On convergence of residual-based extended Kaczmarz method for solving inconsistent linear systems

Abstract

The Kaczmarz-type methods, also called as the algebraic reconstruction techniques, have been studied and applied due to their high computational efficiency and good convergence property for solving very large systems of linear equations. Recently, the residual-based extended Kaczmarz method was proposed by Bao et al. [9] to solve the inconsistent linear systems. In theoretical framework, based on the condition that the \(i_{k-1}\)-th element of the residual vector \(r_k\) in the k-th iteration step is equal to zero, the authors obtained the convergence theorem. However, this element is not always equal to zero, making the basic hypothesis of the convergence theorem be not satisfied, so that the convergence conclusion does not hold true. According to this meticulous observation, we give a correct and detailed analysis for the convergence property of the residual-based extended Kaczmarz method. As a result, we provide a corrected convergence theory for this iteration method. Also, we examine these results by a numerical example.