... Andersson Alexander Andrianov Colin Bennett Emil Cornea Fredrik Ekstedt Fausto Espinal Stephe... more ... Andersson Alexander Andrianov Colin Bennett Emil Cornea Fredrik Ekstedt Fausto Espinal Stephen Fenner Michael Hilton Ralph Howard Terry Huntsberger Björn Jawerth Goran Kronquist Toshiro Kubota Martin Lindberg Yiping Mao Per Gunnar Martinsson Johan Ràde Henrik ...
In this paper, we discuss the performance achieved by several implementations of the recently def... more In this paper, we discuss the performance achieved by several implementations of the recently defined Message Passing Interface (MPI) standard. In particular, performance results for different implementations of the broadcast operation are analyzed and compared on the Delta, ...
... Steven Huss-Lederman Elaine M. Jacobson Anna Tsao ... LetLCM represent the least common multi... more ... Steven Huss-Lederman Elaine M. Jacobson Anna Tsao ... LetLCM represent the least common multiple of P and Q. PUMMA begins by shu ing the rows of B on each node so adjacent rows are the same as the columns of A that will be multiplied and communicated at each stage. ...
In this paper we report on the development of an e cient and portable implementation of Strassen'... more In this paper we report on the development of an e cient and portable implementation of Strassen's matrix multiplication algorithm. Our implementation is designed to be used in place of DGEMM, the Level 3 BLAS matrix multiplication routine. E cient performance will be obtained for all matrix sizes and shapes and the additional memory needed for temporaryvariables has beenminimized. Replacing DGEMM with our routine should provide a signi cant performance gain for large matrices while providing the same performance for small matrices. We measure performance of our code on the IBM RS/6000, CRAY YMP C90, and CRAY T3D single processor, and o er comparisons to other codes. Our performance data recon rms that Strassen's algorithm is practical for realistic size matrices. The usefulness of our implementation is demonstrated by replacing DGEMM with our routine in a large application code.
In this document, we are concerned with the effccts of data layouts for nonsquare processor meshe... more In this document, we are concerned with the effccts of data layouts for nonsquare processor meshes on the implementation of common dense linear algebra kernels such as matrix-matrix multiplication, LU factorizations, or eigenvalue solvers. In particular, we address ease of programming and tunability of the resulting software. We introduce a generalization of the torus wrap data layout that results in a decoupling of "local" and "global" data layout view. As a result, it allows for intuitive programming of linear algebra algorithms and for tuning of the algorithm for a particular mesh aspect ratio or machine characteristics. This layout is as simple as the proposed HPF layout but, in our opinion, enhances ease of programming as well as ease of performance tuning. We emphasize that we do not advocate that all users need be concerned with these issues. We do, however, believe, that for the foreseeable future "assembler coding" (as message-passing code is likely to be viewed from a HPF programmers' perspective) will be needed to deliver high performance for computationally intensive kernels. As a result, we believe that the adoption of this approach not only would accelerate the generation of efficient linear algebra software libraries but also would accelerate the adoption of HPF as a result. We point out, however, that the adoption of this new layout would necessitate that an HPF compiler ensure that data objects are operated on in a consistent fashion across subroutine and function calls.
In this paper, preliminary research results on a new algorithm for finding all the eigenvalues an... more In this paper, preliminary research results on a new algorithm for finding all the eigenvalues and eigenvectors of a real diagonalizable matrix with real eigenvalues are presented. The basic mathematical theory behind this approach is reviewed and is followed by a discussion of the numerical considerations of the actual implementation. The numerical algorithm has been tested on thousands of matrices on both a Cray-2 and an IBM RS/6000 Model 580 workstation. The results of these tests are presented. Finally, issues concerning the parallel implementation of the algorithm are discussed. The algorithm's heavy reliance on matrix-matrix multiplication, coupled with the divide and conquer nature of this algorithm, should yield a highly parallelizable algorithm.
In this paper, we give an overview of the Invariant Subspace Decomposition Algorithm for banded s... more In this paper, we give an overview of the Invariant Subspace Decomposition Algorithm for banded symmetric matrices and describe a sequential implementation of this algorithm. Our implementation uses a specialized routine for performing banded matrix multiplication together with ...
In this paper we report on the development of an e cient and portable implementation of Strassen&... more In this paper we report on the development of an e cient and portable implementation of Strassen's matrix multiplication algorithm for matrices of arbitrary size. Our implementation is designed to be used in place of DGEMM, the Level 3 BLAS matrix multiplication routine. Our code is designed so that e cient performance will be obtained for all matrix sizes and shapes and that the additional memory needed for temporary variables is minimized. Replacing DGEMM with our routine should provide signi cant performance gain for large matrices while providing the same performance for small matrices. We measure performance of our code on the IBM RS/6000, CRAY YMP C90, and CRAY T3D single processor, and o er comparisons to other codes. Our performance data recon rms that Strassen's algorithm is practical for realistic size matrices. The usefulness of our implementation is demonstrated by replacing DGEMM with our routine in a large application code. Our implementation is based on a deta...
ABSTRACT this document relates to the packages developed for solving the symmetric eigenproblem w... more ABSTRACT this document relates to the packages developed for solving the symmetric eigenproblem which is joint work between Argonne National Laboratory, Center for Computing Sciences, and Duke University. The software contains two main packages:
The goal of the PRISM project is the development of infrastructure and algorithms for the paralle... more The goal of the PRISM project is the development of infrastructure and algorithms for the parallel solution of eigenvalue problems. We are currently investigating a complete eigensolver based on the Invariant Subspace Decomposition Algorithm for dense symmetric matrices (SYISDA). After brie y reviewing SYISDA, we discuss the algorithmic highlights of a distributed-memory implementation of this approach. These include a fast matrix-matrix multiplication algorithm, a new approach to parallel band reduction and tridiagonalization, and a harness for coordinating the divide-and-conquer parallelism in the problem. We also present performance results of these kernels as well as the overall SYISDA implementation on the Intel Touchstone Delta prototype.
... Steven Huss-Lederman Elaine M. Jacobson Anna Tsao ... Several algorithms have been proposed f... more ... Steven Huss-Lederman Elaine M. Jacobson Anna Tsao ... Several algorithms have been proposed for paral-lel, distributed memory matrix multiplication includ-ing 1D-systolic [15], 2D-systolic or Cannon's algo-rithm [5, 151, Broadcast-Multiply-Roll [12, 13, 141 and ...
In this paper, preliminary research results on a new algorithm for finding all the eigenvalues an... more In this paper, preliminary research results on a new algorithm for finding all the eigenvalues and eigenvectors of a real diagonalizable matrix with real eigenvalues are presented. The basic mathematical theory behind this approach is reviewed and is followed by a discussion of the numerical considerations of the actual implementation. The numerical algorithm has been tested on thousands of matrices
In this paper we report on the development of an efficient and portable implementation of Strasse... more In this paper we report on the development of an efficient and portable implementation of Strassen's matrix multiplication algorithm for matrices of arbitrary size. Our implementation is designed to be used in place of DGEMM, the Level 3 BLAS matrix multiplication routine. Our code is designed so that efficient performance will be obtained for all matrix sizes and shapes and that the additional memory needed for temporary variables is minimized. Replacing DGEMM with our routine should provide significant performance gain for large matrices while providing the same performance for small matrices. We measure performance of our code on the IBM RS/6000, CRAY YMP C90, and CRAY T3D single processor, and offer comparisons to other codes. Our performance data reconfirms that Strassen's algorithm is practical for realistic size matrices. The usefulness of our implementation is demonstrated by replacing DGEMM with our routine in a large application code. Our implementation is based on...
... Andersson Alexander Andrianov Colin Bennett Emil Cornea Fredrik Ekstedt Fausto Espinal Stephe... more ... Andersson Alexander Andrianov Colin Bennett Emil Cornea Fredrik Ekstedt Fausto Espinal Stephen Fenner Michael Hilton Ralph Howard Terry Huntsberger Björn Jawerth Goran Kronquist Toshiro Kubota Martin Lindberg Yiping Mao Per Gunnar Martinsson Johan Ràde Henrik ...
In this paper, we discuss the performance achieved by several implementations of the recently def... more In this paper, we discuss the performance achieved by several implementations of the recently defined Message Passing Interface (MPI) standard. In particular, performance results for different implementations of the broadcast operation are analyzed and compared on the Delta, ...
... Steven Huss-Lederman Elaine M. Jacobson Anna Tsao ... LetLCM represent the least common multi... more ... Steven Huss-Lederman Elaine M. Jacobson Anna Tsao ... LetLCM represent the least common multiple of P and Q. PUMMA begins by shu ing the rows of B on each node so adjacent rows are the same as the columns of A that will be multiplied and communicated at each stage. ...
In this paper we report on the development of an e cient and portable implementation of Strassen'... more In this paper we report on the development of an e cient and portable implementation of Strassen's matrix multiplication algorithm. Our implementation is designed to be used in place of DGEMM, the Level 3 BLAS matrix multiplication routine. E cient performance will be obtained for all matrix sizes and shapes and the additional memory needed for temporaryvariables has beenminimized. Replacing DGEMM with our routine should provide a signi cant performance gain for large matrices while providing the same performance for small matrices. We measure performance of our code on the IBM RS/6000, CRAY YMP C90, and CRAY T3D single processor, and o er comparisons to other codes. Our performance data recon rms that Strassen's algorithm is practical for realistic size matrices. The usefulness of our implementation is demonstrated by replacing DGEMM with our routine in a large application code.
In this document, we are concerned with the effccts of data layouts for nonsquare processor meshe... more In this document, we are concerned with the effccts of data layouts for nonsquare processor meshes on the implementation of common dense linear algebra kernels such as matrix-matrix multiplication, LU factorizations, or eigenvalue solvers. In particular, we address ease of programming and tunability of the resulting software. We introduce a generalization of the torus wrap data layout that results in a decoupling of "local" and "global" data layout view. As a result, it allows for intuitive programming of linear algebra algorithms and for tuning of the algorithm for a particular mesh aspect ratio or machine characteristics. This layout is as simple as the proposed HPF layout but, in our opinion, enhances ease of programming as well as ease of performance tuning. We emphasize that we do not advocate that all users need be concerned with these issues. We do, however, believe, that for the foreseeable future "assembler coding" (as message-passing code is likely to be viewed from a HPF programmers' perspective) will be needed to deliver high performance for computationally intensive kernels. As a result, we believe that the adoption of this approach not only would accelerate the generation of efficient linear algebra software libraries but also would accelerate the adoption of HPF as a result. We point out, however, that the adoption of this new layout would necessitate that an HPF compiler ensure that data objects are operated on in a consistent fashion across subroutine and function calls.
In this paper, preliminary research results on a new algorithm for finding all the eigenvalues an... more In this paper, preliminary research results on a new algorithm for finding all the eigenvalues and eigenvectors of a real diagonalizable matrix with real eigenvalues are presented. The basic mathematical theory behind this approach is reviewed and is followed by a discussion of the numerical considerations of the actual implementation. The numerical algorithm has been tested on thousands of matrices on both a Cray-2 and an IBM RS/6000 Model 580 workstation. The results of these tests are presented. Finally, issues concerning the parallel implementation of the algorithm are discussed. The algorithm's heavy reliance on matrix-matrix multiplication, coupled with the divide and conquer nature of this algorithm, should yield a highly parallelizable algorithm.
In this paper, we give an overview of the Invariant Subspace Decomposition Algorithm for banded s... more In this paper, we give an overview of the Invariant Subspace Decomposition Algorithm for banded symmetric matrices and describe a sequential implementation of this algorithm. Our implementation uses a specialized routine for performing banded matrix multiplication together with ...
In this paper we report on the development of an e cient and portable implementation of Strassen&... more In this paper we report on the development of an e cient and portable implementation of Strassen's matrix multiplication algorithm for matrices of arbitrary size. Our implementation is designed to be used in place of DGEMM, the Level 3 BLAS matrix multiplication routine. Our code is designed so that e cient performance will be obtained for all matrix sizes and shapes and that the additional memory needed for temporary variables is minimized. Replacing DGEMM with our routine should provide signi cant performance gain for large matrices while providing the same performance for small matrices. We measure performance of our code on the IBM RS/6000, CRAY YMP C90, and CRAY T3D single processor, and o er comparisons to other codes. Our performance data recon rms that Strassen's algorithm is practical for realistic size matrices. The usefulness of our implementation is demonstrated by replacing DGEMM with our routine in a large application code. Our implementation is based on a deta...
ABSTRACT this document relates to the packages developed for solving the symmetric eigenproblem w... more ABSTRACT this document relates to the packages developed for solving the symmetric eigenproblem which is joint work between Argonne National Laboratory, Center for Computing Sciences, and Duke University. The software contains two main packages:
The goal of the PRISM project is the development of infrastructure and algorithms for the paralle... more The goal of the PRISM project is the development of infrastructure and algorithms for the parallel solution of eigenvalue problems. We are currently investigating a complete eigensolver based on the Invariant Subspace Decomposition Algorithm for dense symmetric matrices (SYISDA). After brie y reviewing SYISDA, we discuss the algorithmic highlights of a distributed-memory implementation of this approach. These include a fast matrix-matrix multiplication algorithm, a new approach to parallel band reduction and tridiagonalization, and a harness for coordinating the divide-and-conquer parallelism in the problem. We also present performance results of these kernels as well as the overall SYISDA implementation on the Intel Touchstone Delta prototype.
... Steven Huss-Lederman Elaine M. Jacobson Anna Tsao ... Several algorithms have been proposed f... more ... Steven Huss-Lederman Elaine M. Jacobson Anna Tsao ... Several algorithms have been proposed for paral-lel, distributed memory matrix multiplication includ-ing 1D-systolic [15], 2D-systolic or Cannon's algo-rithm [5, 151, Broadcast-Multiply-Roll [12, 13, 141 and ...
In this paper, preliminary research results on a new algorithm for finding all the eigenvalues an... more In this paper, preliminary research results on a new algorithm for finding all the eigenvalues and eigenvectors of a real diagonalizable matrix with real eigenvalues are presented. The basic mathematical theory behind this approach is reviewed and is followed by a discussion of the numerical considerations of the actual implementation. The numerical algorithm has been tested on thousands of matrices
In this paper we report on the development of an efficient and portable implementation of Strasse... more In this paper we report on the development of an efficient and portable implementation of Strassen's matrix multiplication algorithm for matrices of arbitrary size. Our implementation is designed to be used in place of DGEMM, the Level 3 BLAS matrix multiplication routine. Our code is designed so that efficient performance will be obtained for all matrix sizes and shapes and that the additional memory needed for temporary variables is minimized. Replacing DGEMM with our routine should provide significant performance gain for large matrices while providing the same performance for small matrices. We measure performance of our code on the IBM RS/6000, CRAY YMP C90, and CRAY T3D single processor, and offer comparisons to other codes. Our performance data reconfirms that Strassen's algorithm is practical for realistic size matrices. The usefulness of our implementation is demonstrated by replacing DGEMM with our routine in a large application code. Our implementation is based on...
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