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 wav...
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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.
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 coefficient...
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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.
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...
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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.
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 comp...
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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.
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. Ph...
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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.
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 e...
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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.
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....
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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.
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 c...
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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.
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 tw...
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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.
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 fo...
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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.
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