Direct Methods for Linear Systems

2014 ◽  
pp. 1-209 ◽  
Author(s):  
Åke Björck
Keyword(s):  
Linear Systems ◽  
Direct Methods

scholarly journals Amesos2 and Belos: Direct and Iterative Solvers for Large Sparse Linear Systems

2012 ◽  
Vol 20 (3) ◽  
pp. 241-255 ◽  
Author(s):  
Eric Bavier ◽  
Mark Hoemmen ◽  
Sivasankaran Rajamanickam ◽  
Heidi Thornquist
Keyword(s):  
Linear Systems ◽  
Iterative Methods ◽  
Sparse Matrix ◽  
Sparse Matrices ◽  
Direct Methods ◽  
Iterative Solvers ◽  
Software Project ◽  
Sparse Linear Systems ◽  
Scalar Type ◽  
Mixed Precision

Solvers for large sparse linear systems come in two categories: direct and iterative. Amesos2, a package in the Trilinos software project, provides direct methods, and Belos, another Trilinos package, provides iterative methods. Amesos2 offers a common interface to many different sparse matrix factorization codes, and can handle any implementation of sparse matrices and vectors, via an easy-to-extend C++ traits interface. It can also factor matrices whose entries have arbitrary “Scalar” type, enabling extended-precision and mixed-precision algorithms. Belos includes many different iterative methods for solving large sparse linear systems and least-squares problems. Unlike competing iterative solver libraries, Belos completely decouples the algorithms from the implementations of the underlying linear algebra objects. This lets Belos exploit the latest hardware without changes to the code. Belos favors algorithms that solve higher-level problems, such as multiple simultaneous linear systems and sequences of related linear systems, faster than standard algorithms. The package also supports extended-precision and mixed-precision algorithms. Together, Amesos2 and Belos form a complete suite of sparse linear solvers.

A Study of Sparse Matrix Methods on New Hardware

2015 ◽  
Vol 3 (3) ◽  
pp. 1-19
Author(s):  
Athanasios Fevgas ◽  
Konstantis Daloukas ◽  
Panagiota Tsompanopoulou ◽  
Panayiotis Bozanis
Keyword(s):  
Power Systems ◽  
Linear Systems ◽  
Large Scale ◽  
Comprehensive Evaluation ◽  
Sparse Matrix ◽  
Electrical Power ◽  
Iterative Algorithms ◽  
Direct Methods ◽  
Computer Hardware ◽  
Computing Platforms

Modeling of numerous scientific and engineering problems, such as multi-physic problems and analysis of electrical power systems, amounts to the solution of large scale linear systems. The main characteristics of such systems are the large sparsity ratio and the large number of unknowns that can reach thousands or even millions of equations. As a result, efficient solution of sparse large-scale linear systems is of great importance in order to enable analysis of such problems. Direct and iterative algorithms are the prevalent methods for solution of linear systems. Advances in computer hardware provide new challenges and capabilities for sparse solvers. The authors present a comprehensive evaluation of some, state of the art, sparse methods (direct and iterative) using modern computing platforms, aiming to determine the performance boundaries of each solver on different hardware infrastructures. By identifying the potential performance bottlenecks of out-of-core direct methods, the authors present a series of optimizations that increase their efficiency on flash-based systems.

Direct Methods for the Solution of Linear Systems

2007 ◽  
pp. 57-122
Author(s):  
Alfio Quarteroni ◽  
Riccardo Sacco ◽  
Fausto Saleri
Keyword(s):  
Linear Systems ◽  
Direct Methods

Direct Methods for the Solution of Linear Systems

2006 ◽  
pp. 59-124
Author(s):  
Alfio Quarteroni ◽  
Riccardo Sacco ◽  
Fausto Saleri
Keyword(s):  
Linear Systems ◽  
Direct Methods
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