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89
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 247 (20 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 ...
Spectral Compression of Mesh Geometry
, 2000
"... We show how spectral methods may be applied to 3D mesh data to obtain compact representations. This is achieved by projecting the mesh geometry onto an orthonormal basis derived from the mesh topology. To reduce complexity, the mesh is partitioned into a number of balanced submeshes with minimal int ..."
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Cited by 232 (6 self)
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We show how spectral methods may be applied to 3D mesh data to obtain compact representations. This is achieved by projecting the mesh geometry onto an orthonormal basis derived from the mesh topology. To reduce complexity, the mesh is partitioned into a number of balanced submeshes with minimal interaction, each of which are compressed independently. Our methods may be used for compression and progressive transmission of 3D content, and are shown to be vastly superior to existing methods using spatial techniques, if slight loss can be tolerated.
Sets of matrices all infinite products of which converge. Linear Algebra and its Applications
, 1992
"... An infinite product IIT = lMi of matrices converges (on the right) if limi _ _ M,... Mi exists. A set Z = (Ai: i> l} of n X n matrices is called an RCP set (rightconvergent product set) if all infinite products with each element drawn from Z converge. Such sets of matrices arise in constructing ..."
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Cited by 117 (2 self)
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An infinite product IIT = lMi of matrices converges (on the right) if limi _ _ M,... Mi exists. A set Z = (Ai: i> l} of n X n matrices is called an RCP set (rightconvergent product set) if all infinite products with each element drawn from Z converge. Such sets of matrices arise in constructing selfsimilar objects like von Koch’s snowflake curve, in various interpolation schemes, in constructing wavelets of compact support, and in studying nonhomogeneous Markov chains. This paper gives necessary conditions and also some sufficient conditions for a set X to be an RCP set. These are conditions on the eigenvalues and left eigenspaces of matrices in 2 and finite products of these matrices. Necessary and sufficient conditions are given for a finite set Z to be an RCP set having a limit function M,(d) = rIT = lAd,, where d = (d,,., d,,..>, which is a continuous function on the space of all sequences d with the sequence topology. Finite RCP sets of columnstochastic matrices are completely characterized. Some results are given on the problem of algorithmically
Structured Pseudospectra For Polynomial Eigenvalue Problems With Applications
 SIAM J. MATRIX ANAL. APPL
, 2001
"... Pseudospectra associated with the standard and generalized eigenvalue problems have been widely investigated in recent years. We extend the usual definitions in two respects, by treating the polynomial eigenvalue problem and by allowing structured perturbations of a type arising in control theory. W ..."
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Cited by 62 (10 self)
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Pseudospectra associated with the standard and generalized eigenvalue problems have been widely investigated in recent years. We extend the usual definitions in two respects, by treating the polynomial eigenvalue problem and by allowing structured perturbations of a type arising in control theory. We explore connections between structured pseudospectra, structured backward errors, and structured stability radii. Two main approaches for computing pseudospectra are described. One is based on a transfer function and employs a generalized Schur decomposition of the companion form pencil. The other, specific to quadratic polynomials, finds a solvent of the associated quadratic matrLx equation and thereby factorizes the quadratic hmatrLx. Possible approaches for large, sparse problems are also outlined. A collection of examples from vibrating systems, control theory, acoustics, and fluid mechanics is given to illustrate the techniques.
The conditioning of linearizations of matrix polynomials
 Manchester Institute for Mathematical Sciences, The University of Manchester
, 2005
"... Abstract. The standard way of solving the polynomial eigenvalue problem of degree m in n×n matrices is to “linearize ” to a pencil in mn×mn matrices and solve the generalized eigenvalue problem. For a given polynomial, P, infinitely many linearizations exist and they can have widely varying eigenva ..."
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Cited by 56 (21 self)
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Abstract. The standard way of solving the polynomial eigenvalue problem of degree m in n×n matrices is to “linearize ” to a pencil in mn×mn matrices and solve the generalized eigenvalue problem. For a given polynomial, P, infinitely many linearizations exist and they can have widely varying eigenvalue condition numbers. We investigate the conditioning of linearizations from a vector space DL(P) of pencils recently identified and studied by Mackey, Mackey, Mehl, and Mehrmann. We look for the best conditioned linearization and compare the conditioning with that of the original polynomial. Two particular pencils are shown always to be almost optimal over linearizations in DL(P) for eigenvalues of modulus greater than or less than 1, respectively, provided that the problem is not too badly scaled and that the pencils are linearizations. Moreover, under this scaling assumption, these pencils are shown to be about as well conditioned as the original polynomial. For quadratic eigenvalue problems that are not too heavily damped, a simple scaling is shown to convert the problem to one that is well scaled. We also analyze the eigenvalue conditioning of the widely used first and second companion linearizations. The conditioning of the first companion linearization relative to that of P is shown to depend on the coefficient matrix norms, the eigenvalue, and the left eigenvectors of the linearization and of P. The companion form is found to be potentially much more
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 54 (18 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.
Numerical Computation of Deflating Subspaces of SkewHamiltonian/Hamiltonian Pencils
 SIAM J. Matrix Anal. Appl
, 2002
"... . We discuss the numerical solution of structured generalized eigenvalue problems that arise from linearquadratic optimal control problems, H1 optimization, multibody systems, and many other areas of applied mathematics, physics, and chemistry. The classical approach for these problems requires com ..."
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Cited by 47 (27 self)
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. We discuss the numerical solution of structured generalized eigenvalue problems that arise from linearquadratic optimal control problems, H1 optimization, multibody systems, and many other areas of applied mathematics, physics, and chemistry. The classical approach for these problems requires computing invariant and deating subspaces of matrices and matrix pencils with Hamiltonian and/or skewHamiltonian structure. We extend the recently developed methods for Hamiltonian matrices to the general case of skewHamiltonian/Hamiltonian pencils. The algorithms circumvent problems with skewHamiltonian/Hamiltonian matrix pencils that lack structured Schur forms by embedding them into matrix pencils that always admit a structured Schur form. The rounding error analysis of the resulting algorithms is favorable. For the embedded matrix pencils, the algorithms use structure preserving unitary matrix computations and are strongly backwards stable, i.e., they compute the exact structured Schur form of a nearby matrix pencil with the same structure. Keywords. eigenvalue problem, deating subspace, Hamiltonian matrix, skewHamiltonian matrix, skewHamiltonian/Hamiltonian matrix pencil. AMS subject classication. 49N10, 65F15, 93B40, 93B36. 1.
Analysis and performance of some basic spacetime architectures
 Selected Areas in Communications, IEEE Journal on
, 2003
"... Abstract—In this paper, we discuss some of the most basic architectural superstructures for wireless links with multiple antennas: at the transmit site and at the receive site. Toward leveraging the gains of the last half century of coding theory, we emphasize those structures that can be composed u ..."
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Cited by 45 (1 self)
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Abstract—In this paper, we discuss some of the most basic architectural superstructures for wireless links with multiple antennas: at the transmit site and at the receive site. Toward leveraging the gains of the last half century of coding theory, we emphasize those structures that can be composed using spatially one dimensional coders and decoders. These structures are investigated primarily under a probability of outage constraint. The random matrix channel is assumed to hold steady for such a large number ofdimensional vector symbol transmission times, that an infinite time horizon Shannon analysis provides useful insights. The resulting extraordinary capacities are contrasted for architectures that differ in the way that they manage selfinterference in the presence of additive receiver noise. A universally optimal architecture with a diagonal space–time layering is treated, as is an architecture with horizontal space–time layering and an architecture with a single outer code. Some capacity asymptotes for large numbers of antennas are also included. Some results for frequency selective channels are presented: It is only necessary to feedback rates, one per transmit antenna, to attain capacity. Also, capacity of an () link is, in a certain sense, invariant with respect to signaling format. Index Terms—Multiple antennas, space–time processing, wireless communications.
Backward error of polynomial eigenproblems solved by linearization
 Manchester Institute for Mathematical Sciences, The University of Manchester
, 2006
"... Abstract. The most widely used approach for solving the polynomial eigenvalue problem P(λ)x = ��m i=0 λi � Ai x =0inn × n matrices Ai is to linearize to produce a larger order pencil L(λ) =λX + Y, whose eigensystem is then found by any method for generalized eigenproblems. For a given polynomial P, ..."
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Cited by 44 (11 self)
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Abstract. The most widely used approach for solving the polynomial eigenvalue problem P(λ)x = ��m i=0 λi � Ai x =0inn × n matrices Ai is to linearize to produce a larger order pencil L(λ) =λX + Y, whose eigensystem is then found by any method for generalized eigenproblems. For a given polynomial P, infinitely many linearizations L exist and approximate eigenpairs of P computed via linearization can have widely varying backward errors. We show that if a certain onesided factorization relating L to P can be found then a simple formula permits recovery of right eigenvectors of P from those of L, and the backward error of an approximate eigenpair of P can be bounded in terms of the backward error for the corresponding approximate eigenpair of L. A similar factorization has the same implications for left eigenvectors. We use this technique to derive backward error bounds depending only on the norms of the Ai for the companion pencils and for the vector space DL(P) of pencils recently identified by Mackey, Mackey, Mehl, and Mehrmann. In all cases, sufficient conditions are identified for an optimal backward error for P. These results are shown to be entirely consistent with those of Higham, Mackey, and Tisseur on the conditioning of linearizations of P. Other contributions of this work are a block scaling of the companion pencils