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37
Symmetric linearizations for matrix polynomials
 SIAM J. MATRIX ANAL. APPL
, 2006
"... A standard way of treating the polynomial eigenvalue problem P(λ)x = 0 is to convert it into an equivalent matrix pencil—a process known as linearization. Two vector spaces of pencils L1(P) and L2(P), and their intersection DL(P), have recently been defined and studied by Mackey, Mackey, Mehl, and M ..."
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Cited by 31 (12 self)
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A standard way of treating the polynomial eigenvalue problem P(λ)x = 0 is to convert it into an equivalent matrix pencil—a process known as linearization. Two vector spaces of pencils L1(P) and L2(P), and their intersection DL(P), have recently been defined and studied by Mackey, Mackey, Mehl, and Mehrmann. The aim of our work is to gain new insight into these spaces and the extent to which their constituent pencils inherit structure from P. For arbitrary polynomials we show that every pencil in DL(P) is block symmetric and we obtain a convenient basis for DL(P) built from block Hankel matrices. This basis is then exploited to prove that the first deg(P) pencils in a sequence constructed by Lancaster in the 1960s generate DL(P). When P is symmetric, we show that the symmetric pencils in L1(P) comprise DL(P), while for Hermitian P the Hermitian pencils in L1(P) form a proper subset of DL(P) that we explicitly characterize. Almost all pencils in each of these subsets are shown to be linearizations. In addition to obtaining new results, this work provides a selfcontained treatment of some of the key properties of DL(P) together with some new, more concise proofs.
NLEVP: A collection of nonlinear eigenvalue problems. Users’ guide
, 2011
"... Reports available from: And by contacting: ..."
More on Pseudospectra for Polynomial Eigenvalue Problems and Applications in Control Theory
, 2002
"... Definitions and characterizations of pseudospectra are given for rectangular matrix polynomials expressed in homogeneous form: P(#,#)= # 0 .It is shown that problems with infinite (pseudo)eigenvalues are elegantly treated in this framework. For such problems stereographic projection onto the Riemann ..."
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Cited by 16 (6 self)
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Definitions and characterizations of pseudospectra are given for rectangular matrix polynomials expressed in homogeneous form: P(#,#)= # 0 .It is shown that problems with infinite (pseudo)eigenvalues are elegantly treated in this framework. For such problems stereographic projection onto the Riemann sphere is shown to provide a convenient way to visualize pseudospectra. Lower bounds for the distance to the nearest nonregular polynomial and the nearest uncontrollable dth order system (with equality for standard statespace systems) are obtained in terms of pseudospectra, showing that pseudospectra are a fundamental tool for reasoning about matrix polynomials in areas such as control theory. How and why to incorporate linear structure into pseudospectra is also discussed by example.
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 12 (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, skewHermitian) and the spectral properties of the problem. We classify the available choices of methods and catalogue available software. Key words. quadratic eigenvalue problem, eigenvalue, eigenvector, matrix, matrix polynomial, secondorder differential equation, 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Vibration analysis of structural systems  Modal superpositio...
Nonvariational approximation of discrete eigenvalues of selfadjoint operators
 IMA J. Numer. Anal
"... Abstract. We establish sufficiency conditions in order to achieve approximation to discrete eigenvalues of selfadjoint operators in the secondorder projection method suggested recently by Levitin and Shargorodsky, [15]. We find explicit estimates for the eigenvalue error and study in detail two co ..."
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Cited by 11 (5 self)
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Abstract. We establish sufficiency conditions in order to achieve approximation to discrete eigenvalues of selfadjoint operators in the secondorder projection method suggested recently by Levitin and Shargorodsky, [15]. We find explicit estimates for the eigenvalue error and study in detail two concrete model examples. Our results show that, unlike the majority of the standard methods, secondorder projection strategies combine nonpollution and approximation at a very high level of generality. 1.
Semiclassical analysis and pseudospectra
, 2004
"... accepted for publication in J. Diff. Eqns. We prove an approximate spectral theorem for nonselfadjoint operators and investigate its applications to second order differential operators in the semiclassical limit. This leads to the construction of a twisted FBI transform. We also investigate the c ..."
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Cited by 8 (1 self)
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accepted for publication in J. Diff. Eqns. We prove an approximate spectral theorem for nonselfadjoint operators and investigate its applications to second order differential operators in the semiclassical limit. This leads to the construction of a twisted FBI transform. We also investigate the connections between pseudospectra and boundary conditions in the semiclassical limit.
Pseudospectra of rectangular matrices
 IMA JOURNAL OF NUMERICAL ANALYSIS (2002) 22, 501–519
, 2002
"... Pseudospectra of rectangular matrices vary continuously with the matrix entries, a feature that eigenvalues of these matrices do not have. Some properties of eigenvalues and pseudospectra of rectangular matrices are explored, and an efficient algorithm for the computation of pseudospectra is propose ..."
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Cited by 7 (1 self)
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Pseudospectra of rectangular matrices vary continuously with the matrix entries, a feature that eigenvalues of these matrices do not have. Some properties of eigenvalues and pseudospectra of rectangular matrices are explored, and an efficient algorithm for the computation of pseudospectra is proposed. Applications are given in (square) eigenvalue computation (Lanczos iteration), square pseudospectra approximation (Arnoldi iteration), control theory (nearest uncontrollable system) and game theory.
Backward Error, Condition Numbers, and Pseudospectrum for the Multiparameter
 Linear Algebra Appl
, 2002
"... We define and evaluate the normwise backward error and condition numbers for the multiparameter eigenvalue problem (MEP). The pseudospectrum for the MEP is defined and characterized. We show that the distance from a right definite MEP to the closest non right definite MEP is related to the smallest ..."
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Cited by 7 (3 self)
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We define and evaluate the normwise backward error and condition numbers for the multiparameter eigenvalue problem (MEP). The pseudospectrum for the MEP is defined and characterized. We show that the distance from a right definite MEP to the closest non right definite MEP is related to the smallest unbounded pseudospectrum. Some numerical results are given. Key words. Multiparameter eigenvalue problem, right definiteness, backward error, condition number, pseudospectrum, nearness problem. AMS subject classifications. 65F15, 15A18, 15A69. 1.
Structured Hölder condition numbers for multiple eigenvalues
, 2006
"... The sensitivity of a multiple eigenvalue of a matrix under perturbations can be measured by its Hölder condition number. Various extensions of this concept are considered. A meaningful notion of structured Hölder condition numbers is introduced and it is shown that many existing results on structure ..."
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Cited by 7 (3 self)
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The sensitivity of a multiple eigenvalue of a matrix under perturbations can be measured by its Hölder condition number. Various extensions of this concept are considered. A meaningful notion of structured Hölder condition numbers is introduced and it is shown that many existing results on structured condition numbers for simple eigenvalues carry over to multiple eigenvalues. The structures investigated in more detail include real, Toeplitz, Hankel, symmetric, skewsymmetric, Hamiltonian, and skewHamiltonian matrices. Furthermore, unstructured and structured Hölder condition numbers for multiple eigenvalues of matrix pencils are introduced. Particular attention is given to symmetric/skewsymmetric, Hermitian and palindromic pencils. It is also shown how matrix polynomial eigenvalue problems can be covered within this framework. 1
Linear DifferentialAlgebraic Equations of HigherOrder and the Regularity or Singularity of Matrix Polynomials
 of Matrix Polynomials, PhD thesis, Institut für Mathematik, Technische Universität
, 2004
"... ..."