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In this paper, we consider the standard forms of two kinds of Kaczmarz-Tanabetype methods, one is derived from the Kaczmarz method and the other is derivedfrom the symmetric Kaczmarz method. As a famous image reconstruction method incomputerized tomography, the Kaczmarz method is simple and easy to implement,but its convergence speed is slow, so is the symmetric Kaczmarz method. Whenthe standard forms of the Kaczmarz-Tanabe type methods are obtained, theiriteration matrices can be used continuously in the subsequent iterations.Moreover, the iteration matrices can be stored in the image reconstructiondevices, which enables the Kaczmarz method and the symmetric Kaczmarz method tobe used like the simultaneous iterative reconstructive techniques (SIRT).Meanwhile, theoretical analysis shows that the convergence rate of thesymmetric Kaczmarz-Tanabe method is better than that of the Kaczmarz-Tanabemethod but is slightly worse than that of two-step Kaczmarz-Tanabe method,which is verified numerically. Numerical experiments also show that theconvergence rates of the Kaczmarz-Tanabe method and the symmetricKaczmarz-Tanabe method are better than those of the SIRT methods.

The standard forms and convergence theory of the Kaczmarz-Tanabe type methods for solving linear systems