Results 11  20
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537
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 130 (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 Shifted Block Lanczos Algorithm For Solving Sparse Symmetric Generalized Eigenproblems
, 1994
"... An "industrial strength" algorithm for solving sparse symmetric generalized eigenproblems is described. The algorithm has its foundations in known techniques in solving sparse symmetric eigenproblems, notably the spectral transformation of Ericsson and Ruhe and the block Lanczos algorithm. ..."
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Cited by 106 (7 self)
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An "industrial strength" algorithm for solving sparse symmetric generalized eigenproblems is described. The algorithm has its foundations in known techniques in solving sparse symmetric eigenproblems, notably the spectral transformation of Ericsson and Ruhe and the block Lanczos algorithm. However, the combination of these two techniques is not trivial; there are many pitfalls awaiting the unwary implementor. The focus of this paper is on identifying those pitfalls and avoiding them, leading to a "bombproof" algorithm that can live as a black box eigensolver inside a large applications code. The code that results comprises a robust shift selection strategy and a block Lanczos algorithm that is a novel combination of new techniques and extensions of old techniques.
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 104 (11 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
A JacobiDavidson Iteration Method for Linear Eigenvalue Problems
 SIAM J. Matrix Anal. Appl
, 2000
"... . In this paper we propose a new method for the iterative computation of a few of the extremal eigenvalues of a symmetric matrix and their associated eigenvectors. The method is based on an old and almost unknown method of Jacobi. Jacobi's approach, combined with Davidson's method, leads t ..."
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Cited by 96 (9 self)
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. In this paper we propose a new method for the iterative computation of a few of the extremal eigenvalues of a symmetric matrix and their associated eigenvectors. The method is based on an old and almost unknown method of Jacobi. Jacobi's approach, combined with Davidson's method, leads to a new method that has improved convergence properties and that may be used for general matrices. We also propose a variant of the new method that may be useful for the computation of nonextremal eigenvalues as well. Key words. eigenvalues and eigenvectors, Davidson's method, Jacobi iterations, harmonic Ritz values AMS subject classifications. 65F15, 65N25 PII. S0036144599363084 1. Introduction. Suppose we want to compute one or more eigenvalues and their corresponding eigenvectors of the n n matrix A. Several iterative methods are available: Jacobi's diagonalization method [9], [23], the power method [9], the method of Lanczos [13], [23], Arnoldi's method [1], [26], and Davidson's method [4], ...
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.
Reducedorder modeling of large linear subcircuits via a block Lanczos algorithm
 In Proc. 32nd ACM/IEEE Design Automation Conf
, 1995
"... A method for the e�cient computation of accu� rate reduced�order models of large linear circuits is de� scribed. The method � called MPVL � employs a novel block Lanczos algorithm to compute matrix Pad�e ap� proximations of matrix�valued network transfer func� tions. The reduced�order models � compu ..."
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Cited by 92 (22 self)
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A method for the e�cient computation of accu� rate reduced�order models of large linear circuits is de� scribed. The method � called MPVL � employs a novel block Lanczos algorithm to compute matrix Pad�e ap� proximations of matrix�valued network transfer func� tions. The reduced�order models � computed to the re� quired level of accuracy � are used tospeed up the anal� ysis of circuits containing large linear blocks. The lin� ear blocks are replaced by their reduced�order models� and the resulting smaller circuit can be analyzed with general�purpose simulators � with signi�cant savings in simulation time and � practically � no loss of accuracy. 1
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.
A BlockQMR Algorithm for NonHermitian Linear Systems with Multiple RightHand Sides
, 1997
"... Many applications require the solution of multiple linear systems that have the same coefficient matrix, but differ in their righthand sides. Instead of applying an iterative method to each of these systems individually, it is more efficient to employ a block version of the method that generates it ..."
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Cited by 77 (10 self)
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Many applications require the solution of multiple linear systems that have the same coefficient matrix, but differ in their righthand sides. Instead of applying an iterative method to each of these systems individually, it is more efficient to employ a block version of the method that generates iterates for all the systems simultaneously. In this paper, we propose a block version of Freund and Nachtigal's quasiminimal residual (QMR) method for the iterative solution of nonHermitian linear systems. The blockQMR method uses a novel Lanczostype process for multiple starting vectors, which was recently developed by Aliaga, Boley, Freund, and Hern'andez, to compute suitable basis vectors for the underlying block Krylov subspaces. We describe the basic blockQMR method, and also give important implementation details. In particular, we show how to incorporate deflation to drop converged linear systems, and to delete linearly and almost linearly dependent vectors in the underlying block ...
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.
Scientific Computing on Bulk Synchronous Parallel Architectures
"... We theoretically and experimentally analyse the efficiency with which a wide range of important scientific computations can be performed on bulk synchronous parallel architectures. ..."
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Cited by 75 (16 self)
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We theoretically and experimentally analyse the efficiency with which a wide range of important scientific computations can be performed on bulk synchronous parallel architectures.