Error estimates of Kaczmarz and randomized Kaczmarz methods
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The Kaczmarz method is an iterative projection scheme for solving con-sistent system Ax = b. It is later extended to the inconsistent and ill-posed linear problems. But the classical Kaczmarz method is sensitive to the correlation of the adjacent equations. In order to reduce the impact of correlation on the convergence rate, the randomized Kaczmarz method and randomized block Kaczmarz method are proposed, respectively. In the current literature, the error estimate results of these methods are established based on the error x_k-x_*_2, where x_* is the solution of linear system Ax=b. In this paper, we extend the present error estimates of the Kaczmarz and randomized Kaczmarz methods on the basis of the convergence theorem of Kunio Tanabe, and obtain some general results about the error x_k-P_N(A)x_0-x^†_2.
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