发表机构
Massachusetts Institute of Technology（麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究随机矩阵的增长因子，证明了部分主元选取下高斯矩阵增长因子很少超过根号n，解决了Trefethen的猜想，并给出了无主元选取时的紧估计与渐近分布。
AI 中文摘要
高斯消元法是求解线性系统最古老且最常用的方法。对于给定矩阵，其数值稳定性由增长因子控制，增长因子衡量消元过程中矩阵元素可能变大的程度。本文证明了关于随机矩阵增长因子的若干新结果。首先，我们给出了经Haar正交矩阵预处理且不进行主元选取的矩阵增长因子的紧估计，并刻画了高斯矩阵在不进行主元选取时增长因子的渐近分布。其次，也是最重要的，我们证明了在部分主元选取下，$n \times n$ 高斯矩阵的增长因子很少远大于 $\sqrt{n}$，从而解决了Nick Trefethen的一个旧猜想。用于证明该猜想的相同技术还提供了部分主元选取下增长因子的改进平滑分析。
英文摘要
Gaussian elimination is the oldest and most popular method for solving a linear system. Its numerical stability for a given matrix is controlled by the growth factor, a measure of how large entries can become during elimination. In this work, we prove a number of new results regarding the growth factor of random matrices. First, we provide tight estimates for the growth factor of a matrix preconditioned by a Haar orthogonal matrix without pivoting and characterize the asymptotic distribution of the growth factor of Gaussian matrices without pivoting. Second, and most notably, we prove that the growth factor of an $n \times n$ Gaussian matrix under partial pivoting is rarely much larger than $\sqrt{n}$, resolving an old conjecture of Nick Trefethen. The same techniques used to prove this conjecture also provide an improved smoothed analysis of the growth factor under partial pivoting.
