Results 1  10
of
27
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 ..."
Abstract

Cited by 161 (33 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 68 (17 self)
 Add to MetaCart
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...
Design of a Parallel Nonsymmetric Eigenroutine Toolbox, Part I
, 1993
"... The dense nonsymmetric eigenproblem is one of the hardest linear algebra problems to solve effectively on massively parallel machines. Rather than trying to design a "black box" eigenroutine in the spirit of EISPACK or LAPACK, we propose building a toolbox for this problem. The tools are meant to ..."
Abstract

Cited by 63 (14 self)
 Add to MetaCart
The dense nonsymmetric eigenproblem is one of the hardest linear algebra problems to solve effectively on massively parallel machines. Rather than trying to design a "black box" eigenroutine in the spirit of EISPACK or LAPACK, we propose building a toolbox for this problem. The tools are meant to be used in different combinations on different problems and architectures. In this paper, we will describe these tools which include basic block matrix computations, the matrix sign function, 2dimensional bisection, and spectral divide and conquer using the matrix sign function to find selected eigenvalues. We also outline how we deal with illconditioning and potential instability. Numerical examples are included. A future paper will discuss error analysis in detail and extensions to the generalized eigenproblem.
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. ..."
Abstract

Cited by 34 (5 self)
 Add to MetaCart
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...
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 ..."
Abstract

Cited by 23 (12 self)
 Add to MetaCart
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...
Parallel Tridiagonalization through TwoStep Band Reduction
 In Proceedings of the Scalable HighPerformance Computing Conference
, 1994
"... We present a twostep variant of the "successive band reduction" paradigm for the tridiagonalization of symmetric matrices. Here we reduce a full matrix first to narrowbanded form and then to tridiagonal form. The first step allows easy exploitation of block orthogonal transformations. In the secon ..."
Abstract

Cited by 23 (12 self)
 Add to MetaCart
We present a twostep variant of the "successive band reduction" paradigm for the tridiagonalization of symmetric matrices. Here we reduce a full matrix first to narrowbanded form and then to tridiagonal form. The first step allows easy exploitation of block orthogonal transformations. In the second step, we employ a new blocked version of a banded matrix tridiagonalization algorithm by Lang. In particular, we are able to express the update of the orthogonal transformation matrix in terms of block transformations. This expression leads to an algorithm that is almost entirely based on BLAS3 kernels and has greatly improved data movement and communication characteristics. We also present some performance results on the Intel Touchstone DELTA and the IBM SP1. 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 approachthereof [8] is the method of This work...
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 ..."
Abstract

Cited by 17 (5 self)
 Add to MetaCart
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 ...
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 ..."
Abstract

Cited by 16 (6 self)
 Add to MetaCart
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
The Design of Linear Algebra Libraries for High Performance Computers
, 1993
"... 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, followe ..."
Abstract

Cited by 16 (1 self)
 Add to MetaCart
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 movementbetween di#erent 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 Subgrams #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 ...
Parallel Performance of a Symmetric Eigensolver based on the Invariant Subspace Decomposition Approach
, 1994
"... In this paper, we discuss work in progress on a complete eigensolver based on the Invariant Subspace Decomposition Algorithm for dense symmetric matrices (SYISDA). We describe a recently developed acceleration technique that substantially reduces the overall work required by this algorithm and revie ..."
Abstract

Cited by 15 (0 self)
 Add to MetaCart
In this paper, we discuss work in progress on a complete eigensolver based on the Invariant Subspace Decomposition Algorithm for dense symmetric matrices (SYISDA). We describe a recently developed acceleration technique that substantially reduces the overall work required by this algorithm and review the algorithmic highlights of a distributedmemory implementation of this approach. These include a fast matrixmatrix multiplication algorithm, a new approach to parallel band reduction and tridiagonalization, and a harness for coordinating the divideandconquer parallelism in the problem. We present performance results for the dominant kernel, dense matrix multiplication, as well as for the overall SYISDA implementation on the Intel Touchstone Delta and the Intel Paragon. 1. Introduction Computation of eigenvalues and eigenvectors is an essential kernel in many applications, and several promising parallel algorithms have been investigated [26, 3, 28, 22, 25, 6]. The work presented in t...