Detecting and Solving Hyperbolic Quadratic Eigenvalue Problems

Guo, Chun-Hua and Higham, Nicholas J. and Tisseur, Françoise (2009) Detecting and Solving Hyperbolic Quadratic Eigenvalue Problems. SIAM Journal On Matrix Analysis and Applications., 30 (4). pp. 1593-1613. ISSN 1095-7162

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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: Article
Uncontrolled Keywords: quadratic eigenvalue problem, hyperbolic, overdamped, weakly overdamped, quadratic matrix polynomial, quadratic matrix equation, solvent, cyclic reduction, doubling algorithm
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Nick Higham
Date Deposited: 17 Jan 2009
Last Modified: 20 Oct 2017 14:12

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