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Randomized iterative algorithms have attracted much attention in recent yearsbecause they can approximately solve large-scale linear systems of equationswithout accessing the entire coefficient matrix. In this paper, we propose twonovel pseudoinverse-free randomized block iterative algorithms for solvingconsistent and inconsistent linear systems. The proposed algorithms require twouser-defined random matrices: one for row sampling and the other for columnsampling. We can recover the well-known doubly stochastic Gauss--Seidel,randomized Kaczmarz, randomized coordinate descent, and randomized extendedKaczmarz algorithms by choosing appropriate random matrices used in ouralgorithms. Because our algorithms allow for a much wider selection of thesetwo random matrices, a number of new specific algorithms can be obtained. Weprove the linear convergence in the mean square sense of our algorithms.Numerical experiments for linear systems with synthetic and real-worldcoefficient matrices demonstrate the efficiency of some special cases of ouralgorithms.

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