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 addresses and analyzes 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 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 structured 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起着关键作用，广泛应用于数值技术的鲁棒性评估。为量化任意问题的相对敏感性，需通过分量摄动理论研究结构化条件数。本文针对秩结构矩阵（包括柯西-范德蒙矩阵和{1,1}-拟可分矩阵）的Moore-Penrose逆及最小范数最小二乘解，分析并建立了结构化混合条件数与分量条件数理论框架。该框架可计算秩亏参数化矩阵的混合条件数与分量条件数上界，并显著加快柯西-范德蒙矩阵与{1,1}-拟可分矩阵的结构化条件数上界计算。进一步从理论与实验角度对所得上界进行了比较研究。此外，针对{1,1}-拟可分矩阵的Moore-Penrose逆与最小范数最小二乘解，提出了结构化有效条件数。数值实验验证了所提上界的可靠性，并表明结构化有效条件数在计算成本与数值量级上均显著优于非结构化条件数。