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This paper combines modern numerical computation with theoretical results toimprove our understanding of the growth factor problem for Gaussianelimination. On the computational side we obtain lower bounds for the maximumgrowth for complete pivoting for $n=1:75$ and $n=100$ using the Julia JuMPoptimization package. At $n=100$ we obtain a growth factor bigger than $3n$.The numerical evidence suggests that the maximum growth factor is bigger than$n$ if and only if $n \ge 11$. We also present a number of theoretical results.We show that the maximum growth factor over matrices with entries restricted toa subset of the reals is nearly equal to the maximum growth factor over allreal matrices. We also show that the growth factors under floating pointarithmetic and exact arithmetic are nearly identical. Finally, throughnumerical search, and stability and extrapolation results, we provide improvedlower bounds for the maximum growth factor. Specifically, we find that thelargest growth factor is bigger than $1.0045n$ for $n>10$, and the lim sup ofthe ratio with $n$ is greater than or equal to $3.317$. In contrast to the oldconjecture that growth might never be bigger than $n$, it seems likely that themaximum growth divided by $n$ goes to infinity as $n \rightarrow \infty$.

Some New Results on the Maximum Growth Factor in Gaussian Elimination