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Polynomial Eigenvalue Problems: Theory, Computation, and Structure

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Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory

Abstract

Matrix polynomial eigenproblems arise in many application areas, both directly and as approximations for more general nonlinear eigenproblems. One of the most common strategies for solving a polynomial eigenproblem is via a linearization, which replaces the matrix polynomial by a matrix pencil with the same spectrum, and then computes with the pencil. Many matrix polynomials arising from applications have additional algebraic structure, leading to symmetries in the spectrum that are important for any computational method to respect. Thus it is useful to employ a structured linearization for a matrix polynomial with structure. This essay surveys the progress over the last decade in our understanding of linearizations and their construction, both with and without structure, and the impact this has had on numerical practice.

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Notes

  1. 1.

    Supported by Ministerio de Economía y Competitividad of Spain through grant MTM2012-32542.

  2. 2.

    Supported by Engineering and Physical Sciences Research Council grant EP/I005293.

  3. 3.

    The story is quite different for singular polynomials P. In that case, none of the pencils in \(\mathbb{D}\mathbb{L}(P)\) is ever a linearization for P, even when P has no structure [18].

  4. 4.

    Pun intended.

  5. 5.

    The previous footnote2, and this footnote3 to that footnote2, are here especially for Volker.

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Authors and Affiliations

  1. Department of Mathematics, Western Michigan University, 1903 W Michigan Ave, Kalamazoo, MI, 49008, USA

    D. Steven Mackey & Niloufer Mackey

  2. School of Mathematics, The University of Manchester, Alan Turing Building, M13 9PL, Manchester, UK

    Françoise Tisseur

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  1. D. Steven Mackey
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  2. Niloufer Mackey
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  3. Françoise Tisseur
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Correspondence to Niloufer Mackey .

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Editors and Affiliations

  1. Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany

    Peter Benner

  2. Carl-Friedrich-Gauß Fakultät, TU Braunschweig, Institute Computational Mathematics, Braunschweig, Germany

    Matthias Bollhöfer

  3. MATHICSE, École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland

    Daniel Kressner

  4. Institute of Mathematics, T U Berlin, Berlin, Germany

    Christian Mehl

  5. Institute of Mathematics, University of Augsburg, Augsburg, Germany

    Tatjana Stykel

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Mackey, D.S., Mackey, N., Tisseur, F. (2015). Polynomial Eigenvalue Problems: Theory, Computation, and Structure. In: Benner, P., Bollhöfer, M., Kressner, D., Mehl, C., Stykel, T. (eds) Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-15260-8_12

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