ALAFF LU factorization with pivoting can fail in practice

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Subsection 6.5.2 LU factorization with pivoting can fail in practice

While LU factorization with pivoting is considered to be a numerically stable approach to solving linear systems, the following paper discusses cases where it may fail in practice:

[18] Leslie V. Foster, Gaussian elimination with partial pivoting can fail in practice, SIAM Journal on Matrix Analysis and Applications, 15 (1994), pp. 1354–1362.

Also of interest may be the paper

[50] Stephen J. Wright, A Collection of Problems for Which {G}aussian Elimination with Partial Pivoting is Unstable, SIAM Journal on Scientific Computing, Vol. 14, No. 1, 1993.

which discusses a number of (not necessarily practical) examples where LU factorization with pivoting fails.