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Strong underrelaxation in Kaczmarz's method for inconsistent systems

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Summary

We investigate the behavior of Kaczmarz's method with relaxation for inconsistent systems. We show that when the relaxation parameter goes to zero, the limits of the cyclic subsequences generated by the method approach a weighted least squares solution of the system. This point minimizes the sum of the squares of the Euclidean distances to the hyperplanes of the system. If the starting point is chosen properly, then the limits approach the minimum norm weighted least squares solution. The proof is given for a block-Kaczmarz method.

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Author notes

  1. Yair Censor

    Present address: Department of Computer Studies, University of Haifa, Mt. Carmel, 31999, Haifa, Israel

Authors and Affiliations

  1. Medical Image Processing Group, Department of Radiology, Hospital of the University of Pennsylvania, 19104, Philadelphia, PA, USA

    Yair Censor

  2. Department of Mathematical Sciences, University of Delaware, 19711, Newark, DE, USA

    Paul P. B. Eggermont

  3. Department of Computer Studies, University of Haifa, Mt. Carmel, 31999, Haifa, Israel

    Dan Gordon

Authors
  1. Yair Censor
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  2. Paul P. B. Eggermont
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  3. Dan Gordon
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Censor, Y., Eggermont, P.P.B. & Gordon, D. Strong underrelaxation in Kaczmarz's method for inconsistent systems. Numer. Math. 41, 83–92 (1983). https://doi.org/10.1007/BF01396307

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  • DOI: https://doi.org/10.1007/BF01396307

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