Gaussian elimination with partial pivoting (GEPP) is a widely used 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：部分主元高斯消去法（GEPP）是一种广泛用于求解稠密线性系统的方法。在GEPP的每一步中，如有必要，会通过行转置主元移动来确保剩余未三角化子系统的首列的前导主元在幅度上达到最大值。我们将通过理论和数值方法研究这种主元移动的必要频率。针对特定的Haar随机系综，我们给出了GEPP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的其他常见变换进行了比较。此外，我们还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。对这些随机系综的经验谱密度（ESD）进行实验估算后，我们提出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ESD收敛于复单位圆盘上的一种测度，该测度依赖于α，并介于单位圆盘上的均匀测度与原点处的狄拉克测度之间。