The extensions of convergence rates of Kaczmarz-type methods

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 ‖xk−PN(A)x0−x†‖2 for Kaczmarz-type methods, where x† is the generalized solution of Ax=b and PN(A)x0 is the orthogonal projection of a given initial value x0 onto the null space N(A). It includes the well-known error term ‖xk−x∗‖2 as a special case when x0=0 and x†=x∗, where x∗ is a true solution of Ax=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.