Perturbation theory plays a crucial role in sensitivity analysis, which is extensively used to assess the robustness of numerical techniques. To quantify the relative sensitivity of any problem, it becomes essential to investigate structured condition numbers (CNs) via componentwise perturbation theory. This paper address and analyze structured mixed condition number (MCN) and componentwise condition number (CCN) for the Moore-Penrose (M-P) inverse and the minimum norm least squares (MNLS) solution involving rank-structured matrices, which include the Cauchy-Vandermonde (CV) matrices and $\{1,1\}$-quasiseparable (QS) matrices. A general framework has been developed first to compute the upper bounds for MCN and CCN of rank deficient parameterized matrices. This framework leads to faster computation of upper bounds of structured CNs for CV and $\{1,1\}$-QS matrices. Furthermore, comparisons of obtained upper bounds are investigated theoretically and experimentally. In addition, the structured effective CNs for the M-P inverse and the MNLS solution of $\{1,1\}$-QS matrices are presented. Numerical tests reveal the reliability of the proposed upper bounds as well as demonstrate that the effective CNs are computationally less expensive and can be substantially smaller compared to the unstructured CNs.

翻译：摄动理论在敏感性分析中发挥着关键作用，广泛应用于评估数值技术的鲁棒性。为量化任意问题的相对敏感度，通过分量摄动理论研究结构化条件数（CNs）至关重要。本文针对涉及秩结构矩阵（包括柯西-范德蒙德（CV）矩阵和{1,1}-拟可分（QS）矩阵）的Moore-Penrose（M-P）逆与最小范数最小二乘（MNLS）解，分析并研究了结构化混合条件数（MCN）和分量条件数（CCN）。首先构建了一个通用框架，用于计算秩亏参数化矩阵的MCN与CCN的上界。该框架实现了更快计算CV矩阵与{1,1}-QS矩阵结构化CNs的上界。此外，从理论与实验两方面对所获上界进行了比较分析。同时，针对{1,1}-QS矩阵的M-P逆与MNLS解，提出了结构化有效CNs。数值实验验证了所提上界的可靠性，并表明有效CNs计算成本更低，且相比非结构化CNs可显著减小。