The growth factor in Gaussian elimination is the ratio of the largest number that appears at any stage of the elimination to the largest element in the original matrix. J.H. Wilkinson's backward error analysis of the early 1960s showed that Gaussian elimination is numerically stable as long as the growth factor remains small. In 1968 C. Cryer published the famous conjecture. This insight appears to have been fundamental to his success in breaking the conjecture. As well as settling the complete pivoting conjecture, Gould's work raises the hope that further understanding of the behaviour of the growth factor might be achieved with the help of numerical experiments.
Several questions remain unanswered.
All of these are questions of worst-case behavior. The algorithm almost invariably used in practice is Gaussian elimination with partial pivoting, which uses only row interchanges rather than row and column interchanges. Nevertheless, decades of experience and extensive numerical experiments have shown that in practice, Gaussian elimination with partial pivoting is highly stable, with growth factors of order N or less for all but a tiny proportion of matrices. This phenomenon of average-case stability is still not fully understood. I'm devising a survey to gauge use of reusable code by numerical programmers.