Preconvergence of the randomized extended Kaczmarz method
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In this paper, we analyze the convergence behavior of the randomized extended Kaczmarz (REK) method for all types of linear systems (consistent or inconsistent, overdetermined or underdetermined, full-rank or rank-deficient). The analysis shows that the larger the singular value of A is, the faster the error decays in the corresponding right singular vector space, and as k→∞, x_k-x_⋆ tends to the right singular vector corresponding to the smallest singular value of A, where x_k is the kth approximation of the REK method and x_⋆ is the minimum ℓ_2-norm least squares solution. These results explain the phenomenon found in the extensive numerical experiments appearing in the literature that the REK method seems to converge faster in the beginning. A simple numerical example is provided to confirm the above findings.
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