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，深化了对高斯消元法中增长因子问题的理解。在计算方面，利用Julia JuMP优化包，我们得到了n=1.75和n=100时全主元法最大增长因子的下界。当n=100时，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：限制于实数子集的矩阵的最大增长因子，与全体实矩阵的最大增长因子几乎相等；浮点运算与精确运算下的增长因子也几乎一致。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：最大增长因子大于1.0045n，且其与n的比值上极限不小于3.317。与以往关于增长因子可能永不超过n的猜想相反，最大增长因子除以n的比值很可能随n→∞而趋于无穷。