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19
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 37 (4 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
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 17 (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.
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 11 (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.
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 9 (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 8 (3 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.
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 6 (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 : : : : : : : : : : :...
Structural preserving model reductions
, 2004
"... Abstract A general framework for structural preserving model reductions by Krylov subspace projection methods is developed. The goal is to preserve any substructures of importance in the matrices L, G, C, B that define the model prescribed by transferfunction H(s) = L*(G + sC)1B. As an applicatio ..."
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Cited by 3 (1 self)
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Abstract A general framework for structural preserving model reductions by Krylov subspace projection methods is developed. The goal is to preserve any substructures of importance in the matrices L, G, C, B that define the model prescribed by transferfunction H(s) = L*(G + sC)1B. As an application, quadratic transfer functions targeted by Su and Craig (J. Guidance, Control, and Dynamics, 14 (1991), pp. 260267.) is revisited, which leads to an improved algorithm than Su's and Craig's original interms of achieving the same approximation accuracy with smaller reduced systems. Other contributions include new GramSchmidt orthogonalization process and newArnoldi process that only orthogonalize the prescribed portion of all basis vectors as opposing to whole vectors by existing counterparts. These new processes are designedas one way to numerically realize the idea in the general framework.
Iterative methods for solving Ax = b: GMRES/FOM versus QMR/BiCG
 Advances in Computational Mathematics
, 1996
"... We study the convergence of GMRES/FOM and QMR/BiCG methods for solving nonsymmetric Ax = b. We prove that given the results of a BiCG computation on Ax = b, we can obtain a matrix B with the same eigenvalues as A and a vector c such that the residual norms generated by a FOM computation on Bx = c ar ..."
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Cited by 3 (0 self)
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We study the convergence of GMRES/FOM and QMR/BiCG methods for solving nonsymmetric Ax = b. We prove that given the results of a BiCG computation on Ax = b, we can obtain a matrix B with the same eigenvalues as A and a vector c such that the residual norms generated by a FOM computation on Bx = c are identical to those generated by the BiCG computations. Using a unitary equivalence for each of these methods, we obtain test problems where we can easily vary certain spectral properties of the matrices. We use these test problems to study the effects of nonnormality on the convergence of GMRES and QMR, to study the effects of eigenvalue outliers on the convergence of QMR, and to compare the convergence of restarted GMRES and QMR across a family of normal and nonnormal problems. Our GMRES tests on nonnormal test matrices indicate that nonnormality can have unexpected effects upon the residual norm convergence, giving misleading indications of superior convergence when the error norms for G...