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A comparison of solvers for large eigenvalue problems occuring in the design of resonant cavities

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS 1999年第1期6卷 3-16页

作者： Arbenz, P Geus, R Swiss Fed Inst Technol Inst Comp Sci CH-8092 Zurich Switzerland

We present experiments with various solvers for large sparse generalized symmetric matrix eigenvalue problems. These problems occur in the computation of a few of the lowest frequencies of standing electromagnetic waves in resonant cavities with the finite element method. The solvers investigated are (1) subspace iteration, (2) block lanczos algorithm, (3) implicitly restarted lanczos algorithm and (4) Jacobi-Davidson algorithm. The experiments have been conducted on a Hewlett-Packard Exemplar S-Class system. Copyright (C) 1999 John Wiley & Sons, Ltd.

关键词： Maxwell equations cavity resonator large sparse eigenvalue problem subspace iteration lanczos algorithm restarted lanczos algorithm Jacobi-Davidson algorithm

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Using nonorthogonal lanczos vectors in the computation of matrix functions

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 1998年第1期19卷 38-54页

作者： Druskin, V Greenbaum, A Knizhnerman, L Schlumberger Doll Res Ctr Ridgefield CT 06877 USA NYU Courant Inst Math Sci New York NY 10012 USA Cent Geophys Expedit Moscow 123298 Russia

The lanczos algorithm uses a three-term recurrence to construct an orthonormal basis for the Krylov space corresponding to a symmetric matrix A and a nonzero starting vector phi. The vectors and recurrence coefficients produced by this algorithm can be used for a number of purposes, including solving linear systems Au = phi and computing the matrix exponential e(-tA)phi. Although the vectors produced in finite precision arithmetic are not orthogonal, we show why they can still be used effectively for these purposes.

关键词： lanczos algorithm iterative methods matrix exponential finite precision arithmetic

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Transpose-free formulations of lanczos-type methods for nonsymmetric linear systems

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NUMERICAL algorithmS 1998年第1-2期17卷 51-66页

作者： Chan, TF de Pillis, L van der Vorst, H Univ Calif Los Angeles Dept Math Los Angeles CA 90095 USA Harvey Mudd Coll Dept Math Claremont CA 91711 USA Univ Utrecht Inst Math NL-3508 TA Utrecht Netherlands

We present a transpose-free version of the nonsymmetric scaled lanczos procedure. It generates the same tridiagonal matrix as the classical algorithm, using two matrix-vector products per iteration without accessing A(T). We apply this algorithm to obtain a transpose-free version of the Quasi-minimal residual method of Freund and Nachtigal [15] (without look-ahead), which requires three matrix-vector products per iteration. We also present a related transpose-free version of the bi-conjugate gradients algorithm.

关键词： lanczos algorithm quasi-minimal residual algorithm bi-conjugate gradients algorithm nonsymmetric linear systems Krylov subspace methods

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A fast method to diagonalize a Hankel matrix

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LINEAR ALGEBRA AND ITS APPLICATIONS 1998年第1-3期284卷 41-52页

作者： Boley, DL Luk, FT Vandevoorde, D Univ Minnesota Dept Comp Sci Minneapolis MN 55455 USA Rensselaer Polytech Inst Troy NY 12180 USA

We consider a Vandermonde factorization of a Hankel matrix, and propose a new approach to compute the full decomposition in O(n(2)) operations. The method is based on the use of a variant of the lanczos method to compute a tridiagonal matrix whose eigenvalues are the modes generating the entries in the Hankel matrix. By adapting existing methods to solve for these eigenvalues and then for the coefficients, one arrives at a method to compute the entire decomposition in O(n(2)) operations. The method is illustrated with a simple numerical example. (C) 1998 Published by Elsevier Science Inc. All rights reserved.

关键词： Hankel matrix Vandermonde decomposition Prony's method lanczos algorithm

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Direct calculation of long time correlation functions using an optical potential

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INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 1998年第5期68卷 317-328页

作者： Alacid, M Leforestier, C Univ Montpellier 2 UMR 5636 Lab Struct & Dynam Syst Mol & Solides F-34095 Montpellier 05 France

We present an L-2 method aimed at directly computing autocorrelation functions [Phi(0)/Phi(f)] for systems displaying long time recurrences. By making use of a lanczos scheme, as previously proposed by Wyatt [Chem. Phys. Lett. 121, 301 (1985)], the method avoids explicit time propagation of the wavefunction. The problem associated with spurious recurrences, due to the finite size of the L-2-box, is solved in terms of an optical potential located in the asymptotic region. The resulting complex representation of the Hamiltonian operator is handled by a complex symmetric lanczos scheme, which retains the same basic advantages as its real version. The method is illustrated on the ozone photodissociation process which displays a very detailed recurrence structure over a long time period. It is shown that such a direct calculation of the correlation function is about one order of magnitude faster than an actual wavepacket propagation. The accuracy of the method is assessed by comparison to calculations performed without any optical potential but using a very large box size along the dissociation coordinate. (C) 1998 John Wiley & Sons, Inc.

关键词： autocorrelation function optical potential lanczos algorithm ozone photodissociation hyperspherical coordinates

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On Behaviors, Rational Interpolation, and the lanczos algorithm

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IFAC Proceedings Volumes 1996年第1期29卷 3957-3962页

作者： A.C. Antoulas E.J. Grimme D.C. Sorensen Department of Electrical and Computer Engineering Rice University Houston TX 77251-1892 USA Coordinated Science Laboratory University of Illinois at Urbana-Champaign Urbana IL 61801-2307 USA Department of Computational and Applied Mathematics Rice University Houston TX 77251-1892 USA

This paper explores the interconnections between two methods which can be used to obtain rational interpolants. The first method, the behavioral approach, constructs a generating system in the frequency domain which explains a given data set composed of trajectories. The second method, the rational lanczos algorithm, can be used to construct a rational interpolant for the transfer function of a linear system defined by (potentially very high-order) state-space equations. This paper works to merge the theoretical attributes of the behavioral approach with the theoretical and computational properties of rational lanczos. As a result, it lays the foundation for the computation of reduced-order, stabilizing controllers through rational interpolation.

关键词： linear systems behaviors rational interpolation lanczos algorithm Krylov subspaces

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Analysis of projection methods for solving linear systems with multiple right-hand sides

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 1997年第6期18卷 1698-1721页

作者： Chan, TF Wan, WL Department of Mathematics Univ. of California at Los Angeles Los Angeles CA 90024 United States

We analyze a class of Krylov projection methods but mainly concentrate on a specific conjugate gradient (CG) implementation by Smith, Peterson, and Mittra [IEEE transactions on Antennas and Propogation, 37 (1989), pp. 1490-1493] to solve the linear system AX = B, where A is symmetric positive definite and B is a multiple of right-hand sides, This method generates a Krylov subspace from a set of direction vectors obtained by solving one of the systems, called the seed system, by the CG method and then projects the residuals of other systems orthogonally onto the generated Krylov subspace to get the approximate solutions. The whole process is repeated with another unsolved system as a seed until all the systems are solved. We observe in practice a superconvergence behavior of the CG process of the seed system when compared with the usual CG process. We also observe that only a small number of restarts is required to solve all the systems if the right-hand sides are close to each other. These two features together make the method particularly effective. In this paper, we give theoretical proof to justify these observations. Furthermore, we combine the advantages of this method and the block CG method and propose a block extension of this single seed method.

关键词： linear systems multiple right-hand sides Krylov space lanczos algorithm conjugate gradient method

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Generalized cross-validation for large-scale problems

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JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS 1997年第1期6卷 1-34页

作者： Golub, GH vonMatt, U ETH ZENTRUM SWISS CTR SCI COMPCH-8092 ZURICHSWITZERLAND

Although generalized cross-validation is a popular tool for calculating a regularization parameter, it has been rarely applied to large-scale problems until recently. A major difficulty lies in the evaluation of the cross-validation function that requires the calculation of the trace of an inverse matrix. In the last few years stochastic trace estimators have been proposed to alleviate this problem. This article demonstrates numerical approximation techniques that further reduce the computational complexity. The new approach employs Gauss quadrature to compute lower and upper bounds on the cross-validation function. It only requires the operator form of the system matrix-that is, a subroutine to evaluate matrix-vector products. Thus, the factorization of large matrices can be avoided. The new approach has been implemented in MATLAB. Numerical experiments confirm the remarkable accuracy of the stochastic trace estimator. Regularization parameters are computed for ill-posed problems with 100, 1,000, and 10,000 unknowns.

关键词： Gauss quadrature generalized cross-validation ill-posed problems lanczos algorithm stochastic trace estimator Tikhonov regularization

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THE PARALLEL COMPUTATION OF PARTIAL EIGENSOLUTIONS USING A MODIFIED lanczos METHOD

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Parallel algorithms and Applications 1997年第3-4期11卷 299-323页

作者： K. Murphy[a] M. Clint[a] M. Szularz[b] J. Weston[b] [a] Department of Computer Science The Queen's University of Belfast Belfast U.K [b] School of Information and Software Engineering The University of Ulster at Coleraine Coleraine U.K

The lanczos algorithm is one of the most widely used methods for finding a small number of the extremal eigenvalues and associated eigenvectors of large, sparse, symmetric matrices. In this paper the performance on two parallel machines with different architectures of a modified version of the algorithm which incorporates a novel convergence monitoring method is assessed. The investigation has been carried out using a shared memory Convex C3840 with two processors and a 16-node Intel iPSC/860 hypercube. It is shown that parallel implementations of the modified algorithm can efficiently exploit the facilities provided by both machines. However, there are significant architecture dependent considerations which favour the use of the shared memory machine for the solution of general instances of the problem. These considerations relate to the cost of inter-processor communication and the limited availability of fast memory on the distributed memory machine.

关键词： lanczos algorithm convergence monitoring orthogonalization high performance computing

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Reduced-order modeling of large passive linear circuits by means of the SYPVL algorithm 96

Reduced-order modeling of large passive linear circuits by m...

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Proceedings of the 1996 IEEE/ACM international conference on Computer-aided design

作者： R. W. Freund P. Feldmann Bell Laboratories Lucent Technologies Murray Hill NJ

ISBN: (纸本)9780818675973

Lucent TechnologiesThis paper discusses the analysis of large linear electrical networks consisting of passive components, such as resistors, capacitors, inductors, and transformers. Such networks admit a symmetric formulation of their circuit equations. We introduce SyPVL, an efficient and numerically stable algorithm for the computation of reduced-order models of large, linear, passive networks. SyPVL represents the specialization of the more general PVL algorithm, to symmetric problems. Besides the gain in efficiency over PVL, SyPVL also preserves the symmetry of the problem, and, as a consequence, can often guarantee the stability of the resulting reduced-order models. Moreover, these reduced-order models can be synthesized as actual physical circuits, thus facilitating compatibility with existing analysis tools. The application of SyPVL is illustrated with two interconnect-analysis examples.

关键词： lanczos algorithm

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