Pseudoinverse-free randomized block iterative methods for consistent and inconsistent linear systems

Randomized iterative methods 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 methods for solving consistent and inconsistent linear systems. Our methods require two user-defined random matrices: one for row sampling and the other for column sampling. The well-known doubly stochastic Gauss--Seidel, randomized Kaczmarz, randomized coordinate descent, and randomized extended Kaczmarz methods are special cases of our methods corresponding to specially selected random matrices. Because our methods allow for a much wider selection of these two random matrices, a number of new specific methods can be obtained. We prove the linear convergence (in the mean square sense) of our methods. Numerical experiments for linear systems with synthetic and real-world coefficient matrices demonstrate the efficiency of some special cases of our methods. We remark that due to the pseudoinverse-free nature our methods can be easily implemented for parallel computation to yield further computational gains.