On extended Kaczmarz methods with random sampling and maximum-distance for solving large inconsistent linear systems

In this paper, combining random sampling and maximum-distance, we propose two new extended Kaczmarz methods for solving inconsistent linear systems with full column rank. Based on some inequalities, we prove the convergences of the new extended Kaczmarz methods and give the upper bounds of the expected convergence rates. Numerical experiments show that the two new extended Kaczmarz methods are more effective than the existing methods. Especially, the new method with Z-test has certain advantage over that without it.