On the block Kaczmarz-Tanabe methods with relaxation parameters for solving linear systems

In this paper, we consider the block technology, which is also an implementation method to high-performance computing, of the Kaczmarz-Tanabe type methods. Based on the ideas of block method and relaxation method, we introduce the relaxation parameters into the Kaczmarz-Tanabe method and get the block relaxation Kaczmarz-Tanabe iteration and the relaxation block Kaczmarz-Tanabe iteration, respectively. Subsequently, we provide sufficient conditions for the convergence of these two methods, i.e., \(\bar{\mu }_i\in (0,2)\) ${\overline{\mu }}_i\in (0,2)$ for the block relaxation Kaczmarz-Tanabe method and \(\widetilde{\mu }_i\in (0,1]\) ${\tilde{\mu }}_i\in (0,1]$ for the relaxation block Kaczmarz-Tanabe method. The difference in relaxation parameters between the block relaxation Kaczmarz Tanabe method and the relaxation block Kaczmarz-Tanabe method is an important finding of this work. Numerical experiments show the effectiveness of these convergence conditions for these two methods, as well as the deviation state from the theory when the sufficient conditions of the relaxation block Kaczmarz-Tanabe method fails.