Convergence analysis for Kaczmarz-type methods in a Hilbert space framework

Using the concept of stable Hilbert space splittings, we provide a unified approach to the convergence analysis for multiplicative Schwarz methods (a version of alternating directions methods), and in particular Kaczmarz-type methods for solving linear systems. We consider both fixed cyclic and randomized ordering strategies, and cover block versions as well. For the classical Kaczmarz method with cyclic ordering for solving general linear systems Ax=b, a new convergence rate estimate in terms of the generalized condition number of A and logarithmically depending on the rank of A is presented.