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Mathematics of Computation

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Algorithms for hyperbolic quadratic eigenvalue problems

Authors: Chun-Hua Guo and Peter Lancaster
Journal: Math. Comp. 74 (2005), 1777-1791
MSC (2000): Primary 65F30; Secondary 15A18, 15A24
Published electronically: February 16, 2005
MathSciNet review: 2164096
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Abstract: We consider the quadratic eigenvalue problem (or the QEP) $(\lambda ^2 A+\lambda B + C)x=0$, where $A, B,$ and $C$ are Hermitian with $A$ positive definite. The QEP is called hyperbolic if $(x^*Bx)^2\!>\!4(x^*Ax)(x^*Cx)$ for all nonzero $x\in {\mathbb C}^n$. We show that a relatively efficient test for hyperbolicity can be obtained by computing the eigenvalues of the QEP. A hyperbolic QEP is overdamped if $B$ is positive definite and $C$ is positive semidefinite. We show that a hyperbolic QEP (whose eigenvalues are necessarily real) is overdamped if and only if its largest eigenvalue is nonpositive. For overdamped QEPs, we show that all eigenpairs can be found efficiently by finding two solutions of the corresponding quadratic matrix equation using a method based on cyclic reduction. We also present a new measure for the degree of hyperbolicity of a hyperbolic QEP.

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Additional Information

Chun-Hua Guo
Affiliation: Department of Mathematics and Statistics, University of Regina, Regina, Saskatch- ewan, Canada S4S 0A2

Peter Lancaster
Affiliation: Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada T2N 1N4

Keywords: Quadratic eigenvalue problem, overdamping condition, quadratic matrix equation, solvent, cyclic reduction
Received by editor(s): August 7, 2003
Received by editor(s) in revised form: June 6, 2004
Published electronically: February 16, 2005
Additional Notes: This work was supported in part by grants from the Natural Sciences and Engineering Research Council of Canada.
Article copyright: © Copyright 2005 American Mathematical Society