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The Quadratic Eigenvalue Problem
, 2001
"... . We survey the quadratic eigenvalue problem, treating its many applications, its mathematical properties, and a variety of numerical solution techniques. Emphasis is given to exploiting both the structure of the matrices in the problem (dense, sparse, real, complex, Hermitian, skewHermitian) and t ..."
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Cited by 260 (21 self)
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. We survey the quadratic eigenvalue problem, treating its many applications, its mathematical properties, and a variety of numerical solution techniques. Emphasis is given to exploiting both the structure of the matrices in the problem (dense, sparse, real, complex, Hermitian, skew
QUADRATIC EIGENVALUE PROBLEM ∗
, 2005
"... Abstract. We consider numerical methods for the computation of the eigenvalues of the tridiagonal hyperbolic quadratic eigenvalue problem. The eigenvalues are computed as zeros of the characteristic polynomial using the bisection, Laguerre’s method, the Ehrlich–Aberth method, and the Durand–Kerner m ..."
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Abstract. We consider numerical methods for the computation of the eigenvalues of the tridiagonal hyperbolic quadratic eigenvalue problem. The eigenvalues are computed as zeros of the characteristic polynomial using the bisection, Laguerre’s method, the Ehrlich–Aberth method, and the Durand
The Hyperbolic Quadratic Eigenvalue Problem
, 2014
"... The hyperbolic quadratic eigenvalue problem (HQEP) was shown to admit the CourantFischer type minmax principles in 1955 by Duffin and Cauchy type interlacing inequalities in 2010 by Veselić. It can be regarded as the closest analogue (among all kinds of quadratic eigenvalue problems) to the standa ..."
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Cited by 3 (2 self)
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The hyperbolic quadratic eigenvalue problem (HQEP) was shown to admit the CourantFischer type minmax principles in 1955 by Duffin and Cauchy type interlacing inequalities in 2010 by Veselić. It can be regarded as the closest analogue (among all kinds of quadratic eigenvalue problems
A Survey of the Quadratic Eigenvalue Problem
 SIAM Review
, 2000
"... . We survey the quadratic eigenvalue problem, treating its many applications, its mathematical properties, and a variety of numerical solution techniques. Emphasis is given to exploiting both the structure of the matrices in the problem (dense, sparse, real, complex, Hermitian, skewHermitian) and t ..."
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Cited by 14 (0 self)
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. We survey the quadratic eigenvalue problem, treating its many applications, its mathematical properties, and a variety of numerical solution techniques. Emphasis is given to exploiting both the structure of the matrices in the problem (dense, sparse, real, complex, Hermitian, skew
Algorithms for hyperbolic quadratic eigenvalue problems
 Math. Comp
, 2005
"... Abstract. We consider the quadratic eigenvalue problem (or the QEP) (λ2A + 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) forall nonzero x ∈ Cn. We show that a relatively efficient test for hyperbolicity can be obtai ..."
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Cited by 15 (4 self)
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Abstract. We consider the quadratic eigenvalue problem (or the QEP) (λ2A + 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) forall nonzero x ∈ Cn. We show that a relatively efficient test for hyperbolicity can
A Subspace Approximation Method for the Quadratic Eigenvalue Problem
 SIAM J. Matrix Anal. Appl
, 2005
"... Quadratic eigenvalue problems involving large matrices arise frequently in areas such as the vibration analysis of structures, MEMS simulation, and the solution of quadratically constrained least squares problems. The typical approach is to solve the quadratic eigenvalue problem using a mathematical ..."
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Cited by 6 (0 self)
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Quadratic eigenvalue problems involving large matrices arise frequently in areas such as the vibration analysis of structures, MEMS simulation, and the solution of quadratically constrained least squares problems. The typical approach is to solve the quadratic eigenvalue problem using a
Numerical solution of a quadratic eigenvalue problem
 Linear Algebra Appl
"... Dedicated to Peter Lancaster on the occasion of his 75th birthday We consider the quadratic eigenvalue problem (QEP) (λ 2 M + λG + K)x = 0, where M = M T is positive definite, K = K T is negative definite, and G = −G T. The eigenvalues of the QEP occur in quadruplets (λ, λ, −λ, −λ) or in real or pur ..."
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Cited by 10 (4 self)
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Dedicated to Peter Lancaster on the occasion of his 75th birthday We consider the quadratic eigenvalue problem (QEP) (λ 2 M + λG + K)x = 0, where M = M T is positive definite, K = K T is negative definite, and G = −G T. The eigenvalues of the QEP occur in quadruplets (λ, λ, −λ, −λ) or in real
Alternatives To The Rayleigh Quotient For The Quadratic Eigenvalue Problem
 SIAM J. Sc. Comp
, 2001
"... We consider the quadratic eigenvalue problem 2 Ax + Bx + Cx = 0: Suppose that u is an approximation to an eigenvector x (for instance obtained by a subspace method), and that we want to determine an approximation to the corresponding eigenvalue . The usual approach is to impose the Galerkin condi ..."
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Cited by 2 (0 self)
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We consider the quadratic eigenvalue problem 2 Ax + Bx + Cx = 0: Suppose that u is an approximation to an eigenvector x (for instance obtained by a subspace method), and that we want to determine an approximation to the corresponding eigenvalue . The usual approach is to impose the Galerkin
Elliptic and hyperbolic quadratic eigenvalue problems and associated distance problems
, 2002
"... Two important classes of quadratic eigenvalue problems are composed of elliptic and hyperbolic problems. In [Linear Algebra Appl., 351–352 (2002) 455], the distance to the nearest nonhyperbolic or nonelliptic quadratic eigenvalue problem is obtained using a global minimization problem. This paper ..."
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Cited by 3 (0 self)
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Two important classes of quadratic eigenvalue problems are composed of elliptic and hyperbolic problems. In [Linear Algebra Appl., 351–352 (2002) 455], the distance to the nearest nonhyperbolic or nonelliptic quadratic eigenvalue problem is obtained using a global minimization problem. This paper
Waveguide Propagation Modes and Quadratic Eigenvalue Problems
"... Abstract This paper presents a direct approach to determine numerically the propagation modes in waveguides via a finite element method. Given a pulsation ω, a quadratic eigenvalue problem is solved to obtain the propagation constant β. 1 Propagation modes with finite elements Our goal is to obtain ..."
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Abstract This paper presents a direct approach to determine numerically the propagation modes in waveguides via a finite element method. Given a pulsation ω, a quadratic eigenvalue problem is solved to obtain the propagation constant β. 1 Propagation modes with finite elements Our goal
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