Direct and Iterative Kaczmarz - like Solvers

Starting from an extension of the classical Kaczmarz’s iterative method, presented by the author in a previous work, we construct in the present paper a direct solver for the computation of the minimal norm solution of inconsistent and rank-deficient leastsquares problems. The construction is made by introducing in the original iteration a set of supplementary directions for projection. In the second part of the paper we analyse the possibility to introduce in each iteration of the classical Kaczmarz’s algorithm some supplementary projections, like those used for the construction of the previously mentioned direct solver. In this way we obtain a faster iterative version of Kaczmarz’s method. Numerical experiments, with both clases of methods are described in the last section of the paper.1 1 The classical and extended Kaczmarz algorithms for arbitrary least-squares problems. Fundamental results Let A be an m × n real matrix and b ∈ IR a given vector. In order to simplify the presentation, but without any restriction of the generality, we shall suppose that n ≤ m. By 〈·, ·〉, ‖ · ‖ we shall denote the Euclidean scalar product and associated norm, respectively on some space IR. Let (AMS) Subject Classification 65F10, 65F20.