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ThickRestart Lanczos Method for Symmetric Eigenvalue Problems
 SIAM J. MATRIX ANAL. APPL
, 1998
"... For real symmetric eigenvalue problems, there are a number of algorithms that are mathematically equivalent, for example, the Lanczos algorithm, the Arnoldi method and the unpreconditioned Davidson method. The Lanczos algorithm is often preferred because it uses significantly fewer arithmetic ope ..."
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Cited by 23 (3 self)
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For real symmetric eigenvalue problems, there are a number of algorithms that are mathematically equivalent, for example, the Lanczos algorithm, the Arnoldi method and the unpreconditioned Davidson method. The Lanczos algorithm is often preferred because it uses significantly fewer arithmetic operations per iteration. To limit the maximum memory usage, these algorithms are often restarted. In recent years, a number of effective restarting schemes have been developed for the Arnoldi method and the Davidson method. This paper describes a simple restarting scheme for the Lanczos algorithm. This restarted Lanczos algorithm uses as many arithmetic operations as the original algorithm. Theoretically, this restarted Lanczos method is equivalent to the implicitly restarted Arnoldi method and the thickrestart Davidson method. Because it uses less arithmetic operations than the others, it is an attractive alternative for solving symmetric eigenvalue problems.
1 Parallel Spectral Clustering in Distributed Systems
"... Spectral clustering algorithms have been shown to be more effective in finding clusters than some traditional algorithms such as kmeans. However, spectral clustering suffers from a scalability problem in both memory use and computational time when the size of a data set is large. To perform cluster ..."
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Cited by 22 (0 self)
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Spectral clustering algorithms have been shown to be more effective in finding clusters than some traditional algorithms such as kmeans. However, spectral clustering suffers from a scalability problem in both memory use and computational time when the size of a data set is large. To perform clustering on large data sets, we investigate two representative ways of approximating the dense similarity matrix. We compare one approach by sparsifying the matrix with another by the Nyström method. We then pick the strategy of sparsifying the matrix via retaining nearest neighbors and investigate its parallelization. We parallelize both memory use and computation on distributed computers. Through
IRBL: An implicitly restarted block Lanczos method for largescale Hermitian eigenproblems
 SIAM J. Sci. Comput
"... Abstract. The irbleigs code is an implementation of an implicitly restarted blockLanczos method for computing a few selected nearby eigenvalues and associated eigenvectors of a large, possibly sparse, Hermitian matrix A. The code requires only the evaluation of matrixvector products with A; in par ..."
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Cited by 17 (6 self)
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Abstract. The irbleigs code is an implementation of an implicitly restarted blockLanczos method for computing a few selected nearby eigenvalues and associated eigenvectors of a large, possibly sparse, Hermitian matrix A. The code requires only the evaluation of matrixvector products with A; in particular, factorization of A is not demanded, noris the solution of linearsystems of equations with the matrix A. This, together with a fairly small storage requirement, makes the irbleigs code well suited for largescale problems. Applications of the irbleigs code to certain generalized eigenvalue problems and to the computation of a few singular values and associated singular vectors are also discussed. Numerous computed examples illustrate the performance of the method and provide comparisons with other available codes.
The design of a block rational Lanczos code with partial reorthogonalization and implicit restarting
, 2000
"... We discuss the design and development of a new Fortran code EA16 for the computation of selected eigenvalues and eigenvectors of largescale real symmetric eigenvalue problems. EA16 can be used for either the standard or the generalized eigenvalue problem. The underlying method used by EA16 is the b ..."
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Cited by 2 (0 self)
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We discuss the design and development of a new Fortran code EA16 for the computation of selected eigenvalues and eigenvectors of largescale real symmetric eigenvalue problems. EA16 can be used for either the standard or the generalized eigenvalue problem. The underlying method used by EA16 is the block Lanczos method with partial reorthogonalization plus implicit restarting, combined with purging and locking of converged Ritz pairs. A spectral transformation may optionally be used. The code allows a change of pole via the rational Lanczos method. Particular attention is paid to the solution of generalized eigenvalue problems with a singular mass matrix. Keywords: eigenvalue problem, real symmetric, generalized, largescale, software, Lanczos method, rational Lanczos method Computational Science and Engineering Department Atlas Centre Rutherford Appleton Laboratory Oxon OX11 0QX April 19, 2000 Contents 1 Introduction 1 2 Basic theory 2 2.1 The block Lanczos process . . . . . . ...
Protein Motions through Eigenanalyses: A Set of Study Cases
, 1995
"... The study of collective motions of molecules provides useful insights into the large amplitude conformational changes the molecules experiment during chemical reactions. In particular, theoretical normal modes analyses of proteins taking into account the lowestfrequency modes may help to predict ..."
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Cited by 1 (1 self)
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The study of collective motions of molecules provides useful insights into the large amplitude conformational changes the molecules experiment during chemical reactions. In particular, theoretical normal modes analyses of proteins taking into account the lowestfrequency modes may help to predict the nature of such conformational changes. This work lists a set of proteins for which the theoretical motion history has been examined. Focus is given on the computation of lowfrequency modes (eigenvalues and eigenvectors) using a code based on the Lanczos algorithm. The normal modes approach and the main ideas governing the technique employed to compute the required modes are rst outlined. Then, ve distinct cases ranging from 396 to 8528 atoms are discussed. Finally, guidelines for the eigenanalyses of similar problems are proposed. y CERFACS, Centre Europeen de Recherche et de Formation Avancee en Calcul Scientique, 42 av. G. Coriolis, 31057 Toulouse Cedex, France, email: marques@cerfacs.fr z Laboratoire de Physique Quantique, IRSAMC, Universite PaulSabatier, 118 route de Narbonne, 31062 Toulouse Cedex, France, email: yves@irsamc1.upstlse.fr 1 1
Eigensolvers and Applications
 Advanced Solution Procedures on Innovative Computer Architectures
, 1996
"... This article presents an overview of eigenproblems that arise in current finiteelement computations. We focus on a set of applications that have been studied at CERFACS, Centre Europen de Recherche et de Formation Avance en Calcul Scientifique, and describe the ideas and tools that have been develop ..."
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This article presents an overview of eigenproblems that arise in current finiteelement computations. We focus on a set of applications that have been studied at CERFACS, Centre Europen de Recherche et de Formation Avance en Calcul Scientifique, and describe the ideas and tools that have been developed to deal with them. The main characteristics of tlye different cases are given. We also discuss the trends as well as the research efforts to understand and tackle new applications