In this paper, we investigate the Kaczmarz–Tanabe method for exact and inexact linear systems. The Kaczmarz–Tanabe method is derived from the Kaczmarz method, but is more stable than that. We analyze the convergence and the convergence rate of the Kaczmarz–Tanabe method based on the singular value decomposition theory, and discover two important factors, i.e., the second maximum singular value of $Q$ and the minimum non-zero singular value of $A$, that influence the convergence speed and the amplitude of fluctuation of the Kaczmarz–Tanabe method (even for the Kaczmarz method). Numerical tests verify the theoretical results of the Kaczmarz–Tanabe method.

在本文中，我们研究了精确和不精确线性系统的Kaczmarz-Tanabe方法。Kaczmarz-Tanabe方法是从Kaczmarz方法衍生而来的，但比它更稳定。我们基于奇异值分解理论分析了Kaczmarz-Tanabe方法的收敛性和收敛速度，发现了两个重要因素，即奇异值的第二最大奇异值。$问$ 和最小非零奇异值 $\mathrm{一种}$，这会影响Kaczmarz-Tanabe方法的收敛速度和波动幅度（即使对于Kaczmarz方法也是如此）。数值试验验证了Kaczmarz-Tanabe方法的理论结果。