Journal Article10.1137/1032002
Parallel algorithms for dense linear algebra computations
214
TL;DR: The purpose is to review the current status and to provide an overall perspective of parallel algorithms for solving dense, banded, or block-structured problems arising in the major areas of direct solution of linear systems, least squares computations, eigenvalue and singular value computation, and rapid elliptic solvers.
read more
Abstract: Scientific and engineering research is becoming increasingly dependent upon the development and implementation of efficient parallel algorithms on modern high-performance computers. Numerical linear algebra is an indispensable tool in such research and this paper attempts to collect and describe a selection of some of its more important parallel algorithms. The purpose is to review the current status and to provide an overall perspective of parallel algorithms for solving dense, banded, or block-structured problems arising in the major areas of direct solution of linear systems, least squares computations, eigenvalue and singular value computations, and rapid elliptic solvers. A major emphasis is given here to certain computational primitives whose efficient execution on parallel and vector computers is essential in order to obtain high performance algorithms.
read more
Chat with Paper
AI Agents for this Paper
Find similar papers on Google Scholar, PubMed and Arxiv
Write a critical review of this paper
Analyze citations of this paper to find unaddressed research gaps
Citations
•Book
Proximal Algorithms
Neal Parikh,Stephen Boyd +1 more
- 27 Nov 2013
TL;DR: The many different interpretations of proximal operators and algorithms are discussed, their connections to many other topics in optimization and applied mathematics are described, some popular algorithms are surveyed, and a large number of examples of proxiesimal operators that commonly arise in practice are provided.
4.2K
LAPACK: a portable linear algebra library for high-performance computers
E. Anderson,Zhaojun Bai,Jack Dongarra,Anne Greenbaum,A. McKenney,J. Du Croz,S. Hammarling,James Demmel,Christian Bischof,Danny C. Sorensen +9 more
- 01 Oct 1990
TL;DR: The goal of the LAPACK project is to design and implement a portable linear algebra library for efficient use on a variety of high-performance computers, based on the widely used LINPACK and EISPACK packages, but extends their functionality in a number of ways.
516
LAPACK: A portable linear algebra library for high-performance computers
TL;DR: The goal of the LAPACK project is to design and implement a portable linear algebra library for efficient use on a variety of high-performance computers, based on the widely used LINPACK and EISPACK packages, but extends their functionality in a number of ways.
Design and Implementation of the ScaLAPACK LU, QR, and Cholesky Factorization Routines
TL;DR: The ScaLAPACK library as discussed by the authors provides a set of core factorization routines that allow the factorization and solution of a dense system of linear equations via LU, QR, and Cholesky.
Software libraries for linear algebra computations on high performance computers
Jack Dongarra,David W. Walker +1 more
TL;DR: This paper discusses the design of linear algebra libraries for high performance computers, with particular emphasis on the development of scalable algorithms for multiple instruction multiple data (MIMD) distributed memory concurrent computers.
137
References
Singular value decomposition and least squares solutions
Gene H. Golub,C. Reinsch +1 more
TL;DR: The decomposition of A is called the singular value decomposition (SVD) and the diagonal elements of ∑ are the non-negative square roots of the eigenvalues of A T A; they are called singular values.
3.3K
Gaussian elimination is not optimal
TL;DR: In this paper, Cook et al. gave an algorithm which computes the coefficients of the product of two square matrices A and B of order n with less than 4. 7 n l°g 7 arithmetical operations (all logarithms in this paper are for base 2).
2.9K
•Book
Computer methods for mathematical computations
George E. Forsythe,Michael A. Malcolm,Cleve B. Moler +2 more
- 01 Jan 1977
•Book
Introduction to matrix computations
G. W. Stewart
- 11 Jun 1973
TL;DR: Rounding-Error Analysis of Solution of Triangular Systems and of Gaussian Elimination.
2.5K