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ARPACK Users Guide: Solution of Large Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods.
, 1997
"... this document is intended to provide a cursory overview of the Implicitly Restarted Arnoldi/Lanczos Method that this software is based upon. The goal is to provide some understanding of the underlying algorithm, expected behavior, additional references, and capabilities as well as limitations of the ..."
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Cited by 136 (14 self)
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this document is intended to provide a cursory overview of the Implicitly Restarted Arnoldi/Lanczos Method that this software is based upon. The goal is to provide some understanding of the underlying algorithm, expected behavior, additional references, and capabilities as well as limitations of the software. 1.7 Dependence on LAPACK and BLAS
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. However, ..."
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Cited by 86 (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.
Diffusion Wavelets
, 2004
"... We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their ..."
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Cited by 72 (12 self)
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We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their fast application. Classes of operators satisfying these conditions include diffusionlike operators, in any dimension, on manifolds, graphs, and in nonhomogeneous media. In this case our construction can be viewed as a farreaching generalization of Fast Multipole Methods, achieved through a different point of view, and of the nonstandard wavelet representation of CalderónZygmund and pseudodifferential operators, achieved through a different multiresolution analysis adapted to the operator. We show how the dyadic powers of an operator can be used to induce a multiresolution analysis, as in classical LittlewoodPaley and wavelet theory, and we show how to construct, with fast and stable algorithms, scaling function and wavelet bases associated to this multiresolution analysis, and the corresponding downsampling operators, and use them to compress the corresponding powers of the operator. This allows to extend multiscale signal processing to general spaces (such as manifolds and graphs) in a very natural way, with corresponding fast algorithms.
StructurePreserving Methods for Computing Eigenpairs of Large Sparse SkewHamiltonian/Hamiltonian Pencils
 SIAM J. Sci. Comput
, 2000
"... We study large, sparse generalized eigenvalue problems for matrix pencils, where one of the matrices is Hamiltonian and the other skew Hamiltonian. Problems of this form arise in the numerical simulation of elastic deformation of anisotropic materials, in structural mechanics and in the linearquadr ..."
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Cited by 58 (15 self)
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We study large, sparse generalized eigenvalue problems for matrix pencils, where one of the matrices is Hamiltonian and the other skew Hamiltonian. Problems of this form arise in the numerical simulation of elastic deformation of anisotropic materials, in structural mechanics and in the linearquadratic control problem for partial differential equations. We develop a structurepreserving skewHamiltonian, isotropic, implicitlyrestarted shiftandinvert Arnoldi algorithm (SHIRA). Several numerical examples demonstrate the superiority of SHIRA over a competing unstructured method.
Model reduction of state space systems via an Implicitly Restarted Lanczos method
 Numer. Algorithms
, 1996
"... The nonsymmetric Lanczos method has recently received significant attention as a model reduction technique for largescale systems. Unfortunately, the Lanczos method may produce an unstable partial realization for a given, stable system. To remedy this situation, inexpensive implicit restarts are de ..."
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Cited by 56 (8 self)
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The nonsymmetric Lanczos method has recently received significant attention as a model reduction technique for largescale systems. Unfortunately, the Lanczos method may produce an unstable partial realization for a given, stable system. To remedy this situation, inexpensive implicit restarts are developed which can be employed to stabilize the Lanczos generated model.
An Implicitly Restarted Lanczos Method for Large Symmetric. . .
 ETNA
, 1994
"... . The Lanczos process is a well known technique for computing a few, say k, eigenvalues and associated eigenvectors of a large symmetric nn matrix. However, loss of orthogonality of the computed Krylov subspace basis can reduce the accuracy of the computed approximate eigenvalues. In the implicitly ..."
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Cited by 54 (13 self)
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. The Lanczos process is a well known technique for computing a few, say k, eigenvalues and associated eigenvectors of a large symmetric nn matrix. However, loss of orthogonality of the computed Krylov subspace basis can reduce the accuracy of the computed approximate eigenvalues. In the implicitly restarted Lanczos method studied in the present paper, this problem is addressed by fixing the number of steps in the Lanczos process at a prescribed value, k +p, where p typically is not much larger, and may be smaller, than k. Orthogonality of the k + p basis vectors of the Krylov subspace is secured by reorthogonalizing these vectors when necessary. The implicitly restarted Lanczos method exploits that the residual vector obtained by the Lanczos process is a function of the initial Lanczos vector. The method updates the initial Lanczos vector through an iterative scheme. The purpose of the iterative scheme is to determine an initial vector such that the associated residual vector is tiny....
Rational Krylov, A Practical Algorithm For Large Sparse Nonsymmetric Matrix Pencils
 SIAM J. Sci. Comput
, 1998
"... The Rational Krylov algorithm computes eigenvalues and eigenvectors of a regular not necessarily symmetric matrix pencil. It is a generalization of the shifted and inverted Arnoldi algorithm, where several factorizations with different shifts are used in one run. It computes an orthogonal basis and ..."
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Cited by 47 (0 self)
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The Rational Krylov algorithm computes eigenvalues and eigenvectors of a regular not necessarily symmetric matrix pencil. It is a generalization of the shifted and inverted Arnoldi algorithm, where several factorizations with different shifts are used in one run. It computes an orthogonal basis and a small Hessenberg pencil. The eigensolution of the Hessenberg pencil approximates the solution of the original pencil. Different types of Ritz values and harmonic Ritz values are described and compared. Periodical purging of uninteresting directions reduces the size of the basis, and makes it possible to get many linearly independent eigenvectors and principal vectors to pencils with multiple eigenvalues. Relations to iterative methods are established. Results are reported for two large test examples. One is a symmetric pencil coming from a finite element approximation of a membrane, the other a nonsymmetric matrix modeling an idealized aircraft stability problem.
On restarting the Arnoldi method for large nonsymmetric eigenvalue problems
 Mathematics of Computation
, 1996
"... Abstract. The Arnoldi method computes eigenvalues of large nonsymmetric matrices. Restarting is generally needed to reduce storage requirements and orthogonalization costs. However, restarting slows down the convergence and makes the choice of the new starting vector difficult if several eigenvalues ..."
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Cited by 43 (9 self)
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Abstract. The Arnoldi method computes eigenvalues of large nonsymmetric matrices. Restarting is generally needed to reduce storage requirements and orthogonalization costs. However, restarting slows down the convergence and makes the choice of the new starting vector difficult if several eigenvalues are desired. We analyze several approaches to restarting and show why Sorensen’s implicit QR approach is generally far superior to the others. Ritz vectors are combined in precisely the right way for an effective new starting vector. Also, a new method for restarting Arnoldi is presented. It is mathematically equivalent to the Sorensen approach but has additional uses. 1.
An Arnoldi code for computing selected eigenvalues of sparse real unsymmetric matrices
, 1995
"... Arnoldi methods can be more effective than subspace iteration methods for computing the dominant eigenvalues of a large sparse real unsymmetric matrix. A code, EB12, for the sparse unsymmetric eigenvalue problem based on a subspace iteration algorithm, optionally combined with Chebychev acceleration ..."
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Cited by 28 (4 self)
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Arnoldi methods can be more effective than subspace iteration methods for computing the dominant eigenvalues of a large sparse real unsymmetric matrix. A code, EB12, for the sparse unsymmetric eigenvalue problem based on a subspace iteration algorithm, optionally combined with Chebychev acceleration, has recently been described by Duff and Scott (1993) and is included in the Harwell Subroutine Library (Anon 1993). In this paper we consider variants of the method of Arnoldi and discuss the design and development of a code to implement these methods. The new code, which is called EB13, offers the user the choice of a basic Arnoldi algorithm, an Arnoldi algorithm with Chebychev acceleration, and a Chebychev preconditioned Arnoldi algorithm. Each method is available in blocked and unblocked form. The code may be used to compute either the rightmost eigenvalues, the eigenvalues of largest absolute value, or the eigenvalues of largest imaginary part. The performance of each option in the EB...
Matrix transformations for computing rightmost eigenvalues of large sparse nonsymmetric eigenvalue problems
 IMA J. Numer. Anal
, 1996
"... This paper gives an overview of matrix transformations for finding rightmost eigenvalues of Ax = kx and Ax = kBx with A and B real nonsymmetric and B possibly singular. The aim is not to present new material, but to introduce the reader to the application of matrix transformations to the solution o ..."
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Cited by 26 (7 self)
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This paper gives an overview of matrix transformations for finding rightmost eigenvalues of Ax = kx and Ax = kBx with A and B real nonsymmetric and B possibly singular. The aim is not to present new material, but to introduce the reader to the application of matrix transformations to the solution of largescale eigenvalue problems. The paper explains and discusses the use of Chebyshev polynomials and the shiftinvert and Cayley ^ transforms as matrix transformations for problems that arise from the discretization df partial differential equations. A few other techniques are described. The reliability of iterative methods is also dealt with by introducing the concept of domain of confidence or trust region. This overview gives the reader an idea of the benefits and the drawbacks of several transformation techniques. We also briefly discuss the current software