-Krylov subspace iteration methods have increasingly become useful and successful techniques for solving linear and nonlinear systems and eigenvalue problems, especially since the increase development of the preconditioners~\cite{Saad2003,Meijerink77}. One reason of the popularity of these methods is their generality, simplicity and efficiency to solve systems of equations arising from very large and complex problems. %A Krylov method is based on a projection process onto a Krylov subspace spanned by vectors and it forms a sequence of approximations by minimizing the residual over the subspace formed~\cite{}.
-
-GMRES is one of the most widely used Krylov iterative method for solving sparse and large linear systems. It is developed by Saad and al.~\cite{Saad86} as a generalized method to deal with unsymmetric and non-Hermitian problems, and indefinite symmetric problems too. In its original version called full GMRES, it minimizes the residual over the current Krylov subspace until convergence in at most $n$ iterations, where $n$ is the size of the sparse matrix. It should be noted that full GMRES is too expensive in the case of large matrices since the required orthogonalization process per iteration grows quadratically with the number of iterations. For that reason, in practice GMRES is restarted after each $m\ll n$ iterations to avoid the storage of a large orthonormal basis. However, the convergence behavior of the restarted GMRES, called GMRES($m$), in many cases depends quite critically on the value of $m$~\cite{Huang89}. Therefore in most cases, a preconditioning technique is applied to the restarted GMRES method in order to improve its convergence.
+Krylov subspace iteration methods have increasingly become useful and successful
+techniques for solving linear, nonlinear systems and eigenvalue problems,
+especially since the increase development of the
+preconditioners~\cite{Saad2003,Meijerink77}. One reason of the popularity of
+these methods is their generality, simplicity and efficiency to solve systems of
+equations arising from very large and complex problems.
+
+GMRES is one of the most widely used Krylov iterative method for solving sparse
+and large linear systems. It is developed by Saad and al.~\cite{Saad86} as a
+generalized method to deal with unsymmetric and non-Hermitian problems, and
+indefinite symmetric problems too. In its original version called full GMRES, it
+minimizes the residual over the current Krylov subspace until convergence in at
+most $n$ iterations, where $n$ is the size of the sparse matrix. It should be
+noticed that full GMRES is too expensive in the case of large matrices since the
+required orthogonalization process per iteration grows quadratically with the
+number of iterations. For that reason, in practice GMRES is restarted after each
+$m\ll n$ iterations to avoid the storage of a large orthonormal basis. However,
+the convergence behavior of the restarted GMRES, called GMRES($m$), in many
+cases depends quite critically on the value of $m$~\cite{Huang89}. Therefore in
+most cases, a preconditioning technique is applied to the restarted GMRES method
+in order to improve its convergence.
