Detecting a definite Hermitian pair and a hyperbolic or elliptic quadratic eigenvalue problem, and associated nearness problems

Higham, Nicholas J.; Tisseur, Françoise; Dooren, Paul M. Van

doi:10.1016/s0024-3795(02)00281-1

Cited by 64 publications

(73 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…It has been noted in [13] that testing for hyperbolicity can be reduced to testing for definiteness of a Hermitian pencil of twice the dimension. Here we point out that testing for hyperbolicity can be done more easily.…”

Section: Theorem 2 a Qep With A B C ∈ H N And A > 0 Is Hyperbolic mentioning

confidence: 99%

“…As noted in [13], a hyperbolic problem becomes an overdamped problem if a proper shift is used. Indeed, for any real number θ, The following result about the relationship between hyperbolic QEPs and overdamped QEPs will turn out to be instructive later on.…”

Section: Algorithmmentioning

confidence: 99%

“…We will refer to the solvents X (1) and X (2) as the primary and the secondary solvents. As noted in [13], a hyperbolic QEP becomes an overdamped QEP after a proper shift is applied to the eigenvalue parameter. It is then easy to see that the aforementioned properties for an overdamped QEP also hold for a hyperbolic QEP except that the eigenvalues are not necessarily nonpositive.…”

Section: Introductionmentioning

confidence: 97%

“…Hyperbolic and elliptic QEPs are disjoint classes of QEPs (see [16]). In a recent paper by Higham, Tisseur, and Van Dooren [13], methods for detecting a hyperbolic or elliptic QEP are discussed. They point out that testing for ellipticity can be done straightforwardly by computing the eigenvectors of the QEP, while testing for hyperbolicity is more complicated.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, testing for hyperbolicity is actually half as expensive as testing for ellipticity. Our method is less expensive than the method proposed in [13] since that method needs to compute all 4n eigenvalues of a 2n × 2n QEP with complex matrices (even if the matrices A, B and C are all real). Since hyperbolic QEPs include the class of overdamped QEPs, our method also provides a relatively easy way to check Duffin's overdamping condition [6].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Algorithms for hyperbolic quadratic eigenvalue problems

Guo

Lancaster

2005

Math. Comp.

View full text Add to dashboard Cite

Abstract. We consider the quadratic eigenvalue problem (or the QEP) (λ 2 A + λB + 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 ∈ 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.

show abstract

Section: Theorem 2 a Qep With A B C ∈ H N And A > 0 Is Hyperbolic mentioning

confidence: 99%

Section: Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Algorithms for hyperbolic quadratic eigenvalue problems

Guo

Lancaster

2005

Math. Comp.

View full text Add to dashboard Cite

show abstract

Generalized eigenvalue passivity assessment of descriptor systems with applications to symmetric and singular systems

Knockaert

Ferranti

Dhaene

2012

Int J Numerical Modelling

View full text Add to dashboard Cite

SUMMARYThis paper proposes general eigenvalue-based passivity tests for descriptor state space systems, extending the already available tests for standard state space systems. Generalized eigenvalue tests are proposed to identify passivity violations in the positive-real and bounded-real cases. The important practical case of singular descriptor state space systems is also treated. The proposed passivity tests for descriptor systems can be used in the important case of symmetric (reciprocal) systems, as symmetric systems can always be cast in an explicitly reciprocal descriptor state space format. Also, passivity violation assessment methods are developed in the important case of strictly proper symmetric systems, as in that case the classical test matrices for passivity assessment are undefined. Finally, some pertinent numerical examples are given to demonstrate the usefulness and scope of the different passivity violation assessment tests.

show abstract

Scaling, sensitivity and stability in the numerical solution of quadratic eigenvalue problems

Higham

Mackey

Tisseur

et al. 2007

Numerical Meth Engineering

Self Cite

View full text Add to dashboard Cite

SUMMARYThe most common way of solving the quadratic eigenvalue problem (QEP) (λ 2 M +λD +K)x = 0 is to convert it into a linear problem (λX + Y )z = 0 of twice the dimension and solve the linear problem by the QZ algorithm or a Krylov method. In doing so, it is important to understand the influence of the linearization process on the accuracy and stability of the computed solution. We discuss these issues for three particular linearizations: the standard companion linearization and two linearizations that preserve symmetry in the problem. For illustration we employ a model QEP describing the motion of a beam simply supported at both ends and damped at the midpoint. We show that the above linearizations lead to poor numerical results for the beam problem, but that a two-parameter scaling proposed by Fan, Lin and Van Dooren cures the instabilities. We also show that half of the eigenvalues of the beam QEP are pure imaginary and are eigenvalues of the undamped problem. Our analysis makes use of recently developed theory explaining the sensitivity and stability of linearizations, the main conclusions of which are summarized. As well as arguing that scaling should routinely be used, we give guidance on how to choose a linearization and illustrate the practical value of condition numbers and backward errors.

show abstract

Detecting a definite Hermitian pair and a hyperbolic or elliptic quadratic eigenvalue problem, and associated nearness problems

Cited by 64 publications

References 13 publications

Algorithms for hyperbolic quadratic eigenvalue problems

Algorithms for hyperbolic quadratic eigenvalue problems

Generalized eigenvalue passivity assessment of descriptor systems with applications to symmetric and singular systems

Scaling, sensitivity and stability in the numerical solution of quadratic eigenvalue problems

Contact Info

Product

Resources

About