Gaussian elimination with partial pivoting (GEPP) remains the most common method to solve dense linear systems. Each GEPP step uses a row transposition pivot movement if needed to ensure the leading pivot entry is maximal in magnitude for the leading column of the remaining untriangularized subsystem. We will use theoretical and numerical approaches to study how often this pivot movement is needed. We provide full distributional descriptions for the number of pivot movements needed using GEPP using particular Haar random ensembles, as well as compare these models to other common transformations from randomized numerical linear algebra. Additionally, we introduce new random ensembles with fixed pivot movement counts and fixed sparsity, $\alpha$. Experiments estimating the empirical spectral density (ESD) of these random ensembles leads to a new conjecture on a universality class of random matrices with fixed sparsity whose scaled ESD converges to a measure on the complex unit disk that depends on $\alpha$ and is an interpolation of the uniform measure on the unit disk and the Dirac measure at the origin.

翻译：部分选主元高斯消去法仍是求解稠密线性系统最常用的方法。在每一步GEPP中，若需要，会通过行交换主元移动来确保剩余未三角化子系统的首列中，前导主元幅值最大。我们将通过理论与数值方法研究这种主元移动的发生频率。针对特定Haar随机系综，我们给出了使用GEPP时所需主元移动次数的完整分布描述，并将这些模型与随机数值线性代数中其他常见变换进行了比较。此外，我们引入了具有固定主元移动次数和固定稀疏度α的新随机系综。对这些随机系综经验谱密度的实验估计，催生了一个关于固定稀疏度随机矩阵普适类的新猜想：其缩放后的经验谱密度收敛于复单位圆盘上依赖于α的测度，该测度是单位圆盘上均匀测度与原点处Dirac测度的插值。