## Algorithms for hyperbolic quadratic eigenvalue problems

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- by Chun-Hua Guo and Peter Lancaster PDF
- Math. Comp.
**74**(2005), 1777-1791 Request permission

## 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.## References

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## Additional Information

**Chun-Hua Guo**- Affiliation: Department of Mathematics and Statistics, University of Regina, Regina, Saskatch- ewan, Canada S4S 0A2
- Email: chguo@math.uregina.ca
**Peter Lancaster**- Affiliation: Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada T2N 1N4
- Email: lancaste@ucalgary.ca
- 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.
- © Copyright 2005 American Mathematical Society
- Journal: Math. Comp.
**74**(2005), 1777-1791 - MSC (2000): Primary 65F30; Secondary 15A18, 15A24
- DOI: https://doi.org/10.1090/S0025-5718-05-01748-5
- MathSciNet review: 2164096