Random reordering in SOR-type methods

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.