Gaussian Elimination

Higham, Nicholas J. (2011) Gaussian Elimination. Wiley Interdisciplinary Reviews: Computational Statistics, 3 (3). pp. 230-238. ISSN 1939-0068

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Abstract

As the standard method for solving systems of linear equations, Gaussian elimination (GE) is one of the most important and ubiquitous numerical algorithms. However, its successful use relies on understanding its numerical stability properties and how to organize its computations for efficient execution on modern computers. We give an overview of GE, ranging from theory to computation. We explain why GE computes an LU factorization and the various benefits of this matrix factorization viewpoint. Pivoting strategies for ensuring numerical stability are described. Special properties of GE for certain classes of structured matrices are summarized. How to implement GE in a way that efficiently exploits the hierarchical memories of modern computers is discussed. We also describe block LU factorization, corresponding to the use of pivot blocks instead of pivot elements, and explain how iterative refinement can be used to improve a solution computed by GE. Other topics are GE for sparse matrices and the role GE plays in the TOP500 ranking of the world's fastest computers.

Item Type:	Article
Uncontrolled Keywords:	Gaussian elimination, LU factorization, pivoting, numerical stability, block LU factorization, iterative refinement
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User:	Nick Higham
Date Deposited:	09 May 2011
Last Modified:	20 Oct 2017 14:12
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/1579

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