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.

翻译：Kaczmarz方法是求解相容超定线性系统（如计算机断层扫描中的医学成像）中常用的迭代方法。Kaczmarz迭代按顺序重复扫描所有方程，这导致计算效率较低，尤其在求解大规模问题时更为显著。近期提出的Kaczmarz-Tanabe迭代标准形式有效克服了Kaczmarz方法的计算冗余问题。本文基于松弛Kaczmarz方法，将松弛参数${\bf u}=(\mu_1,\ldots,\mu_m)$引入Kaczmarz-Tanabe方法，并研究该组合的标准形式与收敛性。此外，我们分析并证明了松弛Kaczmarz-Tanabe方法收敛的充分条件，即$\mu_i\in (0,2)$。数值实验展示了该松弛Kaczmarz-Tanabe方法对应这些参数的收敛特性。