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 $1.0045n$ for $n>10$, 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，加深了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 优化包获得了 n=1:75 和 n=100 时完全主元法最大增长因子的下界。当 n=100 时，我们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。我们还给出了一系列理论结果。我们证明，当矩阵元素限于实数子集时，其最大增长因子几乎等于所有实矩阵的最大增长因子。我们还表明，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、稳定性与外推结果，我们改进了最大增长因子的下界。具体而言，我们发现对于 n > 10，最大增长因子大于 1.0045n，且该比值与 n 的上极限大于或等于 3.317。与早期认为增长可能永远不会超过 n 的猜想相反，最大增长除以 n 似乎很可能随着 n → ∞ 而趋于无穷。