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In recent times, a significant amount of effort has been expended towards thedevelopment of stationary iterative techniques for the numerical solution ofgeneralized saddle point (GSP) systems. The condition number (CN) is widelyemployed in perturbation analysis to determine the relative sensitivity of anumerical solution. In order to assess the robustness of numerical solution, inthis paper, we address three types of condition numbers (CNs) for GSP systems:structured normwise, mixed and componentwise, with the assumption thatstructure-preserving perturbations are applied to blocks of the coefficientmatrix of the system. Explicit formulae for the structured CNs are derived inthree 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)-blockof the coefficient matrix. Second, by employing the expressions obtained in thefirst case, the compact formulae for structured CNs are investigated when (1,1)and (2,2)-blocks adhere to the symmetric structures. Third, when the transposeof (1,2)-block equals (2,1)-block. We also compare the obtained formulae ofstructured CNs with their unstructured counterparts. In addition, obtainedresults are used to recover the previous CNs formulae for the weighted leastsquares (WLS) problem and the standard least squares (SLS) problem. Finally,numerical experiments demonstrate that the proposed structured CNs outperformtheir unstructured counterparts, so validating the effectiveness of theproposed CNs.

Structured condition numbers for generalized saddle point systems