An Algorithm for Quadratic Eigenproblems with Low Rank Damping

Taslaman, Leo

doi:10.1137/140969099

Cited by 5 publications

(12 citation statements)

References 20 publications

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“…It is still an open problem how to efficiently exploit the low‐rank property in eigenvalue computation. Recently, in , an algorithm was proposed to compute all eigenpairs of the QEP with low‐rank damping. However, because of the computational complexity, it is not designed for solving large scale problems.…”

Section: Discussionmentioning

confidence: 99%

A Padé approximate linearization algorithm for solving the quadratic eigenvalue problem with low‐rank damping

Ding

Huang

Bai

et al. 2015

Numerical Meth Engineering

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SummaryThe low‐rank damping term appears commonly in quadratic eigenvalue problems arising from physical simulations. To exploit the low‐rank damping property, we propose a Padé approximate linearization (PAL) algorithm. The advantage of the PAL algorithm is that the dimension of the resulting linear eigenvalue problem is only n + ℓm, which is generally substantially smaller than the dimension 2n of the linear eigenvalue problem produced by a direct linearization approach, where n is the dimension of the quadratic eigenvalue problem, and ℓ and m are the rank of the damping matrix and the order of a Padé approximant, respectively. Numerical examples show that by exploiting the low‐rank damping property, the PAL algorithm runs 33–47% faster than the direct linearization approach for solving modest size quadratic eigenvalue problems. Copyright © 2015 John Wiley & Sons, Ltd.

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Section: Discussionmentioning

confidence: 99%

A Padé approximate linearization algorithm for solving the quadratic eigenvalue problem with low‐rank damping

Ding

Huang

Bai

et al. 2015

Numerical Meth Engineering

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“…In particular we have proved a new eigenvalue location result that extends some of Lancaster's early work [8], and furthermore, shown how the eigenvalues move as the damping gets sufficiently strong. In [17], the author used the undamped eigenvalues (which are much easier to compute than the damped eigenvalues) as starting points for an Ehrlich-Aberth iteration that targets the damped eigenvalues. Numerical experiments showed that this worked well, even when the damping was strong.…”

Section: Discussionmentioning

confidence: 99%

“…We then created a strongly damped version of the same problem by multiplying the damping matrix by 10 10 . We used the algorithm described in [17] to compute all eigenvalues of both problems. The eigenvalues were all computed with relative backward errors [18] that were smaller than 10 −14 .…”

Section: (λ)Q(λ) Since P(−λ)q(−λ) = −P(λ)q(λ)mentioning

confidence: 99%

“…Example 4.4. As in Example 3.6, we created several versions of the 100 × 100 damped beam problem by multiplying the original damping matrix by a parameter s. We then used the algorithm in [17] to solve these eigenproblems for a sequence of increasing values of s. The largest backward error of all eigenpairs computed was approximately 2 × 10 −14 . We then used the technique explained in the proof of Proposition 2.5 to compute the normalized eigenvectors y 1 , y 2 , .…”

Section: Eigenspacesmentioning

confidence: 99%

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Strongly Damped Quadratic Matrix Polynomials

Taslaman¹

2015

SIAM J. Matrix Anal. & Appl.

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Abstract. We study the eigenvalues and eigenspaces of the quadratic matrix polynomial Mλ 2 + sDλ + K as s → ∞, where M and K are symmetric positive definite and D is symmetric positive semidefinite. This work is motivated by its application to modal analysis of finite element models with strong linear damping. Our results yield a mathematical explanation of why too strong damping may lead to practically undamped modes such that all nodes in the model vibrate essentially in phase.Key words. quadratic eigenvalue problem, principal angles, canonical angles, matrix polynomial, viscous damping, discrete damper, vibrating system AMS subject classifications. 15A18, 15A22, 65F15, 70J10, 70J30, 70J50 DOI. 10.1137/140959390 1. Introduction. A way to prevent a structure from vibrating violently is to incorporate viscous dampers into the design. A viscous damper is a device that resists motion by producing a force proportional to the velocity of a piston relative to its housing (see Figure 1) raised to a power α. In this paper we consider linear damping, which corresponds to dampers with α = 1. This value of α is the default for certain product lines of seismic dampers [1]. The resisting force produced by a viscous damper arises when fluid, trapped in a cylinder, is forced through small holes.By adjusting the size of these holes, we can make the damper stronger. But stronger is not necessarily better: if a damper is too strong, it resembles a rigid component and hence has little purpose. This suggests that a structure with only very strong dampers should be quite similar to a structure without dampers. The goal of this paper is to investigate this phenomenon more rigorously for discretized structures. We will do this by studying the eigenvalues and eigenspaces of a related quadratic matrix polynomial.Consider a finite element model of a structure with r viscous dampers. If the model vibrates freely, the displacements of its nodes are given by the solutions to the equations of motion:

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“…, p d [λ]) and solves det(P (λ)) = 0 via the polynomial eigenvalue problem P (λ)v = 0. There are various algorithms for solving P (λ)v = 0 including linearization [22,31,43], the Ehrlich-Aberth method [5,21,41], and contour integration [2]. However, regardless of how the polynomial eigenvalue problem is solved in finite precision, the hidden variable resultant method based on the Cayley or the Sylvester matrix is numerically unstable.…”

Section: Matrix Polynomialsmentioning

confidence: 99%

Numerical Instability of Resultant Methods for Multidimensional Rootfinding

Noferini¹,

Townsend²

2016

SIAM J. Numer. Anal.

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Hidden-variable resultant methods are a class of algorithms for solving multidimensional polynomial rootfinding problems. In two dimensions, when significant care is taken, they are competitive practical rootfinders. However, in higher dimensions they are known to miss zeros, calculate roots to low precision, and introduce spurious solutions. We show that the hidden variable resultant method based on the Cayley (Dixon or Bézout) matrix is inherently and spectacularly numerically unstable by a factor that grows exponentially with the dimension. We also show that the Sylvester matrix for solving bivariate polynomial systems can square the condition number of the problem. In other words, two popular hidden variable resultant methods are numerically unstable, and this mathematically explains the difficulties that are frequently reported by practitioners. Regardless of how the constructed polynomial eigenvalue problem is solved, severe numerical difficulties will be present. Along the way, we prove that the Cayley resultant is a generalization of Cramer's rule for solving linear systems and generalize Clenshaw's algorithm to an evaluation scheme for polynomials expressed in a degree-graded polynomial basis.

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An Algorithm for Quadratic Eigenproblems with Low Rank Damping

Cited by 5 publications

References 20 publications

A Padé approximate linearization algorithm for solving the quadratic eigenvalue problem with low‐rank damping

A Padé approximate linearization algorithm for solving the quadratic eigenvalue problem with low‐rank damping

Strongly Damped Quadratic Matrix Polynomials

Numerical Instability of Resultant Methods for Multidimensional Rootfinding

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