This paper addresses structured normwise, mixed, and componentwise condition numbers (CNs) for a linear function of the solution to the generalized saddle point problem (GSPP). We present a general framework enabling us to measure the structured CNs of the individual solution components and derive their explicit formulae when the input matrices have symmetric, Toeplitz, or some general linear structures. In addition, compact formulae for the unstructured CNs are obtained, which recover previous results on CNs for GSPPs for specific choices of the linear function. Furthermore, an application of the derived structured CNs is provided to determine the structured CNs for the weighted Teoplitz regularized least-squares problems and Tikhonov regularization problems, which retrieves some previous studies in the literature.

翻译：本文研究广义鞍点问题（GSPP）解的线性函数的范数型、混合型及分量型结构化条件数（CNs）。我们提出一个通用框架，能够度量各解分量的结构化条件数，并在输入矩阵具有对称、Toeplitz或某些一般线性结构时导出其显式表达式。此外，我们还获得了非结构化条件数的紧凑公式，这些公式通过线性函数的特定选择恢复了先前关于GSPPs条件数的结果。进一步地，我们将推导出的结构化条件数应用于确定加权Toeplitz正则化最小二乘问题与Tikhonov正则化问题的结构化条件数，从而再现了文献中的部分研究成果。