In recent times, a significant amount of effort has been expended towards the development of stationary iterative techniques for the numerical solution of generalized saddle point (GSP) systems. The condition number (CN) is widely employed in perturbation analysis to determine the relative sensitivity of a numerical solution. In order to assess the robustness of numerical solution, in this paper, we address three types of condition numbers (CNs) for GSP systems: structured normwise, mixed and componentwise, with the assumption that structure-preserving perturbations are applied to blocks of the coefficient matrix of the system. Explicit formulae for the structured CNs are derived in three cases. First, when (1,1) and (2,2)-blocks exhibit linear structures (general case) and the transpose of (1,2)-block is not equal to the (2,1)-block of the coefficient matrix. Second, by employing the expressions obtained in the first case, the compact formulae for structured CNs are investigated when (1,1) and (2,2)-blocks adhere to the symmetric structures. Third, when the transpose of (1,2)-block equals (2,1)-block. We also compare the obtained formulae of structured CNs with their unstructured counterparts. In addition, obtained results are used to recover the previous CNs formulae for the weighted least squares (WLS) problem and the standard least squares (SLS) problem. Finally, numerical experiments demonstrate that the proposed structured CNs outperform their unstructured counterparts, so validating the effectiveness of the proposed CNs.

翻译：近年来，大量研究致力于发展用于数值求解广义鞍点系统的定常迭代技术。条件数在扰动分析中被广泛用于确定数值解的相对敏感性。为评估数值解的鲁棒性，本文研究了广义鞍点系统的三类条件数：结构化范数型条件数、混合型条件数和分量型条件数，假设对系统系数矩阵的块施加保结构扰动。我们推导了三种情形下结构化条件数的显式表达式。首先，当(1,1)和(2,2)块具有线性结构（一般情形）且(1,2)块的转置不等于(2,1)块时；其次，利用第一种情形得到的表达式，研究(1,1)和(2,2)块具有对称结构时的结构化条件数紧凑公式；第三，当(1,2)块的转置等于(2,1)块时。我们将得到的结构化条件数公式与其非结构化对应形式进行了比较。此外，利用所得结果恢复了加权最小二乘问题和标准最小二乘问题现有的条件数公式。最后，数值实验表明，所提结构化条件数优于非结构化对应形式，验证了其有效性。