Abstract
For solving the noisy linear systems, we propose a new greedy randomized extended Kaczmarz algorithm by introducing an effective greedy criterion for selecting the working rows and a randomized orthogonal projection for reducing the influence of the noisy term. We prove that the solution of the proposed greedy randomized extended Kaczmarz algorithm converges in expectation to the least squares solution of the given linear system. Theoretical analysis indicate that the convergence rate of the greedy randomized extended Kaczmarz algorithm is much faster than the randomized extended Kaczmarz method, and numerical results also show that the proposed greedy randomized extended Kaczmarz algorithm is superior to the randomized extended Kaczmarz method. Moreover, for noisy linear systems, the proposed algorithm is more effcient than the greedy randomized Kaczmarz algorithm.
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