This paper presents a greedy average block Kaczmarz (GABK) method to solve the large scaled consistent system of linear equations. The GABK method introduces the strategy of extrapolation process to improve the GBK algorithm and to avoid computing the Moore-Penrose inverse of a submatrix of the coefficient matrix determined by the block index set. The GABK method is proved to converge linearly to the least-norm solution of the consistent system of linear equations. Numerical examples show that the GABK method has the best efficiency and effectiveness among all methods compared.

This paper presents a partial block randomized extended Kaczmarz (PBREK) method for solving large overdetermined inconsistent linear system of equations $Ax=b$. The convergence theorem of the PBREK method is derived. Several examples are given to illustrate the effectiveness of the proposed PBREK method compared with the prevuious PREK method and the randomized extended Kaczmarz (REK) method.