Abstract
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.
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The authors would like to express their gratitude to the anonymous referees for their detailed and helpful suggestions that substantially improved the manuscript.
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The work is supported by the National Natural Science Foundation of China (11571004)
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Communicated by: Raymond H. Chan
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Zheng, B., Lv, P. Structured backward error analysis for generalized saddle point problems. Adv Comput Math 46, 34 (2020). https://doi.org/10.1007/s10444-020-09787-x
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DOI: https://doi.org/10.1007/s10444-020-09787-x