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.

When iteratively solving linear systems By = b with Hermitian positive semi-definite B, and in particular when solving least-squares problems for Ax = b by reformulating them as AA * y = b, it is often observed that SOR type methods (Gauß-Seidel, Kaczmarz) perform suboptimally for the given equation ordering, and that random reordering improves the situation on average. This paper is an attempt to provide some additional theoretical support for this phenomenon. We show error bounds for two randomized versions, called shuffled and preshuffled SOR, that improve asymptotically upon the best known bounds for SOR with cyclic ordering. Our results are based on studying the behavior of the triangular truncation of Hermitian matrices with respect to their permutations.

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