On Eigenvector Bounds

Rump, Siegfried M.; Zemke, Jens-Peter M.

doi:10.1023/b:bitn.0000009941.51707.26

Cited by 19 publications

(12 citation statements)

References 7 publications

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“…It is also possible to perform a rigorous numerical test for positive-definiteness of H k using properly rounded IEEE machine arithmetic and MATLAB's (sparse) Cholesky decomposition of a numerically reconditioned H k [36]. We have used the code described in Ref.…”

Section: Verification Of Second-order Jammingmentioning

confidence: 99%

Underconstrained jammed packings of nonspherical hard particles: Ellipses and ellipsoids

et al. 2007

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Continuing on recent computational and experimental work on jammed packings of hard ellipsoids [Donev et al., Science, vol. 303,[990][991][992][993] we consider jamming in packings of smooth strictly convex nonspherical hard particles. We explain why the isocounting conjecture, which states that for large disordered jammed packings the average contact number per particle is twice the number of degrees of freedom per particle (Z = 2d f ), does not apply to nonspherical particles. We develop first-and second-order conditions for jamming, and demonstrate that packings of nonspherical particles can be jammed even though they are hypoconstrained (Z < 2d f ). We apply an algorithm using these conditions to computer-generated hypoconstrained ellipsoid and ellipse packings and demonstrate that our algorithm does produce jammed packings, even close to the sphere point. We also consider packings that are nearly jammed and draw connections to packings of deformable (but stiff) particles. Finally, we consider the jamming conditions for nearly spherical particles and explain quantitatively the behavior we observe in the vicinity of the sphere point.

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Section: Verification Of Second-order Jammingmentioning

confidence: 99%

Underconstrained jammed packings of nonspherical hard particles: Ellipses and ellipsoids

et al. 2007

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“…Also note that an inclusion is only possible if the geometric multiplicity of all included eigenvalues is 1. The reason is again, as for multiple roots of polynomials, that the problem becomes ill-posed for geometric multiplicity greater that one [21].…”

Section: Double Eigenvaluesmentioning

confidence: 99%

“…is proved to be n, and it is easy to see [21] that the eigenvalue λ must be of geometric multiplicity 1. Computational tests show that for dimensions over n = 200 of the matrix inclusions deteriorate.…”

Section: Double Eigenvaluesmentioning

confidence: 99%

Verified error bounds for multiple roots of systems of nonlinear equations

Rump

Graillat

2009

Numer Algor

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It is well known that it is an ill-posed problem to decide whether a function has a multiple root. Even for a univariate polynomial an arbitrary small perturbation of a polynomial coefficient may change the answer from yes to no. Let a system of nonlinear equations be given. In this paper we describe an algorithm for computing verified and narrow error bounds with the property that a slightly perturbed system is proved to have a double root within the computed bounds. For a univariate nonlinear function f we give a similar method also for a multiple root. A narrow error bound for the perturbation is computed as well. Computational results for systems with up to 1000 unknowns demonstrate the performance of the methods.

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“…Most known results treat the special case Ax = λ x where A is an interval matrix (its elements are independent intervals), e.g. [2], [16], [18]. Techniques for finding bounds on the eigenvalues for the still more particular case where A in a symmetric interval matrix can be found in [3] to [5].…”

Section: Problem Statementmentioning

confidence: 99%

Outer Interval Solution of the Eigenvalue Problem under General Form Parametric Dependencies

Kolev

2006

Reliable Comput

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The paper addresses the problem of determining an outer interval solution of the parametric eigenvalue problem A(p)x = λ x, A( p) ∈ R n×n for the general case where the matrix elements aij ( p) are continuous nonlinear functions of the parameter vector p, p belonging to the interval vector p. A method for computing an interval enclosure of each eigenpair (λµ, x (µ) ), µ = 1, …, n, is suggested for the case where λµ is a simple eigenvalue. It is based on the use of an affine interval approximation of aij( p) in p and reduces, essentially, to setting up and solving a real system of n or 2n incomplete quadratic equations for each real or complex eigenvalue, respectively.

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On Eigenvector Bounds

Cited by 19 publications

References 7 publications

Underconstrained jammed packings of nonspherical hard particles: Ellipses and ellipsoids

Underconstrained jammed packings of nonspherical hard particles: Ellipses and ellipsoids

Verified error bounds for multiple roots of systems of nonlinear equations

Outer Interval Solution of the Eigenvalue Problem under General Form Parametric Dependencies

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