Gaussian elimination (GE) is the most used dense linear solver. Error analysis of GE with selected pivoting strategies on well-conditioned systems can focus on studying the behavior of growth factors. Although exponential growth is possible with GE with partial pivoting (GEPP), growth tends to stay much smaller in practice. Support for this behavior was provided recently by Huang and Tikhomirov's average-case analysis of GEPP, which showed GEPP growth factors for Gaussian matrices stay at most polynomial with very high probability. GE with complete pivoting (GECP) has also seen a lot of recent interest, with improvements to both lower and upper bounds on worst-case GECP growth provided by Bisain, Edelman and Urschel in 2023. We are interested in studying how GEPP and GECP behave on the same linear systems as well as studying large growth on particular subclasses of matrices, including orthogonal matrices. Moreover, as a means to better address the question of why large growth is rarely encountered, we further study matrices with a large difference in growth between using GEPP and GECP, and we explore how the smaller growth strategy dominates behavior in a small neighborhood of the initial matrix.

翻译：高斯消去法（GE）是最常用的稠密线性方程组求解器。在良态系统上采用特定选主元策略的GE误差分析可聚焦于增长因子的行为研究。尽管使用部分主元高斯消去法（GEPP）可能出现指数增长，但在实际应用中增长幅度通常显著较小。Huang与Tikhomirov近期对GEPP的平均情况分析为该行为提供了理论支持，证明了高斯矩阵的GEPP增长因子以极高概率保持多项式阶增长。采用完全主元高斯消去法（GECP）近期同样备受关注，Bisain、Edelman和Urschel于2023年改进了最坏情况下GECP增长因子的上下界。本文致力于研究GEPP与GECP在相同线性系统上的行为差异，同时考察特定子类矩阵（包括正交矩阵）中的大增长现象。此外，为深入探究大增长罕见性的根本原因，我们进一步研究GEPP与GECP增长存在显著差异的矩阵，并揭示较小增长策略如何在初始矩阵的局部邻域中主导行为演化。