Randomized Kaczmarz algorithm with averaging and block projection

The randomized Kaczmarz algorithm is a simple iterative method for solving linear systems of equations. This study proposes a variant of the randomized Kaczmarz algorithm by combining block projection and weighted averaging techniques. Here, block projection quickly decreases iterative errors, and averaging reduces randomness and enables parallel computation simultaneously. Their combination can balance the convergence rate, convergence horizon, and computational complexity. In addition, three adaptive weights are designed to balance multiple block calculations and accelerate the proposed method. Exponential convergence is established for general linear systems (overdetermined or underdetermined, full-rank or deficient-rank, and consistent or inconsistent). Numerical simulations explain and verify the results.