Random butterfly matrices were introduced by Parker in 1995 to remove the need for pivoting when using Gaussian elimination. The growing applications of butterfly matrices have often eclipsed the mathematical understanding of how or why butterfly matrices are able to accomplish these given tasks. To help begin to close this gap using theoretical and numerical approaches, we explore the impact on the growth factor of preconditioning a linear system by butterfly matrices. These results are compared to other common methods found in randomized numerical linear algebra. In these experiments, we show preconditioning using butterfly matrices has a more significant dampening impact on large growth factors than other common preconditioners and a smaller increase to minimal growth factor systems. Moreover, we are able to determine the full distribution of the growth factors for a subclass of random butterfly matrices. Previous results by Trefethen and Schreiber relating to the distribution of random growth factors were limited to empirical estimates of the first moment for Ginibre matrices.

翻译：帕克（Parker）于1995年引入随机蝶形矩阵，旨在消除高斯消元过程中的选主元需求。尽管蝶形矩阵的应用日益广泛，但对其如何实现特定任务的数学机理理解仍显不足。为弥合这一理论空白，本研究结合理论分析与数值方法，探讨了以蝶形矩阵预处理线性系统对增长因子的影响，并将结果与随机数值线性代数中其他常用方法进行对比。实验表明，相较于其他常见预处理器，蝶形矩阵预处理能更显著抑制大增长因子，同时对最小增长因子系统的增幅更小。此外，我们完整确定了随机蝶形矩阵子类中增长因子的分布规律。此前Trefethen与Schreiber关于随机增长因子分布的研究，仅限于对Ginibre矩阵一阶矩的经验估计。