Fields of Values of Linear Pencils and Spectral Inclusion Regions

  • Natália Bebiano
  • João da Providência
  • Ana Nata
  • João P. da Providência
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 192)


We propose efficient methods for the numerical approximation of the field of values of the linear pencil \(A-\lambda B\), when one of the matrix coefficients A or B is Hermitian and \(\lambda \in \mathbbm {C}\). Our approach builds on the fact that the field of values can be reduced under compressions to the bidimensional case, for which these sets can be exactly determined. The presented algorithms hold for matrices both of small and large size. Furthermore, we investigate spectral inclusion regions for the pencil based on certain fields of values . The results are illustrated by numerical examples. We point out that the given procedures complement the known ones in the literature.


Field of values Linear pencil Selfadjoint linear pencil 



The authors wish to thank the Referees for most valuable comments. This work was partially supported by the Centre for Mathematics of the University of Coimbra – UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020.


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Natália Bebiano
    • 1
  • João da Providência
    • 2
  • Ana Nata
    • 3
  • João P. da Providência
    • 4
  1. 1.Department of Mathematics, CMUCUniversity of CoimbraCoimbraPortugal
  2. 2.Department of Physics, CFisUCUniversity of CoimbraCoimbraPortugal
  3. 3.Department of Mathematics, CMUCPolytechnic Institute of TomarTomarPortugal
  4. 4.Department of PhysicsUniversity of Beira InteriorCovilhãPortugal

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