Convergence rates of the Kaczmarz–Tanabe method for linear systems

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.