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On the stability of Gauss-Jordan elimination with pivoting

Authors:

G. Peters,

J. H. WilkinsonAuthors Info & Claims

Communications of the ACM, Volume 18, Issue 1

Pages 20 - 24

https://doi.org/10.1145/360569.360653

Published: 01 January 1975 Publication History

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Abstract

The stability of the Gauss-Jordan algorithm with partial pivoting for the solution of general systems of linear equations is commonly regarded as suspect. It is shown that in many respects suspicions are unfounded, and in general the absolute error in the solution is strictly comparable with that corresponding to Gaussian elimination with partial pivoting plus back substitution. However, when A is ill conditioned, the residual corresponding to the Gauss-Jordan solution will often be much greater than that corresponding to the Gaussian elimination solution.

References

[1]

Forsythge, G. E., and Moler, C. B., Computer Solution of Linear Algebraic Systems. Prentice-Hall, Englewood Cliffs, N.J., 1967.

Google Scholar

[2]

Wilkinson, J.H. Error analysis of direct methods of matrix inversion. J. ACM 8, 3 (July 1961), 281-330.

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[3]

Wilkinson, J.H. Rounding Errors in Algebraic Processes. Her Majesty's Stationery Store, London; and Prentice-Hall, Englewood Cliffs, N.J., 1963.

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[4]

Wilkinson, J.H. The Algebraic Eigeuvalue Problem. Oxford University Press, London, 1965.

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Cited By

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Dong YMartinsson P(2023)Simpler is better: a comparative study of randomized pivoting algorithms for CUR and interpolative decompositionsAdvances in Computational Mathematics10.1007/s10444-023-10061-z49:4Online publication date: 7-Aug-2023
https://dl.acm.org/doi/10.1007/s10444-023-10061-z
Tran NJustino JBerg I(2022)The explicit formula for Gauss-Jordan elimination applied to flexible systemsSpecial Matrices10.1515/spma-2022-016810:1(366-393)Online publication date: 10-Jun-2022
https://doi.org/10.1515/spma-2022-0168
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Index Terms

On the stability of Gauss-Jordan elimination with pivoting
1. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
2. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Error-correcting codes

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 January 1975

Published in CACM Volume 18, Issue 1

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Dong YMartinsson P(2023)Simpler is better: a comparative study of randomized pivoting algorithms for CUR and interpolative decompositionsAdvances in Computational Mathematics10.1007/s10444-023-10061-z49:4Online publication date: 7-Aug-2023
https://dl.acm.org/doi/10.1007/s10444-023-10061-z
Tran NJustino JBerg I(2022)The explicit formula for Gauss-Jordan elimination applied to flexible systemsSpecial Matrices10.1515/spma-2022-016810:1(366-393)Online publication date: 10-Jun-2022
https://doi.org/10.1515/spma-2022-0168
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Cited By

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Recommendations

An Implementation of Gaussian Elimination with Partial Pivoting for Sparse Systems

Large Growth Factors in Gaussian Elimination with Pivoting

Some Features of Gaussian Elimination with Rook Pivoting

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