Recently, the structured backward errors for the generalized saddle point problems with some different structures have been studied by some authors, but their results involve some Kronecker products, the vec-permutation matrices, and the orthogonal projection of a large block matrix which make them very expensive to compute when utilized for testing the stability of a practical algorithm or as an effective stopping criteria. In this paper, adopting a new technique, we present the explicit and computable formulae of the normwise structured backward errors for the generalized saddle point problems with five different structures. Our analysis can be viewed as a unified or general treatment for the structured backward errors for all kinds of saddle point problems and the derived results also can be seen as the generalizations of the existing ones for standard saddle point problems, including some Karush-Kuhn-Tucker systems. Some numerical experiments are performed to illustrate that our results can be easily used to test the stability of practical algorithms when applied some physical problems. We also show that the normwise structured and unstructured backward errors can be arbitrarily far apart in some certain cases.

中文翻译：

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广义鞍点问题的结构化后向误差分析

最近，一些作者研究了具有不同结构的广义鞍点问题的结构化后向误差，但其结果涉及一些Kronecker乘积，vec置换矩阵以及大块矩阵的正交投影，这使得它们非常当用于测试实用算法的稳定性或作为有效的停止标准时，计算成本很高。本文采用一种新技术，针对具有五个不同结构的广义鞍点问题，给出了范式结构后向误差的显式和可计算公式。我们的分析可以看作是对各种鞍点问题的结构化后向误差的统一或一般处理，得出的结果也可以看作是对标准鞍点问题（包括一些Karush-Kuhn-塔克系统。进行了一些数值实验，以说明当应用某些物理问题时，我们的结果可以轻松用于测试实用算法的稳定性。我们还表明，在某些情况下，规范的结构化错误和非结构化的后向错误可以任意分开。