A review of the Kaczmarz method

Abstract: The Kaczmarz method is an iterative method for solving linear systems of equations. The Kaczmarz method has been around since it was developed by Kaczmarz 1937. The main idea behind the original Kaczmarz method is to orthogonally project the previous x_k onto the solution space given by a row of the system. The block Kaczmarz on the other hand orthogonally projects the previous x_k onto the solution space given by a sub system of equations. Both the original Kaczmarz method and block Kaczmarz method can only solve consistent systems, however the extended Kaczmarz method is an adaptation that makes it possible to solve inconsistent systems. We will look at both deterministic and randomized, row and block selection processes then compare them on both consistent and inconsistent systems of equations.