Gaussian elimination is the most popular technique for solving a dense linear system. Large errors in this procedure can occur in floating point arithmetic when the matrix's growth factor is large. We study this potential issue and how perturbations can improve the robustness of the Gaussian elimination algorithm. In their 1989 paper, Higham and Higham characterized the complete set of real n by n matrices that achieves the maximum growth factor under partial pivoting. This set of matrices serves as the critical focus of this work. Through theoretical insights and empirical results, we illustrate the high sensitivity of the growth factor of these matrices to perturbations and show how subtle changes can be strategically applied to matrix entries to significantly reduce the growth, thus enhancing computational stability and accuracy.

翻译：高斯消元法是求解稠密线性系统最常用的技术。当矩阵的增长因子较大时，该过程在浮点运算中可能出现较大误差。我们研究了这一潜在问题以及扰动如何提高高斯消元算法的鲁棒性。在1989年的论文中，Higham与Higham刻画了在部分选主元下达到最大增长因子的所有实n×n矩阵的完整集合。该矩阵集合是本文的核心研究对象。通过理论分析和实证结果，我们阐明了这些矩阵的增长因子对扰动的高度敏感性，并展示了如何策略性地对矩阵元素施加细微改变以显著降低增长因子，从而提升计算稳定性与精度。