  Randomized iterative algorithms have attracted much attention in recent years
because they can approximately solve large-scale linear systems of equations
without accessing the entire coefficient matrix. In this paper, we propose two
novel pseudoinverse-free randomized block iterative algorithms for solving
consistent and inconsistent linear systems. The proposed algorithms require two
user-defined random matrices: one for row sampling and the other for column
sampling. We can recover the well-known doubly stochastic Gauss--Seidel,
randomized Kaczmarz, randomized coordinate descent, and randomized extended
Kaczmarz algorithms by choosing appropriate random matrices used in our
algorithms. Because our algorithms allow for a much wider selection of these
two random matrices, a number of new specific algorithms can be obtained. We
prove the linear convergence in the mean square sense of our algorithms.
Numerical experiments for linear systems with synthetic and real-world
coefficient matrices demonstrate the efficiency of some special cases of our
algorithms.
