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Gaussian elimination

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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. 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

Illustration of how rook pivoting searches for the first pivot for a particular 6 × 6 matrix (with the positive integer entries shown). Each dot denotes a putative pivot that is tested to see if it is the largest in magnitude in both its row and its column.

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Algorithms and Computational Methods > Numerical Methods
Algorithms and Computational Methods > Numerical Finite Arithmetic
Applications of Computational Statistics > Computational Mathematics

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