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 that enables us to measure the structured CNs of the individual components of the solution. Then, we 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, applications of the derived structured CNs are provided to determine the structured CNs for the weighted Toeplitz regularized least-squares problems and Tikhonov regularization problems, which retrieves some previous studies in the literature.

翻译：本文针对广义鞍点问题解的线性函数，研究了其结构化范数型、混合型及分量型条件数。我们提出了一个通用框架，用于度量解的各分量的结构化条件数。随后，当输入矩阵具有对称、Toeplitz 或某些一般线性结构时，我们导出了这些条件数的显式表达式。此外，我们还得到了非结构化条件数的紧凑表达式，这些表达式通过线性函数的特定选择，恢复了先前关于广义鞍点问题条件数的结果。进一步地，我们将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和 Tikhonov 正则化问题的结构化条件数，从而复现了文献中的一些已有研究。