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