The standard form and convergence theory of the relaxation Kaczmarz-Tanabe method for solving linear systems

The Kaczmarz method is a popular iterative method for solving consistent, overdetermined linear system such as medical imaging in computerized tomography. The Kaczmarz's iteration repeatedly scans all equations in order, which leads to lower computational efficiency especially in solving a large scale problem. The standard form of Kaczmarz-Tanabe's iteration proposed recently effectively overcomes the computational redundancy problem of the Kaczmarz method. In this paper, we introduce relaxation parameters ${\bf u}=(\mu_1,\ldots,\mu_m)$ into the Kaczmarz-Tanabe method based on the relaxation Kaczmarz method, and consider the standard form and convergence of this combination. Moreover, we analyze and prove the sufficient conditions for convergence of the relaxation Kaczmarz-Tanabe method, i.e., $\mu_i\in (0,2)$. Numerical experiments show the convergence characteristics of the relaxation Kaczmarz-Tanabe method corresponding to these parameters.