高斯消元中最大增长因子的若干新结果
Some New Results on the Maximum Growth Factor in Gaussian Elimination
AI总结：
本文结合 Julia JuMP 数值优化与理论分析，研究高斯消元完全选主元的最大增长因子，给出 n≤100 的下界及渐近下界，表明其与 n 之比可能趋于无穷。
摘要：
本文将现代数值计算与理论结果相结合，以增进我们对高斯消元增长因子问题的理解。在计算方面，我们使用 Julia JuMP 优化包，针对 n=1:75 和 n=100 获得了完全选主元最大增长的下界。在 n=100 时，我们得到大于 3n 的增长因子。数值证据表明，最大增长因子大于 n 当且仅当 n≥11。我们还给出了若干理论结果。我们证明，元素限制在实数的某个子集内的矩阵上的最大增长因子，几乎等于所有实矩阵上的最大增长因子。我们还证明，浮点运算与精确运算下的增长因子几乎相同。最后，通过数值搜索以及稳定性和外推结果，我们给出了最大增长因子的改进下界。具体而言，我们发现对于 n>10，最大增长因子大于 1.0045n，并且该量与 n 之比的上极限大于或等于 3.317。与增长可能永远不会大于 n 的旧猜想相反，看起来当 n→∞ 时，最大增长除以 n 会趋于无穷大。
英文摘要：
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 $n \rightarrow \infty$.
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