Large Growth Factors in Gaussian Elimination with Pivoting

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.