On Kaczmarz method with oblique projection for solving large overdetermined linear systems

Abstract

In this paper, an extension of Kaczmarz method, the Kaczmarz method with oblique projection (KO), is introduced and analyzed. Using this method, a number of iteration steps to solve the over-determined systems of linear equations are significantly reduced, and the the computing time is much saved, especially for those problems that contain some linear equations with near-linear correlation. Simultaneously, a randomized version--randomized Kaczmarz method with oblique projection (RKO) is established. The convergence proofs of these two methods are given and numerical experiments show the effectiveness of the two methods for uniformly distributed random data. Especially when the system has correlated rows, the improvement of experimental results is very prominent.