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Gaussian elimination
Author(s) -
Higham Nicholas J.
Publication year - 2011
Publication title -
wiley interdisciplinary reviews: computational statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.693
H-Index - 38
eISSN - 1939-0068
pISSN - 1939-5108
DOI - 10.1002/wics.164
Subject(s) - gaussian elimination , factorization , computer science , numerical stability , lu decomposition , gaussian , computation , block (permutation group theory) , stability (learning theory) , numerical analysis , algorithm , numerical linear algebra , linear system , computational complexity theory , theoretical computer science , matrix decomposition , mathematics , combinatorics , mathematical analysis , eigenvalues and eigenvectors , physics , quantum mechanics , machine learning
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. WIREs Comp Stat 2011 3 230–238 DOI: 10.1002/wics.164 This article is categorized under: Applications of Computational Statistics > Computational Mathematics Algorithms and Computational Methods > Numerical Finite Arithmetic Algorithms and Computational Methods > Numerical Methods

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