Gaussian Elimination with Partial Pivoting Can Fail in Practice

Even though Gaussian elimination with partial pivoting is very widely used, $n \times n$ matrices can be constructed where the error growth in the algorithm is proportional to $2^{n-1}$. Thus for moderate or large $n$, in theory, there is a potential for disastrous error growth. However, prior to 1993 no reports of such an example in a practical application had appeared in the literature. Examples are presented that arise naturally from integral and differential equations and that lead to disastrous error growth in Gaussian elimination with partial pivoting.