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

Abstract

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