Strong underrelaxation in Kaczmarz's method for inconsistent systems |
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| Authors: | Yair Censor Paul P B Eggermont Dan Gordon |
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| Institution: | (1) Medical Image Processing Group, Department of Radiology, Hospital of the University of Pennsylvania, 19104 Philadelphia, PA, USA;(2) Department of Mathematical Sciences, University of Delaware, 19711 Newark, DE, USA;(3) Present address: Department of Computer Studies, University of Haifa, Mt. Carmel, 31999 Haifa, Israel |
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| Abstract: | 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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| Keywords: | AMS (MOS): 65 F 10 CR: 5 14 |
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