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## 2007.117: Detecting and Solving Hyperbolic Quadratic Eigenvalue Problems

2007.117: Chun-Hua Guo, Nicholas J. Higham and Françoise Tisseur (2007) Detecting and Solving Hyperbolic Quadratic Eigenvalue Problems.

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## Abstract

Hyperbolic quadratic matrix polynomials $Q(\lambda) = \lambda^2 A + \lambda B + C$ are an important class of Hermitian matrix polynomials with real eigenvalues, among which the overdamped quadratics are those with nonpositive eigenvalues. Neither the definition of overdamped nor any of the standard characterizations provides an efficient way to test if a given $Q$ has this property. We show that a quadratically convergent matrix iteration based on cyclic reduction, previously studied by Guo and Lancaster, provides necessary and sufficient conditions for $Q$ to be overdamped. For weakly overdamped $Q$ the iteration is shown to be generically linearly convergent with constant at worst 1/2, which implies that the convergence of the iteration is reasonably fast in almost all cases of practical interest. We show that the matrix iteration can be implemented in such a way that when overdamping is detected a scalar $\mu<0$ is provided that lies in the gap between the $n$ largest and $n$ smallest eigenvalues of the $n\times n$ quadratic eigenvalue problem (QEP) $Q(\lambda)x = 0$. Once such a $\mu$ is known, the QEP can be solved by linearizing to a definite pencil that can be reduced using already available Cholesky factorizations to a standard Hermitian eigenproblem. By incorporating an initial preprocessing stage that shifts a hyperbolic $Q$ so that it is overdamped, we obtain an efficient algorithm that identifies and solves a hyperbolic or overdamped QEP maintaining symmetry throughout and guaranteeing real computed eigenvalues.

Item Type: MIMS Preprint To appear in the SIAM Journal On Matrix Analysis and Applications quadratic eigenvalue problem, hyperbolic, overdamped, weakly overdamped, quadratic matrix polynomial, quadratic matrix equation, solvent, cyclic reduction, doubling algorithm MSC 2000 > 15 Linear and multilinear algebra; matrix theoryMSC 2000 > 65 Numerical analysis 2007.117 Nick Higham 17 September 2008