The extensions of convergence rates of Kaczmarz-type methods

Abstract

Abstract Kaczmarz-type methods, such as the randomized Kaczmarz method, the block Kaczmarz method and the Cimmino method, can be derived from the Kaczmarz method. In this paper, we introduce a new error term ‖ x k − P N ( A ) x 0 − x † ‖ 2 for Kaczmarz-type methods, where x † is the generalized solution of A x = b and P N ( A ) x 0 is the orthogonal projection of a given initial value x 0 onto the null space N ( A ) . It includes the well-known error term ‖ x k − x ∗ ‖ 2 as a special case when x 0 = 0 and x † = x ∗ , where x ∗ is a true solution of A x = b . We investigate the behavior of the new error term and establish the corresponding convergence rates for Kaczmarz-type methods. Especially, from the estimate of new error term for the Kaczmarz method, we can get a more simple proof for the convergence of the Kaczmarz method.