonalization of vectors (indefinite Gram-Schmidt). The projected problem is also an indefinite matrix pair. The next step is to write a specialized, memory-efficient version that exploits the block structure of A and B, referring only to the original problem matrices MCK as in the Q-Arnoldi method. This results in what we have called the Q-Lanczos method. Furthermore, we define a stabilized variant analog of the TOAR method. We show results obtained with parallel implementations in SLEPc.

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Acknowledgments

The authors are grateful to the reviewers for their constructive comments that helped improve the presentation. The computational experiments of Sect. 7 were carried out on the supercomputer Tirant at Universitat de València.

Author information

Correspondence to Jose E. Roman.

Additional information

This work was supported by the Spanish Ministry of Economy and Competitiveness under Grant TIN2013-41049-P. Carmen Campos was supported by the Spanish Ministry of Education, Culture and Sport through an FPU Grant with reference AP2012-0608.

Communicated by Daniel Kressner.

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Campos, C., Roman, J.E. Restarted Q-Arnoldi-type methods exploiting symmetry in quadratic eigenvalue problems. Bit Numer Math 56, 1213–1236 (2016). https://doi.org/10.1007/s10543-016-0601-5

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Keywords

  • Quadratic eigenvalue problem
  • Pseudo-Lanczos
  • Q-Arnoldi
  • TOAR
  • Thick-restart
  • SLEPc

Mathematics Subject Classification

  • 65F15
  • 15A18
  • 65F50