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23
ABLE: an adaptive block Lanczos method for nonHermitian eigenvalue problems
 SIAM Journal on Matrix Analysis and Applications
, 1999
"... Abstract. This work presents an adaptive block Lanczos method for largescale nonHermitian Eigenvalue problems (henceforth the ABLE method). The ABLE method is a block version of the nonHermitian Lanczos algorithm. There are three innovations. First, an adaptive blocksize scheme cures (near) break ..."
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Cited by 52 (5 self)
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Abstract. This work presents an adaptive block Lanczos method for largescale nonHermitian Eigenvalue problems (henceforth the ABLE method). The ABLE method is a block version of the nonHermitian Lanczos algorithm. There are three innovations. First, an adaptive blocksize scheme cures (near) breakdown and adapts the blocksize to the order of multiple or clustered eigenvalues. Second, stopping criteria are developed that exploit the semiquadratic convergence property of the method. Third, a wellknown technique from the Hermitian Lanczos algorithm is generalized to monitor the loss of biorthogonality and maintain semibiorthogonality among the computed Lanczos vectors. Each innovation is theoretically justified. Academic model problems and real application problems are solved to demonstrate the numerical behaviors of the method. Key words. method nonHermitian matrices, eigenvalue problem, spectral transformation, Lanczos AMS subject classifications. 65F15, 65F10 PII. S0895479897317806
Lanczostype solvers for nonsymmetric linear systems of equations
 Acta Numer
, 1997
"... Among the iterative methods for solving large linear systems with a sparse (or, possibly, structured) nonsymmetric matrix, those that are based on the Lanczos process feature short recurrences for the generation of the Krylov space. This means low cost and low memory requirement. This review article ..."
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Cited by 40 (11 self)
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Among the iterative methods for solving large linear systems with a sparse (or, possibly, structured) nonsymmetric matrix, those that are based on the Lanczos process feature short recurrences for the generation of the Krylov space. This means low cost and low memory requirement. This review article introduces the reader not only to the basic forms of the Lanczos process and some of the related theory, but also describes in detail a number of solvers that are based on it, including those that are considered to be the most efficient ones. Possible breakdowns of the algorithms and ways to cure them by lookahead are also discussed. www.DownloadPaper.ir
QMRBased Projection Techniques for the Solution of NonHermitian Systems with Multiple RightHand Sides
, 2001
"... . In this work we consider the simultaneous solution of large linear systems of the form Ax (j) = b (j) ; j = 1; : : : ; K where A is sparse and nonHermitian. We describe singleseed and blockseed projection approaches to these multiple righthand side problems that are based on the QMR and bl ..."
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Cited by 21 (1 self)
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. In this work we consider the simultaneous solution of large linear systems of the form Ax (j) = b (j) ; j = 1; : : : ; K where A is sparse and nonHermitian. We describe singleseed and blockseed projection approaches to these multiple righthand side problems that are based on the QMR and block QMR algorithms, respectively. We use (block) QMR to solve the (block) seed system and generate the relevant biorthogonal subspaces. Approximate solutions to the nonseed systems are simultaneously generated by minimizing their appropriately projected (block) residuals. After the initial (block) seed has converged, the process is repeated by choosing a new (block) seed from among the remaining nonconverged systems and using the previously generated approximate solutions as initial guesses for the new seed and nonseed systems. We give theory for the singleseed case that helps explain the convergence behavior under certain conditions. Implementation details for both the singleseed and b...
A Parallel Version of the Unsymmetric Lanczos Algorithm and its Application to QMR
, 1996
"... A new version of the unsymmetric Lanczos algorithm without lookahead is described combining elements of numerical stability and parallel algorithm design. Firstly, stability is obtained by a coupled twoterm procedure that generates Lanczos vectors scaled to unit length. Secondly, the algorithm is ..."
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Cited by 16 (3 self)
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A new version of the unsymmetric Lanczos algorithm without lookahead is described combining elements of numerical stability and parallel algorithm design. Firstly, stability is obtained by a coupled twoterm procedure that generates Lanczos vectors scaled to unit length. Secondly, the algorithm is derived by making all inner products of a single iteration step independent such that global synchronization on parallel distributed memory computers is reduced. Among the algorithms using the Lanczos process as a major component, the quasiminimal residual (QMR) method for the solution of systems of linear equations is illustrated by an elegant derivation. The resulting QMR algorithm maintains the favorable properties of the Lanczos algorithm while not increasing computational costs as compared with its corresponding original version.
Analysis of the finite precision BiConjugate Gradient algorithm for nonsymmetric linear systems
 Math. Comp
, 1995
"... Abstract. In this paper we analyze the biconjugate gradient algorithm in finite precision arithmetic, and suggest reasons for its often observed robustness. By using a tridiagonal structure, which is preserved by the finite precision biconjugate gradient iteration, we are able to bound its residua ..."
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Cited by 13 (4 self)
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Abstract. In this paper we analyze the biconjugate gradient algorithm in finite precision arithmetic, and suggest reasons for its often observed robustness. By using a tridiagonal structure, which is preserved by the finite precision biconjugate gradient iteration, we are able to bound its residual norm by a minimum polynomial of a perturbed matrix (i.e. the residual norm of the exact GMRES applied to a perturbed matrix) multiplied by an amplification factor. This shows that occurrence of nearbreakdowns or loss of biorthogonality does not necessarily deter convergence of the residuals provided that the amplification factor remains bounded. Numerical examples are given to gain insights into these bounds. 1.
Structurepreserving model reductions using a Krylov subspace projection formulation
, 2004
"... Abstract. A general framework for structurepreserving model reduction by Krylov subspace projection methods is developed. It not only matches as many moments as possible but also preserves substructures of importance in the coefficient matrices L,G,C, and B that define a dynamical system prescribed ..."
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Cited by 12 (4 self)
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Abstract. A general framework for structurepreserving model reduction by Krylov subspace projection methods is developed. It not only matches as many moments as possible but also preserves substructures of importance in the coefficient matrices L,G,C, and B that define a dynamical system prescribed by the transfer function of the form H(s)=L ∗ (G+sC) −1 B. Many existing structurepreserving modelorder reduction methods for linear and secondorder dynamical systems can be derived under this general framework. Furthermore, it also offers insights into the development of new structurepreserving model reduction methods.
Progress in the numerical solution of the nonsymmetric eigenvalue problem
, 1993
"... With the growing demands from disciplinary and interdisciplinary fields of science and engineering for the numerical solution of the nonsymmetric eigenvalue problem, competitive new techniques have been developed for solving the problem. In this paper we examine the state of the art of the algorithm ..."
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Cited by 10 (1 self)
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With the growing demands from disciplinary and interdisciplinary fields of science and engineering for the numerical solution of the nonsymmetric eigenvalue problem, competitive new techniques have been developed for solving the problem. In this paper we examine the state of the art of the algorithmic techniques and the software scene for the problem. Some current developments are also outlined. KEY WORDS nonsymmetric matrices; MACK sparse matrices; eigenvalue problem; EISPACK, 1.
Arnoldi versus Nonsymmetric Lanczos Algorithms for Solving Nonsymmetric Matrix Eigenvalue Problems
 BIT
, 1996
"... We obtain several results which may be useful in determining the convergence behavior of eigenvalue algorithms based upo n Arnoldi and nonsymmetric Lanczos recursions. We derive a relationship between nonsymmetric Lanczos eigenvalue procedures and Arnoldi eigenvalue procedures. We demonstrate that t ..."
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Cited by 7 (1 self)
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We obtain several results which may be useful in determining the convergence behavior of eigenvalue algorithms based upo n Arnoldi and nonsymmetric Lanczos recursions. We derive a relationship between nonsymmetric Lanczos eigenvalue procedures and Arnoldi eigenvalue procedures. We demonstrate that the Arnoldi recursions preserve a property which characterizes normal matrices, and that if we could determine the appropriate starting vectors, we could mimic the nonsymmetric Lanczos eigenvalue convergence on a general diagonalizable matrix by its convergence on related normal matrices. Using a unitary equivalence for each of these Krylov subspace methods, we define sets of test problems where we can easily vary certain spectral properties of the matrices. We use these and other test problems to examine the behavior of an Arnoldi and of a nonsymmetric Lanczos procedure. Mathematical Sciences Department, IBM Research Division, T.J. Watson Research Center, Yorktown Heights, NY 10598, USA, a...
Parallel Application Software on High Performance Computers  Parallel Diagonalisation Routines.
, 1996
"... In this report we list diagonalisation routines available for parallel computers. The methodology of each routine is outlined together with benchmark results on a typical matrix where available. Storage requirements and advantages and disadvantages of the method are also compared. The vast majority ..."
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Cited by 7 (1 self)
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In this report we list diagonalisation routines available for parallel computers. The methodology of each routine is outlined together with benchmark results on a typical matrix where available. Storage requirements and advantages and disadvantages of the method are also compared. The vast majority of these routines are available for real dense symmetric matrices only, although there is a known requirement for other data types  such as Hermitian or structured sparse matrices. We will report on new codes as they become available. This report is available from http://www.dl.ac.uk/TCSC/HPCI/ c fl1996, Daresbury Laboratory. We do not accept any responsibility for loss or damage arising from the use of information contained in any of our reports or in any communication about our tests or investigations. ii CONTENTS iii Contents 1 Summary 1 1.1 Test Results : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 1.2 Recommendations : : : : : : : : : : :...