On eigenvalues of matrices dependent on a parameter

Lancaster, Peter

doi:10.1007/bf01386087

Cited by 252 publications

(128 citation statements)

References 5 publications

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“…In this case, the formula for the second derivative of λ max can be found, for example, in [17] and (less explicitly) in [16]. If, on the other hand, the maximum eigenvalue of X has multiplicity r > 1, the maximum eigenvalue function is smooth near X only if it is restricted to the submanifold of S n consisting of matrices whose maximum eigenvalue has multiplicity r. Thus, the key idea for second-order methods is to model the second-order behaviour of the maximum eigenvalue function on such a manifold.…”

Section: A Second-order Methods To Minimize Fmentioning

confidence: 99%

The spectral bundle method with second-order information

Helmberg

Overton

Rendl

2014

Optimization Methods and Software

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The spectral bundle (SB) method was introduced by Helmberg and Rend [A spectral bundle method for semidefinite programming. SIAM J. Optim. 10 (2000), pp. 673-696] to solve a class of eigenvalue optimization problems that is equivalent to the class of semidefinite programs with the constant trace property. We investigate the feasibility and effectiveness of including full or partial second-order information in the SB method, building on work of Overton [On minimizing the maximum eigenvalue of a symmetric matrix. SIAM J. Matrix Anal. Appl. 9(2) (1988), pp. 256-268] and Overton and Womersley [Second derivatives for optimizing eigenvalues of symmetric matrices. SIAM J. Matrix Anal. Appl. 16 (1995), pp. 697-718]. We propose several variations that include second-order information in the SB method and describe efficient implementations. One of these, namely diagonal scaling based on a low-rank approximation of the second-order model for λ max , improves the standard SB method both with respect to accuracy requirements and computation time.

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Section: A Second-order Methods To Minimize Fmentioning

confidence: 99%

The spectral bundle method with second-order information

Helmberg

Overton

Rendl

2014

Optimization Methods and Software

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“…By Lemma 3.2 all eigenvalues in S out are semisimple and hence differentiable with respect to s [7,Theorem 6]. Consider an eigenvalue λ ∈ S out for an arbitrary s. Since λ is real, the corresponding eigenspace of L(s) has a real basis.…”

Section: Eigenvaluesmentioning

confidence: 99%

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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“…For a given ε, we have k (t) = (ε 2 + 1) dλ ε dt − 1 = 0 at the optimum t. We can differentiate both sides with respect to t and solve for dε dt to get (19). The formula for λ ε can be found in [24,25,31], i.e. using…”

Section: Corollary 1 the Derivativesmentioning

confidence: 99%

Regularization using a parameterized trust region subproblem

Grodzevich

Wolkowicz²

2007

Math. Program.

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We present a new method for regularization of ill-conditioned problems, such as those that arise in image restoration or mathematical processing of medical data. The method extends the traditional trust-region subproblem, TRS, approach that makes use of the L-curve maximum curvature criterion, a strategy recently proposed to find a good regularization parameter. We apply a parameterized trust region approach to estimate the region of maximum curvature of the L-curve and find the regularized solution. This exploits the close connections between various parameters used to solve TRS. A MATLAB code for the algorithm is tested and a comparison to the conjugate gradient least squares, CGLS, approach is given and analysed.

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On eigenvalues of matrices dependent on a parameter

Cited by 252 publications

References 5 publications

The spectral bundle method with second-order information

The spectral bundle method with second-order information

Strongly Damped Quadratic Matrix Polynomials

Regularization using a parameterized trust region subproblem

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