关于Higham平方最大高斯消元增长矩阵的扰动分析
On a perturbation analysis of Higham squared maximum Gaussian elimination growth matrices
AI总结：
针对Higham和Higham 1989年提出的部分选主元下达到最大增长因子的n阶实矩阵，本研究通过理论与实证分析其扰动敏感性，揭示细微元素改变可显著降低增长因子。
摘要：
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，浮点运算中该过程可能出现大误差。在数值线性代数研究中，研究和刻画最坏情况示例往往很有价值。为此，Higham和Higham在1989年的论文中刻画了在部分选主元下达到最大增长因子的所有n阶实矩阵的完整集合。未完成的是这些矩阵在扰动下的敏感性分析。这些矩阵及其邻近矩阵的增长因子是本研究的主题。通过理论洞察和实证结果，我们说明了这些矩阵的增长因子对扰动的高敏感性，并展示了如何策略性地对矩阵元素进行细微改变以显著降低增长。
英文摘要：
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. In the study of numerical linear algebra, it is often valuable to study and characterize the worst case examples. To this end, 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. Left undone is a sensitivity analysis for these matrices under perturbations. The growth factor of these and nearby matrices is the subject 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
