This paper combines modern numerical computation with theoretical results to improve our understanding of the growth factor problem for Gaussian elimination. On the computational side we obtain lower bounds for the maximum growth for complete pivoting for $n=1:75$ and $n=100$ using the Julia JuMP optimization 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 to a subset of the reals is nearly equal to the maximum growth factor over all real matrices. We also show that the growth factors under floating point arithmetic and exact arithmetic are nearly identical. Finally, through numerical search, and stability and extrapolation results, we provide improved lower bounds for the maximum growth factor. Specifically, we find that the largest growth factor is bigger than $1.0045n$, and the lim sup of the ratio with $n$ is greater than or equal to $3.317$. In contrast to the old conjecture that growth might never be bigger than $n$, it seems likely that the maximum growth divided by $n$ goes to infinity as $n \rightarrow \infty$.

翻译：本文结合现代数值计算与理论结果，以增进对高斯消元法中增长因子问题的理解。在计算方面，我们利用Julia JuMP优化包，针对n=1:75和n=100的情况，获得了完全主元消元法下最大增长因子的下界。当n=100时，得到的增长因子大于3n。数值证据表明，最大增长因子大于n当且仅当n≥11。我们还提出了一系列理论结果。我们证明，在矩阵元素限制为实数子集的情况下，最大增长因子几乎等于所有实数矩阵下的最大增长因子。同时，我们证实浮点算术与精确算术下的增长因子几乎相同。最后，通过数值搜索、稳定性分析及外推结果，我们给出了最大增长因子的改进下界。具体而言，我们发现最大增长因子大于1.0045n，且其与n之比的极限上确界大于等于3.317。与过去认为增长因子永远不会大于n的猜想相反，最大增长因子除以n的比值似乎很可能随着n→∞而趋于无穷。