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On the Meany inequality with applications to convergence analysis of several row-action iteration methods

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Abstract

The Meany inequality gives an upper bound in the Euclidean norm for a product of rank-one projection matrices. In this paper we further derive a lower bound related to this inequality. We discuss the internal relationship between the upper bounds given by the Meany inequality and by the inequality in Smith et al. (Bull Am Math Soc 83:1227–1270, 1977) in the finite dimensional real linear space. We also generalize the Meany inequality to the block case. In addition, by making use of the block Meany inequality, we improve existing results and establish new convergence theorems for row-action iteration schemes such as the block Kaczmarz and the Householder–Bauer methods used to solve large linear systems and least-squares problems.

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Authors and Affiliations

  1. State Key Laboratory of Scientific/Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, P. O. Box 2719, Beijing, 100190, People’s Republic of China

    Zhong-Zhi Bai

  2. Department of Mathematics, Ocean University of China, Qingdao, People’s Republic of China

    Xin-Guo Liu

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  1. Zhong-Zhi Bai
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  2. Xin-Guo Liu
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Correspondence to Zhong-Zhi Bai.

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Supported by The National Natural Science Foundation for Creative Research Groups (No. 11021101), The Hundred Talent Project of Chinese Academy of Sciences, The National Basic Research Program (No. 2011CB309703), and The Natural Science Foundation of Shandong Province (No. 2008A07), People’s Republic of China.

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Bai, ZZ., Liu, XG. On the Meany inequality with applications to convergence analysis of several row-action iteration methods. Numer. Math. 124, 215–236 (2013). https://doi.org/10.1007/s00211-012-0512-6

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  • DOI: https://doi.org/10.1007/s00211-012-0512-6

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