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90
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 334 (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.
An Implementation of the LookAhead Lanczos Algorithm for NonHermitian Matrices Part I
, 1991
"... ..."
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 119 (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.
Iterative Solution of Linear Systems
 Acta Numerica
, 1992
"... this paper is as follows. In Section 2, we present some background material on general Krylov subspace methods, of which CGtype algorithms are a special case. We recall the outstanding properties of CG and discuss the issue of optimal extensions of CG to nonHermitian matrices. We also review GMRES ..."
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Cited by 100 (8 self)
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this paper is as follows. In Section 2, we present some background material on general Krylov subspace methods, of which CGtype algorithms are a special case. We recall the outstanding properties of CG and discuss the issue of optimal extensions of CG to nonHermitian matrices. We also review GMRES and related methods, as well as CGlike algorithms for the special case of Hermitian indefinite linear systems. Finally, we briefly discuss the basic idea of preconditioning. In Section 3, we turn to Lanczosbased iterative methods for general nonHermitian linear systems. First, we consider the nonsymmetric Lanczos process, with particular emphasis on the possible breakdowns and potential instabilities in the classical algorithm. Then we describe recent advances in understanding these problems and overcoming them by using lookahead techniques. Moreover, we describe the quasiminimal residual algorithm (QMR) proposed by Freund and Nachtigal (1990), which uses the lookahead Lanczos process to obtain quasioptimal approximate solutions. Next, a survey of transposefree Lanczosbased methods is given. We conclude this section with comments on other related work and some historical remarks. In Section 4, we elaborate on CGNR and CGNE and we point out situations where these approaches are optimal. The general class of Krylov subspace methods also contains parameterdependent algorithms that, unlike CGtype schemes, require explicit information on the spectrum of the coefficient matrix. In Section 5, we discuss recent insights in obtaining appropriate spectral information for parameterdependent Krylov subspace methods. After that, 4 R.W. Freund, G.H. Golub and N.M. Nachtigal
A restarted GMRES method augmented with eigenvectors
 SIAM J. Matrix Anal. Appl
, 1995
"... Abstract. The GMRES method for solving nonsymmetric linear equations is generally used with restarting to reduce storage and orthogonalization costs. Restarting slows down the convergence. However, it is possible to save some important information at the time of the restart. It is proposed that appr ..."
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Cited by 77 (9 self)
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Abstract. The GMRES method for solving nonsymmetric linear equations is generally used with restarting to reduce storage and orthogonalization costs. Restarting slows down the convergence. However, it is possible to save some important information at the time of the restart. It is proposed that approximate eigenvectors corresponding to a few of the smallest eigenvalues be formed and added to the subspace for GMRES. The convergence can be much faster, and the minimum residual property is retained. Key words. GMRES, conjugate gradient, Krylov subspaces, iterative methods, nonsymmetric systems AMS subject classifications. 65F15, 15A18
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 69 (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...
ReducedOrder Modeling Techniques Based on Krylov Subspaces and Their Use in Circuit Simulation
 Applied and Computational Control, Signals, and Circuits
, 1998
"... In recent years, reducedorder modeling techniques based on Krylovsubspace iterations, especially the Lanczos algorithm and the Arnoldi process, have become popular tools to tackle the largescale timeinvariant linear dynamical systems that arise in the simulation of electronic circuits. This pape ..."
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Cited by 53 (10 self)
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In recent years, reducedorder modeling techniques based on Krylovsubspace iterations, especially the Lanczos algorithm and the Arnoldi process, have become popular tools to tackle the largescale timeinvariant linear dynamical systems that arise in the simulation of electronic circuits. This paper reviews the main ideas of reducedorder modeling techniques based on Krylov subspaces and describes the use of reducedorder modeling in circuit simulation. 1 Introduction Krylovsubspace methods, most notably the Lanczos algorithm [81, 82] and the Arnoldi process [5], have long been recognized as powerful tools for largescale matrix computations. Matrices that occur in largescale computations usually have some special structures that allow to compute matrixvector products with such a matrix (or its transpose) much more efficiently than for a dense, unstructured matrix. The most common structure is sparsity, i.e., only few of the matrix entries are nonzero. Computing a matrixvector pr...
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 48 (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.
Krylov space methods on statespace control models
 Circuits, Systems, and Signal Processing
, 1994
"... We give an overview of various Lanczos/Krylov space methods and how they are being used for solving certain problems in Control Systems Theory based on statespace models. The matrix methods used are based on Krylov sequences and are closely related to modern iterative methods for standard matrix pr ..."
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Cited by 43 (4 self)
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We give an overview of various Lanczos/Krylov space methods and how they are being used for solving certain problems in Control Systems Theory based on statespace models. The matrix methods used are based on Krylov sequences and are closely related to modern iterative methods for standard matrix problems such as sets of linear equations and eigenvalue calculations. We show how these methods can be applied to problems in Control Theory such as controllability, observability and model reduction. All the methods are based on the use of statespace models, which may be very sparse and of high dimensionality. For example, we show how one may compute an approximate solution to a Lyapunov equation arising from discretetime linear dynamic system with a large sparse system matrix by the use of the Arnoldi Algorithm, and so obtain an approximate Grammian matrix. This has applications in model reduction. The close relation between the matrix Lanczos algorithm and the algebraic structure of linear control systems is also explored. 1
On restarting the Arnoldi method for large nonsymmetric eigenvalue problems
 Mathematics of Computation
, 1996
"... Abstract. The Arnoldi method computes eigenvalues of large nonsymmetric matrices. Restarting is generally needed to reduce storage requirements and orthogonalization costs. However, restarting slows down the convergence and makes the choice of the new starting vector difficult if several eigenvalues ..."
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Cited by 43 (9 self)
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Abstract. The Arnoldi method computes eigenvalues of large nonsymmetric matrices. Restarting is generally needed to reduce storage requirements and orthogonalization costs. However, restarting slows down the convergence and makes the choice of the new starting vector difficult if several eigenvalues are desired. We analyze several approaches to restarting and show why Sorensen’s implicit QR approach is generally far superior to the others. Ritz vectors are combined in precisely the right way for an effective new starting vector. Also, a new method for restarting Arnoldi is presented. It is mathematically equivalent to the Sorensen approach but has additional uses. 1.