research-article

Large Growth Factors in Gaussian Elimination with Pivoting

Authors:

Nicholas J. Higham,

Desmond J. HighamAuthors Info & Claims

SIAM Journal on Matrix Analysis and Applications, Volume 10, Issue 2

Pages 155 - 164

https://doi.org/10.1137/0610012

Published: 01 April 1989 Publication History

Abstract

The growth factor plays an important role in the error analysis of Gaussian elimination. It is well known that when partial pivoting or complete pivoting is used the growth factor is usually small, but it can be large. The examples of large growth usually quoted involve contrived matrices that are unlikely to occur in practice. We present real and complex $n \times n$ matrices arising from practical applications that, for any pivoting strategy, yield growth factors bounded below by $n / 2$ and n, respectively. These matrices enable us to improve the known lower bounds on the largest possible growth factor in the case of complete pivoting. For partial pivoting, we classify the set of real matrices for which the growth factor is $2^{n - 1} $. Finally, we show that large element growth does not necessarily lead to a large backward error in the solution of a particular linear system, and we comment on the practical implications of this result.

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Cited By

Liu QZhang QXu X(2023)Accuracy and stability of quaternion Gaussian eliminationNumerical Algorithms10.1007/s11075-023-01531-y94:3(1159-1183)Online publication date: 28-Apr-2023
https://dl.acm.org/doi/10.1007/s11075-023-01531-y
Baboulin MDongarra JHerrmann JTomov S(2013)Accelerating Linear System Solutions Using Randomization TechniquesACM Transactions on Mathematical Software10.1145/2427023.242702539:2(1-13)Online publication date: 1-Feb-2013
https://dl.acm.org/doi/10.1145/2427023.2427025
Higham N(2011)Gaussian eliminationWIREs Computational Statistics10.1002/wics.1643:3(230-238)Online publication date: 4-Apr-2011
https://dl.acm.org/doi/10.1002/wics.164
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Large Growth Factors in Gaussian Elimination with Pivoting

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Published In

cover image SIAM Journal on Matrix Analysis and Applications

SIAM Journal on Matrix Analysis and Applications Volume 10, Issue 2

Apr 1989

143 pages

ISSN:0895-4798

DOI:10.1137/sjmael.1989.10.issue-2

Issue’s Table of Contents

Copyright © 1989 Society for Industrial and Applied Mathematics.

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Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 April 1989

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Cited By

Liu QZhang QXu X(2023)Accuracy and stability of quaternion Gaussian eliminationNumerical Algorithms10.1007/s11075-023-01531-y94:3(1159-1183)Online publication date: 28-Apr-2023
https://dl.acm.org/doi/10.1007/s11075-023-01531-y
Baboulin MDongarra JHerrmann JTomov S(2013)Accelerating Linear System Solutions Using Randomization TechniquesACM Transactions on Mathematical Software10.1145/2427023.242702539:2(1-13)Online publication date: 1-Feb-2013
https://dl.acm.org/doi/10.1145/2427023.2427025
Higham N(2011)Gaussian eliminationWIREs Computational Statistics10.1002/wics.1643:3(230-238)Online publication date: 4-Apr-2011
https://dl.acm.org/doi/10.1002/wics.164
Scott J(2010)Scaling and pivoting in an out-of-core sparse direct solverACM Transactions on Mathematical Software10.1145/1731022.173102937:2(1-23)Online publication date: 23-Apr-2010
https://dl.acm.org/doi/10.1145/1731022.1731029
Favati PLeoncini MMartinez A(2000)On the Robustness of Gaussian Elimination with Partial PivotingBIT10.1023/A:102231420148440:1(62-73)Online publication date: 1-Mar-2000
https://dl.acm.org/doi/10.1023/A%3A1022314201484
Amodio PMazzia F(1999)A New Approach to Backward Error Analysis of Lu FactorizationBIT10.1023/A:102235830051739:3(385-402)Online publication date: 1-Sep-1999
https://dl.acm.org/doi/10.1023/A%3A1022358300517
Wright S(1993)A Collection of Problems for Which Gaussian Elimination with Partial Pivoting is UnstableSIAM Journal on Scientific Computing10.1137/091401314:1(231-238)Online publication date: 1-Jan-1993
https://dl.acm.org/doi/10.1137/0914013

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