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75
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 156 (17 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, skewHermitian) and the spectral properties of the problem. We classify numerical methods and catalogue available software. Key words. quadratic eigenvalue problem, eigenvalue, eigenvector, matrix, matrix polynomial, secondorder differential equation, vibration, Millennium footbridge, overdamped system, gyroscopic system, linearization, backward error, pseudospectrum, condition number, Krylov methods, Arnoldi method, Lanczos method, JacobiDavidson method AMS subject classifications. 65F30 Contents 1 Introduction 2 2 Applications of QEPs 4 2.1 Secondorder differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Vibration analysis of structural systems ...
Detecting a Definite Hermitian Pair and a Hyperbolic or Elliptic Quadratic Eigenvalue Problem, and Associated Nearness Problems
, 2001
"... An important class of generalized eigenvalue problems Ax = Bx is those in which A and B are Hermitian and some real linear combination of them is definite. For the quadratic eigenvalue problem (QEP) ( 2 A+B+C)x = 0 with Hermitian A, B and C and positive denite A, particular interest focuses on pr ..."
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Cited by 27 (11 self)
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An important class of generalized eigenvalue problems Ax = Bx is those in which A and B are Hermitian and some real linear combination of them is definite. For the quadratic eigenvalue problem (QEP) ( 2 A+B+C)x = 0 with Hermitian A, B and C and positive denite A, particular interest focuses on problems in which (x Bx) 2 4(x Ax)(x Cx) is onesigned for all nonzero xfor the positive sign these problems are called hyperbolic and for the negative sign elliptic. The important class of overdamped problems arising in mechanics is a subclass of the hyperbolic problems. For each of these classes of generalized and quadratic eigenvalue problems we show how to check that a putative member has the required properties and we derive the distance to the nearest problem outside the class. For definite pairs (A; B) the distance is the Crawford number, and we derive bisection and level set algorithms both for testing its positivity and for computing it. Testing hyperbolicity of a QEP is shown to reduce to testing a related pair for deniteness. The distance to the nearest nonhyperbolic or nonelliptic nn QEP is shown to be the solution of a global minimization problem with n 1 degrees of freedom. Numerical results are given to illustrate the theory and algorithms.
Nonselfadjoint differential operators
, 2002
"... We describe methods which have been used to analyze the spectrum of nonselfadjoint differential operators, emphasizing the differences from the selfadjoint theory. We find that even in cases in which the eigenfunctions can be determined explicitly, they often do not form a basis; this is closely ..."
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Cited by 25 (6 self)
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We describe methods which have been used to analyze the spectrum of nonselfadjoint differential operators, emphasizing the differences from the selfadjoint theory. We find that even in cases in which the eigenfunctions can be determined explicitly, they often do not form a basis; this is closely related to a high degree of instability of the eigenvalues under small perturbations of the operator.
NLEVP: A collection of nonlinear eigenvalue problems. Users’ guide
, 2011
"... Reports available from: And by contacting: ..."
Spectral Asymmetry, Zeta Functions and the Noncommutative Residue
"... Abstract. In this paper, motivated by an approach developed by Wodzicki, we look at the spectral asymmetry of elliptic ΨDO’s in terms of theirs zeta functions. First, using asymmetry formulas of Wodzicki we study the spectral asymmetry of odd elliptic ΨDO’s and of geometric Dirac operators. In parti ..."
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Cited by 18 (7 self)
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Abstract. In this paper, motivated by an approach developed by Wodzicki, we look at the spectral asymmetry of elliptic ΨDO’s in terms of theirs zeta functions. First, using asymmetry formulas of Wodzicki we study the spectral asymmetry of odd elliptic ΨDO’s and of geometric Dirac operators. In particular, we show that the eta function of a selfadjoint elliptic odd ΨDO is regular at every integer point when the dimension and the order have opposite parities (this generalizes a well known result of BransonGilkey for Dirac operators), and we relate the spectral asymmetry of a Dirac operator on a Clifford bundle to the Riemmanian geometric data, which yields a new spectral interpretation of the Einstein action from gravity. We also obtain a large class of examples of elliptic ΨDO’s for which the regular values at the origin of the (local) zeta functions can easily be seen to be independent of the spectral cut. On the other hand, we simplify the proofs of two wellknown and difficult results of Wodzicki: (i) The independence with respect to the spectral cut of the regular value at the origin of the zeta function of an elliptic ΨDO; (ii) The vanishing of the noncommutative residue of a zero’th order ΨDO projector. These results were proved by Wodzicki using a quite difficult and involved characterization of local invariants of spectral asymmetry, which we can bypass here. Finally, in an appendix we give a new proof of the aforementioned asymmetry formulas of Wodzicki. 1.
Detecting and solving hyperbolic quadratic eigenvalue problems
, 2007
"... Reports available from: And by contacting: ..."
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 11 (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 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. Wealsopresentanewmeasureforthedegreeofhyperbolicityofa hyperbolic QEP. 1.
Refined Analytic Torsion as an Element of the Determinant Line
, 2005
"... We construct a canonical element, called the refined analytic torsion, of the determinant line of the cohomology of a closed oriented odddimensional manifold M with coefficients in a flat complex vector bundle E. We compute the RaySinger norm of the refined analytic torsion. In particular, if th ..."
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Cited by 10 (0 self)
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We construct a canonical element, called the refined analytic torsion, of the determinant line of the cohomology of a closed oriented odddimensional manifold M with coefficients in a flat complex vector bundle E. We compute the RaySinger norm of the refined analytic torsion. In particular, if there exists a flat Hermitian metric on E, we show that this norm is equal to 1. We prove a duality theorem, establishing a relationship between the refined analytic torsions corresponding to a flat connection and its dual.
REFINED ANALYTIC TORSION
"... Given an acyclic representation α of the fundamental group of a compact oriented odddimensional manifold, which is close enough to an acyclic unitary representation, we define a refinement Tα of the RaySinger torsion associated to α, which can be viewed as the analytic counterpart of the refined c ..."
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Cited by 9 (4 self)
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Given an acyclic representation α of the fundamental group of a compact oriented odddimensional manifold, which is close enough to an acyclic unitary representation, we define a refinement Tα of the RaySinger torsion associated to α, which can be viewed as the analytic counterpart of the refined combinatorial torsion introduced by Turaev. Tα is equal to the graded determinant of the odd signature operator up to a correction term, the metric anomaly, needed to make it independent of the choice of the Riemannian metric. Tα is a holomorphic function on the space of such representations of the fundamental group. When α is a unitary representation, the absolute value of Tα is equal to the RaySinger torsion and the phase of Tα is proportional to the ηinvariant of the odd signature operator. The fact that the RaySinger torsion and the ηinvariant can be combined into one holomorphic function allows one to use methods of complex analysis to study both invariants. In particular, using these methods we compute the quotient of the refined analytic torsion and Turaev’s refinement of the combinatorial torsion generalizing in this way the classical CheegerMüller theorem. As an application, we extend and improve a result of Farber about the relationship between the FarberTuraev absolute torsion and the ηinvariant. As part of our construction of Tα we prove several new results about determinants and ηinvariants of non selfadjoint elliptic operators. 1.