Kaczmarz-type inner-iteration preconditioned flexible GMRES methods for consistent linear systems

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We propose using greedy and randomized Kaczmarz inner-iterations as preconditioners for the right-preconditioned flexible GMRES method to solve consistent linear systems, with a parameter tuning strategy for adjusting the number of inner iterations and the relaxation parameter.

• We also present theoretical justifications of the right-preconditioned flexible GMRES for solving consistent linear systems.
• Numerical experiments on overdetermined and underdetermined linear systems show that the proposed method is superior to the GMRES method preconditioned by NE-SOR inner iterations in terms of total CPU time.