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Perturbation theory plays a crucial role in sensitivity analysis, which isextensively used to assess the robustness of numerical techniques. To quantifythe relative sensitivity of any problem, it becomes essential to investigatestructured condition numbers (CNs) via componentwise perturbation theory. Thispaper addresses and analyzes structured mixed condition number (MCN) andcomponentwise condition number (CCN) for the Moore-Penrose (M-P) inverse andthe minimum norm least squares (MNLS) solution involving rank-structuredmatrices, which include the Cauchy-Vandermonde (CV) matrices and$\{1,1\}$-quasiseparable (QS) matrices. A general framework has been developedto compute the upper bounds for MCN and CCN of rank deficient parameterizedmatrices. This framework leads to faster computation of upper bounds ofstructured CNs for CV and $\{1,1\}$-QS matrices. Furthermore, comparisons ofobtained upper bounds are investigated theoretically and experimentally. Inaddition, the structured effective CNs for the M-P inverse and the MNLSsolution of $\{1,1\}$-QS matrices are presented. Numerical tests reveal thereliability of the proposed upper bounds as well as demonstrate that thestructured effective CNs are computationally less expensive and can besubstantially smaller compared to the unstructured CNs.

Condition numbers for the Moore-Penrose inverse and the least squares problem involving rank-structured matrices