发表机构
School of Engineering and Applied Sciences, Harvard University（哈佛大学工程与应用科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明部分主元高斯消去法在随机矩阵中不稳定性（增长因子大）呈指数级罕见，针对角元素给出部分结果。
AI 中文摘要
部分主元高斯消去法是求解稠密 $n\times n$ 线性方程组 $Ax=b$ 的标准算法。该过程的“增长因子”$\rho$ 可能高达 $2^{n-1}$，当 $\rho$ 较大时，算法不稳定。然而，长期经验表明，高斯消去法在实践中极其稳定。实验表明，在具有独立正态分布元素的随机矩阵中，增长因子 $\rho \gg n^{1/2}$ 在精确意义上是呈指数级罕见的。这里我们朝着证实这一观察迈出了一步，证明的不是依赖于上三角因子 $U$ 所有元素的完整增长因子，而是针对角元素 $u_{nn}$ 的结论。
英文摘要
Gaussian elimination with partial pivoting is the standard algorithm for solving a dense $n\times n$ system of linear equations $Ax=b$. The ``growth factor'' $ρ\,$ for this process may be as large as $2^{n-1}$, and when $ρ$ is large, the algorithm is unstable. Nevertheless, long experience has shown that Gaussian elimination is resoundingly stable in practice. Experiments indicate that among random matrices with independent normally distributed entries, growth factors $ρ\gg n^{1/2}$ are in a precise sense exponentially rare. Here we take a step toward confirming this observation by proving it not for the full growth factor, which depends on all the entries of the upper-triangular factor $U$, but for the corner entry $u_{nn}^{}$.
