In this diploma seminar thesis, we study the Kaczmarz iterative method for solving systems of linear equations. We present the original cyclic method and prove its convergence, and then describe several newer variants that differ in the way the rows of the matrix are selected, such as the randomized, greedy, and block Kaczmarz methods, as well as the methods with a selectable set and with averaging. We pay special attention to the application of the method in computed tomography, where we reduce the problem of image reconstruction to a system of linear equations and derive the algebraic reconstruction technique. We also address the effect of noise, the relaxation parameter, and semiconvergence. We include several examples implemented in MATLAB, with which we compare the convergence of the discussed variants.
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