中心对称矩阵 QX 分解的严格扰动界
Rigorous Perturbation Bounds for the QX Decomposition for Centrosymmetric Matrices
AI总结：
本文针对中心对称矩阵的 QX 分解，利用矩阵方程方法、Lyapunov 主函数和 Banach 不动点定理，推导了范数型和分量型扰动下的严格扰动界及条件数上界，并通过数值实验验证了理论结果。
摘要：
Konrad Burnik 提出了一种针对中心对称矩阵的保结构 QR 分解，称为 QX 分解。本文在原始矩阵受到范数型或分量型扰动时，获得了 QX 分解严格扰动界的显式表达式。首先，利用矩阵方程方法推导出弱严格扰动界。然后，通过将改进的矩阵-向量方程方法与 Lyapunov 主函数策略及 Banach 不动点定理相结合，得到了强严格扰动界。混合条件数和分量型条件数及其上界也被显式表达。数值实验验证了所得结果的有效性。
英文摘要：
Konrad Burnik suggests a structure-preserving QR factorization for centrosymmetric matrices, known as QX factorization. In this article, we obtain the explicit expressions for rigorous perturbation bounds of the QX factorization when the original matrix is perturbed, either norm-wise or component-wise. First, using the matrix-equation approach, weak rigorous perturbation bounds are derived. Then, strong rigorous perturbation bounds are obtained by combining the modified matrix-vector equation approach with the strategy for the Lyapunov majorant function and the Banach fixed-point theorem. The mixed and component-wise condition numbers and their upper bounds are also explicitly expressed. Numerical tests illustrate the validity of the obtained results.
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。
