Structured condition numbers for generalized saddle point systems

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.