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