On the modified iterative methods for $M$-matrix linear systems

‎This paper deals with scrutinizing the convergence properties of iterative methods to solve linear system of equations‎. ‎Recently‎, ‎several types of the preconditioners have been applied for ameliorating the rate of convergence of the Accelerated Overrelaxation (AOR) method‎. ‎In this paper‎, ‎we study the applicability of a general class of the preconditioned iterative methods under certain conditions‎. ‎More precisely‎, ‎it is demonstrated that the preconditioned Mixed-Type Splitting (MTS) iterative methods can surpass the preconditioned AOR iterative methods for an entirely general class of preconditioners handled by Wang and Song [J‎. ‎Comput‎. ‎Appl‎. ‎Math‎. ‎226 (2009)‎, ‎no‎. ‎1‎, ‎114--124]‎. ‎Finally some numerical results are elaborated which confirm the validity of the established results‎. ‎