Projections and preconditioning for inconsistent least-squares problems

in this paper we describe two iterative algorithms for the numerical solution of linear least-squares problems. They are based on a combination between an extension of the classical Kaczmarz's projections method (Popa [5]) and an approximate orthogonalization technique due to Kovarik. We prove that both new algorithms converge to any solution of an inconsistent and rank-defficient least-squares problem (with respect to the choice of the initial approximation), the convergence being much faster than for the classical Kaczmarz - like methods. Some numerical experiments on a first kind integral equation are described.