Journal ArticleDOI
A survey of direct methods for sparse linear systems
TLDR
The goal of this survey article is to impart a working knowledge of the underlying theory and practice of sparse direct methods for solving linear systems and least-squares problems, and to provide an overview of the algorithms, data structures, and software available to solve these problems.Abstract:
Wilkinson defined a sparse matrix as one with enough zeros that it pays to take advantage of them.1 This informal yet practical definition captures the essence of the goal of direct methods for solving sparse matrix problems. They exploit the sparsity of a matrix to solve problems economically: much faster and using far less memory than if all the entries of a matrix were stored and took part in explicit computations. These methods form the backbone of a wide range of problems in computational science. A glimpse of the breadth of applications relying on sparse solvers can be seen in the origins of matrices in published matrix benchmark collections (Duff and Reid 1979a, Duff, Grimes and Lewis 1989a, Davis and Hu 2011). The goal of this survey article is to impart a working knowledge of the underlying theory and practice of sparse direct methods for solving linear systems and least-squares problems, and to provide an overview of the algorithms, data structures, and software available to solve these problems, so that the reader can both understand the methods and know how best to use them.read more
Citations
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High Performance Computing
TL;DR: The key elements of the Core Program will be described including the construction of a UK e-Science Grid and the need to develop a data architecture for the Grid that will allow federated access to relational databases as well as flat files.
Journal ArticleDOI
Randomized numerical linear algebra: Foundations and algorithms
TL;DR: This survey describes probabilistic algorithms for linear algebraic computations, such as factorizing matrices and solving linear systems, that have a proven track record for real-world problems and treats both the theoretical foundations of the subject and practical computational issues.
Journal ArticleDOI
Algorithm 1000: SuiteSparse:GraphBLAS: Graph Algorithms in the Language of Sparse Linear Algebra
TL;DR: SuiteSparse:GraphBLAS is a full implementation of the GraphBLAS standard, which defines a set of sparse matrix operations on an extended algebra of semirings using an almost unlimited variety of operators and types.
Posted Content
Randomized Numerical Linear Algebra: Foundations & Algorithms.
TL;DR: This survey describes probabilistic algorithms for linear algebra computations, such as factorizing matrices and solving linear systems, that have a proven track record for real-world problem instances and treats both the theoretical foundations and the practical computational issues.
References
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Book
Introduction to Algorithms
TL;DR: The updated new edition of the classic Introduction to Algorithms is intended primarily for use in undergraduate or graduate courses in algorithms or data structures and presents a rich variety of algorithms and covers them in considerable depth while making their design and analysis accessible to all levels of readers.
Book ChapterDOI
Introduction to Algorithms
TL;DR: This chapter provides an overview of the fundamentals of algorithms and their links to self-organization, exploration, and exploitation.
Journal ArticleDOI
Depth-First Search and Linear Graph Algorithms
TL;DR: The value of depth-first search or “backtracking” as a technique for solving problems is illustrated by two examples of an improved version of an algorithm for finding the strongly connected components of a directed graph.
Journal ArticleDOI
A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs
George Karypis,Vipin Kumar +1 more
TL;DR: This work presents a new coarsening heuristic (called heavy-edge heuristic) for which the size of the partition of the coarse graph is within a small factor of theSize of the final partition obtained after multilevel refinement, and presents a much faster variation of the Kernighan--Lin (KL) algorithm for refining during uncoarsening.