标题： 秩结构矩阵相关的Moore-Penrose逆和最小二乘问题的条件数 显示英文标题

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

摘要： 摄动理论在灵敏度分析中起着关键作用，被广泛用于评估数值技术的鲁棒性。 为了量化任何问题的相对灵敏度，通过分量摄动理论研究结构条件数（CNs）变得至关重要。 本文研究并分析了涉及秩结构矩阵的Moore-Penrose（M-P）逆和最小范数最小二乘（MNLS）解的结构混合条件数（MCN）和分量条件数（CCN），这些矩阵包括Cauchy-Vandermonde（CV）矩阵和{1, 1}-拟分离（QS）矩阵。 已经开发了一个通用框架来计算参数化秩亏矩阵的MCN和CCN的上界。 该框架使得CV和{1, 1}-QS矩阵的结构CNs的上界计算更快。 此外，对获得的上界进行了理论和实验比较。 此外，还给出了{1, 1}-QS矩阵的M-P逆和MNLS解的结构有效CNs。 数值测试表明所提出的上界具有可靠性，并且表明结构有效CNs在计算上更便宜，并且与非结构CNs相比可以显著更小。 显示英文摘要

摘要： 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.