Preconditioned Kaczmarz-Extended algorithm with relaxation parameters

We analyse in this paper the possibility of using preconditioning techniques as for square non-singular systems, also in the case of inconsistent least-squares problems. We find conditions in which the minimal norm solution of the preconditioned least-squares problem equals that of the original problem. We also find conditions such that the Kaczmarz-Extended algorithm with relaxation parameters (analysed by the author in 4]), can be adapted to the preconditioned least-squares problem. In the last section of the paper we present numerical experiments, with two variants of preconditioning, applied to an inconsistent linear least-squares model problem.