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21
Applied Numerical Linear Algebra
 Society for Industrial and Applied Mathematics
, 1997
"... We survey general techniques and open problems in numerical linear algebra on parallel architectures. We rst discuss basic principles of parallel processing, describing the costs of basic operations on parallel machines, including general principles for constructing e cient algorithms. We illustrate ..."
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Cited by 532 (26 self)
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We survey general techniques and open problems in numerical linear algebra on parallel architectures. We rst discuss basic principles of parallel processing, describing the costs of basic operations on parallel machines, including general principles for constructing e cient algorithms. We illustrate these principles using current architectures and software systems, and by showing how one would implement matrix multiplication. Then, we present direct and iterative algorithms for solving linear systems of equations, linear least squares problems, the symmetric eigenvalue problem, the nonsymmetric eigenvalue problem, and the singular value decomposition. We consider dense, band and sparse matrices.
A Scalable Linear Algebra Library for Distributed Memory Concurrent Computers
, 1992
"... This paper describes ScaLAPACK, a distributed memory version of the LAPACK software package for dense and banded matrix computations. Key design features are the use of distributed versions of the Level LAS as building blocks, and an ob ectbased interface to the library routines. The square block s ..."
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Cited by 161 (33 self)
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This paper describes ScaLAPACK, a distributed memory version of the LAPACK software package for dense and banded matrix computations. Key design features are the use of distributed versions of the Level LAS as building blocks, and an ob ectbased interface to the library routines. The square block scattered decomposition is described. The implementation of a distributed memory version of the rightlooking LU factorization algorithm on the Intel Delta multicomputer is discussed, and performance results are presented that demonstrated the scalability of the algorithm.
Software libraries for linear algebra computations on high performance computers
 SIAM REVIEW
, 1995
"... This paper discusses the design of linear algebra libraries for high performance computers. Particular emphasis is placed on the development of scalable algorithms for MIMD distributed memory concurrent computers. A brief description of the EISPACK, LINPACK, and LAPACK libraries is given, followed b ..."
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Cited by 68 (17 self)
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This paper discusses the design of linear algebra libraries for high performance computers. Particular emphasis is placed on the development of scalable algorithms for MIMD distributed memory concurrent computers. A brief description of the EISPACK, LINPACK, and LAPACK libraries is given, followed by an outline of ScaLAPACK, which is a distributed memory version of LAPACK currently under development. The importance of blockpartitioned algorithms in reducing the frequency of data movement between different levels of hierarchical memory is stressed. The use of such algorithms helps reduce the message startup costs on distributed memory concurrent computers. Other key ideas in our approach are the use of distributed versions of the Level 3 Basic Linear Algebra Subprograms (BLAS) as computational building blocks, and the use of Basic Linear Algebra Communication Subprograms (BLACS) as communication building blocks. Together the distributed BLAS and the BLACS can be used to construct highe...
The Design of a Parallel Dense Linear Algebra Software Library: Reduction to Hessenberg, Tridiagonal, and Bidiagonal Form
, 1995
"... This paper discusses issues in the design of ScaLAPACK, a software library for performing dense linear algebra computations on distributed memory concurrent computers. These issues are illustrated using the ScaLAPACK routines for reducing matrices to Hessenberg, tridiagonal, and bidiagonal forms. ..."
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Cited by 34 (5 self)
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This paper discusses issues in the design of ScaLAPACK, a software library for performing dense linear algebra computations on distributed memory concurrent computers. These issues are illustrated using the ScaLAPACK routines for reducing matrices to Hessenberg, tridiagonal, and bidiagonal forms. These routines are important in the solution of eigenproblems. The paper focuses on how building blocks are used to create higherlevel library routines. Results are presented that demonstrate the scalability of the reduction routines. The most commonlyused building blocks used in ScaLAPACK are the sequential BLAS, the Parallel BLAS (PBLAS) and the Basic Linear Algebra Communication Subprograms (BLACS). Each of the matrix reduction algorithms consists of a series of steps in each of which one block column (or panel), and/or block row, of the matrix is reduced, followed by an update of the portion of the matrix that has not been factorized so far. This latter phase is performed usin...
An Object Oriented Design for High Performance Linear Algebra on Distributed Memory Architectures
, 1993
"... We describe the design of ScaLAPACK++, an object oriented C++ library for implementing linear algebra computations on distributed memory multicomputers. This package, when complete, will support distributed matrix operations for symmetric, positivedefinite, and nonsymmetric cases. In ScaLAPACK++ w ..."
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Cited by 26 (10 self)
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We describe the design of ScaLAPACK++, an object oriented C++ library for implementing linear algebra computations on distributed memory multicomputers. This package, when complete, will support distributed matrix operations for symmetric, positivedefinite, and nonsymmetric cases. In ScaLAPACK++ we have employed object oriented design methods to enchance scalability, portability, flexibility, and easeofuse. We illustrate some of these points by describing the implementation of basic algorithms and comment on tradeoffs between elegance, generality, and performance.
The Design and Implementation of the ScaLAPACK LU, QR, and Cholesky Factorization Routines
, 1994
"... This paper discusses the core factorization routines included in the ScaLAPACK library. These routines allow the factorization and solution of a dense system of linear equations via LU, QR, and Cholesky. They are implemented using a block cyclic data distribution, and are built using de facto standa ..."
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Cited by 24 (11 self)
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This paper discusses the core factorization routines included in the ScaLAPACK library. These routines allow the factorization and solution of a dense system of linear equations via LU, QR, and Cholesky. They are implemented using a block cyclic data distribution, and are built using de facto standard kernels for matrix and vector operations (BLAS and its parallel counterpart PBLAS) and message passing communication (BLACS). In implementing the ScaLAPACK routines, a major objective was to parallelize the corresponding sequential LAPACK using the BLAS, BLACS, and PBLAS as building blocks, leading to straightforward parallel implementations without a significant loss in performance. We present the details of the implementation of the ScaLAPACK factorization routines, as well as performance and scalability results on the Intel iPSC/860, Intel Touchstone Delta, and Intel Paragon systems.
Scalability Issues Affecting the Design of a Dense Linear Algebra Library
 JOURNAL OF PARALLEL AND DISTRIBUTED COMPUTING
, 1994
"... This paper discusses the scalability of Cholesky, LU, and QR factorization routines on MIMD distributed memory concurrent computers. These routines form part of the ScaLAPACK mathematical software library that extends the widelyused LAPACK library to run efficiently on scalable concurrent computers ..."
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Cited by 23 (12 self)
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This paper discusses the scalability of Cholesky, LU, and QR factorization routines on MIMD distributed memory concurrent computers. These routines form part of the ScaLAPACK mathematical software library that extends the widelyused LAPACK library to run efficiently on scalable concurrent computers. To ensure good scalability and performance, the ScaLAPACK routines are based on blockpartitioned algorithms that reduce the frequency of data movement between different levels of the memory hierarchy, and particularly between processors. The block cyclic data distribution, that is used in all three factorization algorithms, is described. An outline of the sequential and parallel blockpartitioned algorithms is given. Approximate models of algorithms' performance are presented to indicate which factors in the design of the algorithm have an impact upon scalability. These models are compared with timings results on a 128node Intel iPSC/860 hypercube. It is shown that the routines are highl...
A parallel algorithm for the reduction of a nonsymmetric matrix to block upperHessenberg form
 Parallel Comput
, 1995
"... In this paper, we present an algorithm for the reduction to block upperHessenberg form which can be used to solve the nonsymmetric eigenvalue problem on messagepassing multicomputers. On such multicomputers, a nonsymmetric matrix can be distributed across processing nodes con gured into a network ..."
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Cited by 17 (5 self)
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In this paper, we present an algorithm for the reduction to block upperHessenberg form which can be used to solve the nonsymmetric eigenvalue problem on messagepassing multicomputers. On such multicomputers, a nonsymmetric matrix can be distributed across processing nodes con gured into a network of twodimensional mesh processor array using a blockscattered decomposition. Based on the matrix partitioning and mapping, the algorithm employs both Householder re ectors and Givens rotations within each reduction step. We analyze the arithmetic and communication complexities and describe the implementation details of the algorithm on messagepassing multicomputers. We discuss two di erent implementationssynchronous and asynchronousand present performance results on the Intel iPSC/860 and DELTA. We conclude with an evaluation of the algorithm's communication cost, and suggest areas for further improvement. 1
Two Dimensional Basic Linear Algebra Communication Subprograms
, 1991
"... this paper, we describe extensions to a proposed set of linear algebra communication routines for communicating and manipulating data structures that are distributed among the memories of a distributed memory MIMD computer. In particular, recent experience shows that higher performance can be attain ..."
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Cited by 16 (6 self)
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this paper, we describe extensions to a proposed set of linear algebra communication routines for communicating and manipulating data structures that are distributed among the memories of a distributed memory MIMD computer. In particular, recent experience shows that higher performance can be attained on such architectures when parallel dense matrix algorithms utilize a data distribution that views the computational nodes as a logical two dimensional mesh. The motivation for the BLACS continues to be to increase portability, efficiency and modularity at a high level. The audience of the BLACS are mathematical software experts and people with large scale scientific computation to perform. A systematic effort must be made to achieve a de facto standard for the BLACS. ntroduction