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301
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 262 (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, 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 ...
Recent computational developments in Krylov subspace methods for linear systems
 NUMER. LINEAR ALGEBRA APPL
, 2007
"... Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are metho ..."
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Cited by 86 (12 self)
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Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are methods specifically tailored to systems with special properties such as special forms of symmetry and those depending on one or more parameters.
An efficient surface intersection algorithm based on the lower dimensional formulation
 ACM TRANSACTIONS ON GRAPHICS
, 1997
"... We present an efficient algorithm to compute the intersection of algebraic and NURBS surfaces. Our approach is based on combining the marching methods with the algebraic formulation. In particular, we propose a matrix representation for the intersection curve and compute it accurately using matrix c ..."
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Cited by 72 (18 self)
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We present an efficient algorithm to compute the intersection of algebraic and NURBS surfaces. Our approach is based on combining the marching methods with the algebraic formulation. In particular, we propose a matrix representation for the intersection curve and compute it accurately using matrix computations. We present algorithms to compute a start point oneach component of the intersection curve (both open and closed components), detect the presence of singularities, and find all the curve branches near the singularity. We also suggest methods to compute the step size during tracing to prevent component jumping. The algorithm runs an order of magnitude faster than previously published robust algorithms. The complexity of the algorithm is output sensitive.
Structured polynomial eigenvalue problems: Good vibrations from good linearizations
 SIAM J. Matrix Anal. Appl
"... Abstract. Many applications give rise to nonlinear eigenvalue problems with an underlying structured matrix polynomial. In this paper several useful classes of structured polynomial (e.g., palindromic, even, odd) are identified and the relationships between them explored. A special class of lineariz ..."
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Cited by 72 (22 self)
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Abstract. Many applications give rise to nonlinear eigenvalue problems with an underlying structured matrix polynomial. In this paper several useful classes of structured polynomial (e.g., palindromic, even, odd) are identified and the relationships between them explored. A special class of linearizations that reflect the structure of these polynomials, and therefore preserve symmetries in their spectra, is introduced and investigated. We analyze the existence and uniqueness of such linearizations, and show how they may be systematically constructed.
Algorithms for Intersecting Parametric and Algebraic Curves I: Simple Intersections
 ACM Transactions on Graphics
, 1995
"... : The problem of computing the intersection of parametric and algebraic curves arises in many applications of computer graphics and geometric and solid modeling. Previous algorithms are based on techniques from elimination theory or subdivision and iteration. The former is however, restricted to low ..."
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Cited by 71 (19 self)
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: The problem of computing the intersection of parametric and algebraic curves arises in many applications of computer graphics and geometric and solid modeling. Previous algorithms are based on techniques from elimination theory or subdivision and iteration. The former is however, restricted to low degree curves. This is mainly due to issues of efficiency and numerical stability. In this paper we use elimination theory and express the resultant of the equations of intersection as a matrix determinant. The matrix itself rather than its symbolic determinant, a polynomial, is used as the representation. The problem of intersection is reduced to computing the eigenvalues and eigenvectors of a numeric matrix. The main advantage of this approach lies in its efficiency and robustness. Moreover, the numerical accuracy of these operations is well understood. For almost all cases we are able to compute accurate answers in 64 bit IEEE floating point arithmetic. Keywords: Intersection, curves, a...
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 64 (8 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 54 (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
Efficient Inverse Kinematics for General 6R Manipulators
 IEEE Transactions on Robotics and Automation
, 1994
"... The inverse kinematics of serial manipulators is a central problem in the automatic control of robot manipulators. The main interest has been in inverse kinematics of a six revolute (6R) jointed manipulator with arbitrary geometry. It has been recently shown that the joints of a general 6R manipula ..."
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Cited by 52 (1 self)
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The inverse kinematics of serial manipulators is a central problem in the automatic control of robot manipulators. The main interest has been in inverse kinematics of a six revolute (6R) jointed manipulator with arbitrary geometry. It has been recently shown that the joints of a general 6R manipulator can orient themselves in 16 different configurations (at most), for a given pose of the endeffector. However, there are no good practical solutions available, which give a level of performance expected of industrial manipulators. In this paper, we present an algorithm and implementation for efficient inverse kinematics for a general 6R manipulator. When stated mathematically, the problem reduces to solving a system of multivariate equations. We make use of the algebraic properties of the system and the symbolic formulation used for reducing the problem to solving a univariate polynomial. However, the polynomial is expressed as a matrix determinant and its roots are computed by reducin...
NLEVP: A Collection of Nonlinear Eigenvalue Problems
, 2010
"... We present a collection of 46 nonlinear eigenvalue problems in the form of a MATLAB toolbox. The collection contains problems from models of reallife applications as well as ones constructed specifically to have particular properties. A classification is given of polynomial eigenvalue problems acco ..."
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Cited by 49 (12 self)
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We present a collection of 46 nonlinear eigenvalue problems in the form of a MATLAB toolbox. The collection contains problems from models of reallife applications as well as ones constructed specifically to have particular properties. A classification is given of polynomial eigenvalue problems according to their structural properties. Identifiers based on these and other properties can be used to extract particular types of problems from the collection. A brief description of each problem is given. NLEVP serves both to illustrate the tremendous variety of applications of nonlinear Eigenvalue problems and to provide representative problems for testing, tuning, and benchmarking of algorithms and codes.