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Large-Scale Structured Eigenvalue Problems

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

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Abstract

Eigenvalue problems involving large, sparse matrices with Hamiltonian or related structure arise in numerous applications. Hamiltonian problems can be transformed to symplectic or skew-Hamiltonian problems and then solved. This chapter focuses on the transformation to skew-Hamiltonian form and solution by the SHIRA method. Related to, but more general than, Hamiltonian matrices are alternating and palindromic pencils. A SHIRA-like method that operates on alternating (even) pencils M −λ N and can be used even when N is singular, is presented.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Washington State University, Pullman, WA, 99164-3113, USA

    David S. Watkins

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  1. David S. Watkins
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Correspondence to David S. Watkins .

Editor information

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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Watkins, D.S. (2015). Large-Scale Structured Eigenvalue Problems. 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_2

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