In the realm of numerical analysis, the study of structured backward errors (BEs) in saddle point problems (SPPs) has shown promising potential for development. However, these investigations overlook the inherent sparsity pattern of the coefficient matrix of the SPP. Moreover, the existing techniques are not applicable when the block matrices have circulant, Toeplitz, or symmetric-Toeplitz structures and do not even provide structure preserving minimal perturbation matrices for which the BE is attained. To overcome these limitations, we investigate the structured BEs of SPPs when the perturbation matrices exploit the sparsity pattern as well as circulant, Toeplitz, and symmetric-Toeplitz structures. Furthermore, we construct minimal perturbation matrices that preserve the sparsity pattern and the aforementioned structures. Applications of the developed frameworks are utilized to compute BEs for the weighted regularized least squares problem. Finally, numerical experiments are performed to validate our findings, showcasing the utility of the obtained structured BEs in assessing the strong backward stability of numerical algorithms.

翻译：在数值分析领域，鞍点问题的结构化后向误差研究已展现出良好的发展潜力。然而，现有研究忽略了鞍点问题系数矩阵固有的稀疏模式。此外，当分块矩阵具有循环、Toeplitz或对称-Toeplitz结构时，现有技术并不适用，甚至无法提供达到后向误差的结构保持最小扰动矩阵。为克服这些局限，本文研究了当扰动矩阵同时利用稀疏模式以及循环、Toeplitz和对称-Toeplitz结构时鞍点问题的结构化后向误差。进一步地，我们构建了能保持稀疏模式及上述结构的最小扰动矩阵。所建立框架被应用于计算加权正则化最小二乘问题的后向误差。最后，通过数值实验验证了研究结果，展示了所得结构化后向误差在评估数值算法强后向稳定性方面的实用价值。