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A block-symmetric linearization of odd degree matrix polynomials with optimal eigenvalue condition number and backward error

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Abstract

The standard way of solving numerically a polynomial eigenvalue problem (PEP) is to use a linearization and solve the corresponding generalized eigenvalue problem (GEP). In addition, if the PEP possesses one of the structures arising very often in applications, then the use of a linearization that preserves such structure combined with a structured algorithm for the GEP presents considerable numerical advantages. Block-symmetric linearizations have proven to be very useful for constructing structured linearizations of structured matrix polynomials. In this scenario, we analyze the eigenvalue condition numbers and backward errors of approximated eigenpairs of a block symmetric linearization that was introduced by Fiedler (Linear Algebra Appl 372:325–331, 2003) for scalar polynomials and generalized to matrix polynomials by Antoniou and Vologiannidis (Electron J Linear Algebra 11:78–87, 2004). This analysis reveals that such linearization has much better numerical properties than any other block-symmetric linearization analyzed so far in the literature, including those in the well known vector space \(\mathbb {DL}(P)\) of block-symmetric linearizations. The main drawback of the analyzed linearization is that it can be constructed only for matrix polynomials of odd degree, but we believe that it will be possible to extend its use to even degree polynomials via some strategies in the near future.

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Acknowledgements

We would like to thank the two anonymous referees for reading our manuscript so thoroughly and providing such constructive feedback. They asked very interesting questions that helped us improve our paper significantly.

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Authors and Affiliations

  1. Department of Mathematics and College of Creative Studies, University of California Santa Barbara, Santa Barbara, CA, USA

    M. I. Bueno

  2. Department of Mathematics, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911, Leganés, Spain

    F. M. Dopico

  3. Centro de Análise Funcional, Estruturas Lineares e Aplicaçŏes da Universidade de Lisboa and Faculdade de Economia do Porto, Rua Dr. Roberto Frias, 4200-464, Porto, Portugal

    S. Furtado

  4. Boston University, One Silber Way, Boston, MA, 02215, USA

    L. Medina

Authors
  1. M. I. Bueno
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  2. F. M. Dopico
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  3. S. Furtado
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  4. L. Medina
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Corresponding author

Correspondence to M. I. Bueno.

Additional information

The research of M. I. Bueno was partially supported by NSF Grant DMS-1358884 and partially supported by “Ministerio de Economía, Industria y Competitividad of Spain” and “Fondo Europeo de Desarrollo Regional (FEDER) of EU” through Grant MTM-2015-65798-P (MINECO/FEDER, UE). The research of F. M. Dopico was partially supported by “Ministerio de Economía, Industria y Competitividad of Spain” and “Fondo Europeo de Desarrollo Regional (FEDER) of EU” through Grants MTM-2015-68805-REDT and MTM-2015-65798-P (MINECO/FEDER, UE). The research of S. Furtado was partially supported by Project UID/MAT/04721/2013. The research of L. Medina was partially supported by NSF Grant DMS-1358884.

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Bueno, M.I., Dopico, F.M., Furtado, S. et al. A block-symmetric linearization of odd degree matrix polynomials with optimal eigenvalue condition number and backward error. Calcolo 55, 32 (2018). https://doi.org/10.1007/s10092-018-0273-4

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