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164
QMR: a QuasiMinimal Residual Method for NonHermitian Linear Systems
, 1991
"... ... In this paper, we present a novel BCGlike approach, the quasiminimal residual (QMR) method, which overcomes the problems of BCG. An implementation of QMR based on a lookahead version of the nonsymmetric Lanczos algorithm is proposed. It is shown how BCG iterates can be recovered stably from t ..."
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Cited by 395 (26 self)
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... In this paper, we present a novel BCGlike approach, the quasiminimal residual (QMR) method, which overcomes the problems of BCG. An implementation of QMR based on a lookahead version of the nonsymmetric Lanczos algorithm is proposed. It is shown how BCG iterates can be recovered stably from the QMR process. Some further properties of the QMR approach are given and an error bound is presented. Finally, numerical experiments are reported.
Efficient Linear Circuit Analysis by Pade Approximation via the Lanczos Process,”
 IEEE Trans. ComputerAided Design,
, 1995
"... ..."
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, 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 ...
Krylov Projection Methods For Model Reduction
, 1997
"... This dissertation focuses on efficiently forming reducedorder models for large, linear dynamic systems. Projections onto unions of Krylov subspaces lead to a class of reducedorder models known as rational interpolants. The cornerstone of this dissertation is a collection of theory relating Krylov p ..."
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Cited by 213 (3 self)
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This dissertation focuses on efficiently forming reducedorder models for large, linear dynamic systems. Projections onto unions of Krylov subspaces lead to a class of reducedorder models known as rational interpolants. The cornerstone of this dissertation is a collection of theory relating Krylov projection to rational interpolation. Based on this theoretical framework, three algorithms for model reduction are proposed. The first algorithm, dual rational Arnoldi, is a numerically reliable approach involving orthogonal projection matrices. The second, rational Lanczos, is an efficient generalization of existing Lanczosbased methods. The third, rational power Krylov, avoids orthogonalization and is suited for parallel or approximate computations. The performance of the three algorithms is compared via a combination of theory and examples. Independent of the precise algorithm, a host of supporting tools are also developed to form a complete modelreduction package. Techniques for choosing the matching frequencies, estimating the modeling error, insuring the model's stability, treating multipleinput multipleoutput systems, implementing parallelism, and avoiding a need for exact factors of large matrix pencils are all examined to various degrees.
On Krylov subspace approximations to the matrix exponential operator
 SIAM J. NUMER. ANAL
, 1997
"... Krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper. We derive error bounds via a functional calculus of Arnoldi and Lanczos methods that reduces the study of Krylov subspace approximations of functions of matrices to that of linear systems of equ ..."
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Cited by 183 (6 self)
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Krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper. We derive error bounds via a functional calculus of Arnoldi and Lanczos methods that reduces the study of Krylov subspace approximations of functions of matrices to that of linear systems of equations. As a side result, we obtain error bounds for Galerkintype Krylov methods for linear equations, namely the biconjugate gradient method and the full orthogonalization method. For Krylov approximations to matrix exponentials, we show superlinear error decay from relatively small iteration numbers onwards, depending on the geometry of the numerical range, the spectrum or the pseudospectrum. The convergence to exp(A)v is faster than that of corresponding Krylov methods for the solution of linear equations (IA)x = v, which usually arise in the numerical solution of stiff ordinary differential equations. We therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use Krylov approximations to the exponential function of the Jacobian instead of solving linear or nonlinear systems of equations in every time step.
Krylov Subspace Techniques for ReducedOrder Modeling of Nonlinear Dynamical Systems
 Appl. Numer. Math
, 2002
"... Means of applying Krylov subspace techniques for adaptively extracting accurate reducedorder models of largescale nonlinear dynamical systems is a relatively open problem. There has been much current interest in developing such techniques. We focus on a bilinearization method, which extends Kry ..."
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Cited by 93 (5 self)
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Means of applying Krylov subspace techniques for adaptively extracting accurate reducedorder models of largescale nonlinear dynamical systems is a relatively open problem. There has been much current interest in developing such techniques. We focus on a bilinearization method, which extends Krylov subspace techniques for linear systems. In this approach, the nonlinear system is first approximated by a bilinear system through Carleman bilinearization. Then a reducedorder bilinear system is constructed in such a way that it matches certain number of multimoments corresponding to the first few kernels of the VolterraWiener representation of the bilinear system. It is shown that the twosided Krylov subspace technique matches significant more number of multimoments than the corresponding oneside technique.
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 85 (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.
Asymptotic waveform evaluation via a Lanczos method
 Appl. Math. Lett
, 1994
"... AbstractIn this paper we show that the twosided Lanczos procedure combined with implicit restarts, offers significant advantages over Pad6 approximations used typically for model reduction in circuit simulation. ..."
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Cited by 76 (4 self)
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AbstractIn this paper we show that the twosided Lanczos procedure combined with implicit restarts, offers significant advantages over Pad6 approximations used typically for model reduction in circuit simulation.
An Implementation Of The Qmr Method Based On Coupled TwoTerm Recurrences
, 1992
"... . Recently, the authors have proposed a new Krylov subspace iteration, the quasiminimal residual algorithm (QMR), for solving nonHermitian linear systems. In the original implementation of the QMR method, the Lanczos process with lookahead is used to generate basis vectors for the underlying Kryl ..."
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Cited by 73 (14 self)
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. Recently, the authors have proposed a new Krylov subspace iteration, the quasiminimal residual algorithm (QMR), for solving nonHermitian linear systems. In the original implementation of the QMR method, the Lanczos process with lookahead is used to generate basis vectors for the underlying Krylov subspaces. In the Lanczos algorithm, these basis vectors are computed by means of threeterm recurrences. It has been observed that, in finite precision arithmetic, vector iterations based on threeterm recursions are usually less robust than mathematically equivalent coupled twoterm vector recurrences. This paper presents a lookahead algorithm that constructs the Lanczos basis vectors by means of coupled twoterm recursions. Implementation details are given, and the lookahead strategy is described. A new implementation of the QMR method, based on this coupled twoterm algorithm, is proposed. A simplified version of the QMR algorithm without lookahead is also presented, and the specia...