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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.
Parallel Bandreduction and Tridiagonalization
 Proceedings, Sixth SIAM Conference on Parallel Processing for Scientific Computing
, 1993
"... This paper presents a parallel implementation of a blocked band reduction algorithm for symmetric matrices suggested by Bischof and Sun. The reduction to tridiagonal or block tridiagonal form is a special case of this algorithm. A blocked double torus wrap mapping is used as the underlying data dist ..."
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Cited by 17 (5 self)
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This paper presents a parallel implementation of a blocked band reduction algorithm for symmetric matrices suggested by Bischof and Sun. The reduction to tridiagonal or block tridiagonal form is a special case of this algorithm. A blocked double torus wrap mapping is used as the underlying data distribution and the socalled WY representation is employed to represent block orthogonal transformations. Preliminary performance results on the Intel Delta indicate that the algorithm is wellsuited to a MIMD computing environment and that the use of a block approach significantly improves performance. 1 Introduction Reduction to tridiagonal form is a major step in eigenvalue computations for symmetric matrices. If the matrix is full, the conventional Householder tridiagonalization approach [13, p. 276] or block variants thereof [12] is the method of choice. These two approaches also underlie the parallel implementations described for example in [15] and [10]. The approach described in this ...
Block Implementations of Symmetric QR and Jacobi Algorithms
, 1992
"... A common approach to solve problems in numerical linear algebra efficiently on modern high speed computers is to redesign the classical algorithm, which was originally developed for serial computers. In this paper, we discuss block variants of QR and Jacobi algorithms for the computation of the comp ..."
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Cited by 5 (1 self)
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A common approach to solve problems in numerical linear algebra efficiently on modern high speed computers is to redesign the classical algorithm, which was originally developed for serial computers. In this paper, we discuss block variants of QR and Jacobi algorithms for the computation of the complete spectral decomposition of symmetric matrices. We report on numerical tests, which have been performed on a CRAY YMP and an ALLIANT FX/80. 1 Introduction The QRalgorithm is the most widely used algorithm for finding the eigenvalues and eigenvectors of a symmetric matrix. Subroutines, which implement this algorithm (e.g. TRED2 and TQL2 from EISPACK [21]) perform very well on sequential computers. But, the classical algorithm may be reorganized to exploit the hardware of high performance computers. On modern supercomputers with vector and/or parallelprocessing capabilities, it is very important to avoid unnecessary memory references, as moving data between different levels of memory ...
Numerical Linear Algebra and Computer Architecture: An Evolving Interaction
, 1993
"... The aim of this report is to highlight the evolving interaction between computer architecture on the one hand and the design and analysis of algorithms for numerical linear algebra on the other. This report can best be described as a collection of pointers to aspects of the development processes of ..."
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Cited by 3 (0 self)
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The aim of this report is to highlight the evolving interaction between computer architecture on the one hand and the design and analysis of algorithms for numerical linear algebra on the other. This report can best be described as a collection of pointers to aspects of the development processes of numerical linear algebra, and to various existing surveys and software. The rich variety of activity in design, analysis, comparison and evaluation of algorithms, new and old, is examined in the context of rapid technological and architectural advance. The characteristics of algorithms for linear algebra are discussed without going into mathematical details: There are references to important surveys that provide such details. Reflections on the practices of numerical linear algebra will, it is hoped, point to possibilities for and limitations of automating the "mapping" of algorithms onto high performance computers.
A Jacobi Method By Blocks On A Mesh Of Processors
, 1997
"... In this paper, we study the parallelization of the Jacobi method to solve the symmetric eigenvalue problem on a mesh of processors. To solve this problem obtaining a theoretical efficiency of 100% it is necessary to exploit the symmetry of the matrix. The only previous algorithm we know exploiti ..."
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Cited by 1 (1 self)
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In this paper, we study the parallelization of the Jacobi method to solve the symmetric eigenvalue problem on a mesh of processors. To solve this problem obtaining a theoretical efficiency of 100% it is necessary to exploit the symmetry of the matrix. The only previous algorithm we know exploiting the symmetry on multicomputers is that in [21], but that algorithm uses a storage scheme adequate for a logical ring of processors, so having a low scalability. In this paper we show how matrix symmetry can be exploited on a logical mesh of processors obtaining a higher scalability than that obtained with the algorithm in [21]. In addition, we show how the storage scheme exploiting the symmetry can be combined with a scheme by blocks to obtain a highly efficient and scalable Jacobi method for solving the symmetric eigenvalue problem for distributed memory parallel computers. We report performance results from the Intel Touchstone DELTA, the iPSC/860, the Alliant FX/80 and the PA...
A Jacobi Method By Blocks To Solve The Symmetric Eigenvalue Problem
, 1997
"... In this paper, we demonstrate how considerable improvement in the performance of the Jacobi method for solving the symmetric eigenvalue problem can be obtained by reformulating the algorithm to be rich in matrixmatrix multiplication. To achieve this, we borrow techniques developed for parallel J ..."
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In this paper, we demonstrate how considerable improvement in the performance of the Jacobi method for solving the symmetric eigenvalue problem can be obtained by reformulating the algorithm to be rich in matrixmatrix multiplication. To achieve this, we borrow techniques developed for parallel Jacobi methods to enhance data locality. By basing the algorithm on the Basic Linear Algebra Subprograms (BLAS) we have achieved an implementation that will perform well independent of the machine where it will be executed, as long as optimized BLAS exist for that machine. Keywords: Jacobi methods, symmetric eigenvalue problem, algorithms by blocks. 1 Introduction Many different methods exist for computing the eigenvalues of a symmetric matrix, including bisection methods, the QR algorithm, Divide and Conquer (Cuppen's algorithm), and Jacobi's method. For a complete survey and references, we suggest the reader turn to [12]. Jacobi's method is the oldest, dating back to the mid1800's. ...
Accelerating the Convergence of Blocked Jacobi Methods
"... In this work we study the possible combination of two techniques to reduce the execution time when solving the Symmetric Eigenvalue Problem by Jacobi methods: acceleration of convergence, and work by blocks. INTRODUCTION The Symmetric Eigenvalue Problem (SEP) appears in the solution of a lot of pr ..."
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In this work we study the possible combination of two techniques to reduce the execution time when solving the Symmetric Eigenvalue Problem by Jacobi methods: acceleration of convergence, and work by blocks. INTRODUCTION The Symmetric Eigenvalue Problem (SEP) appears in the solution of a lot of problems on science and engineering [5]. In some of these applications the problems to solve are of great size, making it neccessary to use highly efficient methods. The Jacobi method was the most widely used to solve the SEP for more than a century [9], but in the 60's it was surpassed by methods based on reduction of the initial matrix to tridiagonal form [6]. More recently Jacobi methods have become important again due to better stability properties [4] and straightforward parallelization [13,8], and in some cases the Jacobi methods can surpass other methods based on reduction to tridiagonal form [7]. A Jacobi method for the SEP consists in the generation of a succession fA s g through: ...