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 last year by Huang and Tikhomirov's average-case analysis of GEPP, which showed GEPP growth factors stay at most polynomial with very high probability when using small Gaussian perturbations. GE with complete pivoting (GECP) has also seen a lot of recent interest, with recent improvements to lower bounds on worst-case GECP growth provided by Edelman and Urschel earlier this year. 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. We will also study systems when GECP leads to larger growth than GEPP, which will lead to new empirical lower bounds on how much worse GECP can behave compared to GEPP in terms of growth. We also present an empirical study on a family of exponential GEPP growth matrices whose polynomial behavior in small neighborhoods limits to the initial GECP growth factor.

翻译：高斯消元（GE）是最常用的稠密线性求解器。针对采用所选选主元策略的良态系统，GE的误差分析可聚焦于研究增长因子的行为。尽管采用部分选主元的GE（GEPP）可能出现指数级增长，但实际中增长往往远小于此。去年Huang和Tikhomirov对GEPP的平均情形分析支持了这一行为，该分析表明，当使用微小高斯扰动时，GEPP的增长因子以极高概率保持在多项式级别。采用全选主元的GE（GECP）近期也备受关注，今年早些时候Edelman和Urschel改进了最坏情形GECP增长的下界。我们关注研究GEPP和GECP在同一线性系统上的行为差异，以及特定矩阵子类（包括正交矩阵）中出现大增长的情况。我们还将研究GECP导致增长大于GEPP的系统，这将在增长方面为GECP相较GEPP可能表现更差的程度提供新的经验下界。此外，我们针对指数级GEPP增长矩阵族进行了实证研究，这些矩阵在小邻域内的多项式行为会收敛至初始GECP增长因子。