In this paper we present a "mixture" between a classical Kaczmarz's algorit
hm with relaxation parameter and an approximate orthogonalization procedure
dye to Z. Kovarik. We prove that the sequence of approximations generated
by the new algorithm so obtained converges, in the case of consistent least
squares problems, to a solution (for values of the relaxation parameter in
the interval (0, 2)). The numerical experiments described at the end of th
e paper on a discretization of a (model) first kind Fredholm integral equat
ion, show the fact that the convergence properties of our algorithm are ind
ependent on the dimensions of the discretization matrix and can be improved
by an apropriate choice of the relaxation parameter.