LU和QR分解的新的严格扰动界
New rigorous perturbation bounds for the LU and QR factorizations
AI总结：
本文结合矩阵向量方程方法与Lyapunov主函数技术及巴纳赫不动点原理，推导出LU和QR分解在矩阵扰动下的新严格扰动界，改进了Chang和Stehlé的结果，并给出了最优的一阶扰动界表达式。
摘要：
结合修改的矩阵向量方程方法与Lyapunov主函数技术及巴纳赫不动点原理，我们获得了LU和QR分解在给定矩阵中的规范扰动或组件扰动下的新严格扰动界，其中组件扰动形式为标准分解算法产生的回溯误差。每个新的严格扰动界都是文献中通过矩阵向量方程方法推导出的一阶扰动界的一种严格版本，我们给出了其显式表达式。这些界改进了Chang和Stehlé[SIAM Journal on Matrix Analysis and Applications 2010; 31:2841--2859]的结果。此外，我们推导了新的更紧的一阶扰动界，包括LU分解的两个最优界，并给出了LU和QR分解的最优一阶扰动界的显式表达式。
英文摘要：
Combining the modified matrix-vector equation approach with the technique of Lyapunov majorant function and the Banach fixed point principle, we obtain new rigorous perturbation bounds for the LU and QR factorizations with normwise or componentwise perturbations in the given matrix, where the componentwise perturbations have the form of backward errors resulting from the standard factorization algorithms. Each of the new rigorous perturbation bounds is a rigorous version of the first-order perturbation bound derived by the matrix-vector equation approach in the literature, and we present their explicit expressions. These bounds improve the results given by Chang and Stehlé [SIAM Journal on Matrix Analysis and Applications 2010; 31:2841--2859]. Moreover, we derive new tighter first-order perturbation bounds including two optimal ones for the LU factorization, and provide the explicit expressions of the optimal first-order perturbation bounds for the LU and QR factorizations.
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。
