Multi-step greedy Kaczmarz algorithms with simple random sampling for solving large linear systems

By exploiting the simple random sampling and the greedy technique for capturing large residual entries, we put forth two multi-step greedy Kaczmarz algorithms for solving large linear systems. Both algorithms employ simple random sampling ab initio yet differ in the rules for choosing the working row. The new algorithms are proved to be convergent when the linear system is consistent. Some numerical experiments are carried out to verify the effectiveness of the proposed algorithms.