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17
PRIMME: PReconditioned Iterative Multimethod Eigensolver: METHODS AND SOFTWARE DESCRIPTION
, 2006
"... This paper describes the PRIMME software package for the solving large, sparse Hermitian and real symmetric eigenvalue problems. The difficulty and importance of these problems have increased over the years, necessitating the use of preconditioning and near optimally converging iterative methods. O ..."
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Cited by 15 (6 self)
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This paper describes the PRIMME software package for the solving large, sparse Hermitian and real symmetric eigenvalue problems. The difficulty and importance of these problems have increased over the years, necessitating the use of preconditioning and near optimally converging iterative methods. On the other hand, the complexity of tuning or even using such methods has kept them outside the reach of many users. Responding to this problem, our goal was to develop a general purpose software that requires minimal or no tuning, yet it provides the best possible robustness and efficiency. PRIMME is a comprehensive package that brings stateoftheart methods from “bleeding edge ” to production, with a flexible, yet highly usable interface. We review the theory that gives rise to the near optimal methods GD+k and JDQMR, and present the various algorithms that constitute the basis of PRIMME. We also describe the software implementation, interface, and provide some sample experimental results.
Anasazi software for the numerical solution of largescale eigenvalue problems
 ACM TOMS
"... Anasazi is a package within the Trilinos software project that provides a framework for the iterative, numerical solution of largescale eigenvalue problems. Anasazi is written in ANSI C++ and exploits modern software paradigms to enable the research and development of eigensolver algorithms. Furt ..."
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Anasazi is a package within the Trilinos software project that provides a framework for the iterative, numerical solution of largescale eigenvalue problems. Anasazi is written in ANSI C++ and exploits modern software paradigms to enable the research and development of eigensolver algorithms. Furthermore, Anasazi provides implementations for some of the most recent eigensolver methods. The purpose of our paper is to describe the design and development of the Anasazi framework. A performance comparison of Anasazi and the popular FORTRAN 77 code ARPACK is given.
and skinny qr factorizations in mapreduce architectures
 in Proceedings of the second international workshop on MapReduce and its applications
, 2011
"... The QR factorization is one of the most important and useful matrix factorizations in scientific computing. A recent communicationavoiding version of the QR factorization trades flops for messages and is ideal for MapReduce, where computationally intensive processes operate locally on subsets of ..."
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The QR factorization is one of the most important and useful matrix factorizations in scientific computing. A recent communicationavoiding version of the QR factorization trades flops for messages and is ideal for MapReduce, where computationally intensive processes operate locally on subsets of the data. We present an implementation of the tall and skinny QR (TSQR) factorization in the MapReduce framework, and we provide computational results for nearly terabytesized datasets. These tasks run in just a few minutes under a variety of parameter choices. Categories and Subject Descriptors G.1.3 [Numerical analysis]: Numerical Linear Algebra—
Amesos2 and Belos: Direct and iterative solvers for large sparse linear systems
 Scientific Programming
, 2012
"... Solvers for large sparse linear systems come in two categories: direct and iterative. Amesos2, a package in the Trilinos software project, provides direct methods, and Belos, another Trilinos package, provides iterative methods. Amesos2 offers a common interface to many different sparse matrix fact ..."
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Cited by 7 (6 self)
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Solvers for large sparse linear systems come in two categories: direct and iterative. Amesos2, a package in the Trilinos software project, provides direct methods, and Belos, another Trilinos package, provides iterative methods. Amesos2 offers a common interface to many different sparse matrix factorization codes, and can handle any implementation of sparse matrices and vectors, via an easytoextend C++ traits interface. It can also factor matrices whose entries have arbitrary “Scalar ” type, enabling extendedprecision and mixedprecision algorithms. Belos includes many different iterative methods for solving large sparse linear systems and leastsquares problems. Unlike competing iterative solver libraries, Belos completely decouples the algorithms from the implementations of the underlying linear algebra objects. This lets Belos exploit the latest hardware without changes to the code. Belos favors algorithms that solve higherlevel problems, such as multiple simultaneous linear systems and sequences of related linear systems, faster than standard algorithms. The package also supports extendedprecision and mixedprecision algorithms. Together, Amesos2 and Belos form a complete suite of sparse linear solvers. 1
Direct QR factorizations for tallandskinny matrices
 in MapReduce architectures, arXiv:1301.1071 [cs.DC], 2013
"... Abstract—The QR factorization and the SVD are two fundamental matrix decompositions with applications throughout scientific computing and data analysis. For matrices with many more rows than columns, socalled “tallandskinny matrices, ” there is a numerically stable, efficient, communicationavoi ..."
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Abstract—The QR factorization and the SVD are two fundamental matrix decompositions with applications throughout scientific computing and data analysis. For matrices with many more rows than columns, socalled “tallandskinny matrices, ” there is a numerically stable, efficient, communicationavoiding algorithm for computing the QR factorization. It has been used in traditional high performance computing and grid computing environments. For MapReduce environments, existing methods to compute the QR decomposition use a numerically unstable approach that relies on indirectly computing the Q factor. In the best case, these methods require only two passes over the data. In this paper, we describe how to compute a stable tallandskinny QR factorization on a MapReduce architecture in only slightly more than 2 passes over the data. We can compute the SVD with only a small change and no difference in performance. We present a performance comparison between our new direct TSQR method, a standard unstable implementation for MapReduce (Cholesky QR), and the classic stable algorithm implemented for MapReduce (Householder QR). We find that our new stable method has a large performance advantage over the Householder QR method. This holds both in a theoretical performance model as well as in an actual implementation. Keywordsmatrix factorization, QR, SVD, TSQR, MapReduce, Hadoop
IMPLEMENTING COMMUNICATIONOPTIMAL PARALLEL AND SEQUENTIAL QR FACTORIZATIONS
, 2008
"... We present parallel and sequential dense QR factorization algorithms for tall and skinny matrices and general rectangular matrices that both minimize communication, and are as stable as Householder QR. The sequential and parallel algorithms for tall and skinny matrices lead to significant speedups ..."
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We present parallel and sequential dense QR factorization algorithms for tall and skinny matrices and general rectangular matrices that both minimize communication, and are as stable as Householder QR. The sequential and parallel algorithms for tall and skinny matrices lead to significant speedups in practice over some of the existing algorithms, including LAPACK and ScaLAPACK, for example up to 6.7x over ScaLAPACK. The parallel algorithm for general rectangular matrices is estimated to show significant speedups over ScaLAPACK, up to 22x over ScaLAPACK.
A communicationavoiding thickrestart Lanczos method on a distributedmemory system
 Workshop on Algorithms and Programming Tools for NextGeneration HighPerformance Scientific and Software (HPCC
, 2011
"... Abstract. The ThickRestart Lanczos (TRLan) method is an effective method for solving largescale Hermitian eigenvalue problems. On a modern computer, communication can dominate the solution time of TRLan. To enhance the performance of TRLan, we develop CATRLan that integrates communicationavoid ..."
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Abstract. The ThickRestart Lanczos (TRLan) method is an effective method for solving largescale Hermitian eigenvalue problems. On a modern computer, communication can dominate the solution time of TRLan. To enhance the performance of TRLan, we develop CATRLan that integrates communicationavoiding techniques into TRLan. To study the numerical stability and solution time of CATRLan, we conduct numerical experiments using both synthetic diagonal matrices and matrices from the University of Florida sparse matrix collection. Our experimental results on up to 1, 024 processors of a distributedmemory system demonstrate that CATRLan can achieve speedups of up to three over TRLan while maintaining numerical stability.
Preconditioners for Avoiding Communication
"... dirigée par Laura GRIGORI Soutenue le * * *** * 2014 devant un jury compose ́ de: M. Prénom NOM Universite ́ président ..."
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dirigée par Laura GRIGORI Soutenue le * * *** * 2014 devant un jury compose ́ de: M. Prénom NOM Universite ́ président
unknown title
, 2007
"... www.elsevier.com/locate/jpdc Parallel solution of largescale eigenvalue problem formaster equation in protein folding dynamics ..."
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www.elsevier.com/locate/jpdc Parallel solution of largescale eigenvalue problem formaster equation in protein folding dynamics