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THE HYPERBOLIC QUADRATIC EIGENVALUE PROBLEM

Part of: Basic linear algebra Numerical linear algebra Error analysis and interval analysis

Published online by Cambridge University Press:  13 August 2015

XIN LIANG  and
REN-CANG LI
XIN LIANG
Affiliation:
Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg, Germany; liang@mpi-magdeburg.mpg.de
REN-CANG LI
Affiliation:
Department of Mathematics, University of Texas at Arlington, PO Box 19408, Arlington, TX 76019, USA; rcli@uta.edu

Abstract

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The hyperbolic quadratic eigenvalue problem (HQEP) was shown to admit Courant–Fischer type min–max principles in 1955 by Duffin and Cauchy type interlacing inequalities in 2010 by Veselić. It can be regarded as the closest analog (among all kinds of quadratic eigenvalue problem) to the standard Hermitian eigenvalue problem (among all kinds of standard eigenvalue problem). In this paper, we conduct a systematic study on the HQEP both theoretically and numerically. On the theoretical front, we generalize Wielandt–Lidskii type min–max principles and, as a special case, Fan type trace min/max principles and establish Weyl type and Wielandt–Lidskii–Mirsky type perturbation results when an HQEP is perturbed to another HQEP. On the numerical front, we justify the natural generalization of the Rayleigh–Ritz procedure with existing principles and our new optimization principles, and, as consequences of these principles, we extend various current optimization approaches—steepest descent/ascent and nonlinear conjugate gradient type methods for the Hermitian eigenvalue problem—to calculate a few extreme eigenvalues (of both positive and negative type). A detailed convergence analysis is given for the steepest descent/ascent methods. The analysis reveals the intrinsic quantities that control convergence rates and consequently yields ways of constructing effective preconditioners. Numerical examples are presented to demonstrate the proposed theory and algorithms.

MSC classification

Primary: 15A42: Inequalities involving eigenvalues and eigenvectors 65F15: Eigenvalues, eigenvectors
Secondary: 15A18: Eigenvalues, singular values, and eigenvectors 65F08: Preconditioners for iterative methods 65F50: Sparse matrices 65G99: None of the above, but in this section
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 3 , 2015 , e13
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2015

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