A = M_l - N_l,~l\in\{1,\ldots,L\},
\label{eq01}
\end{equation}
-where $M_l$ is a nonsingular matrix, and then solving the linear system by the iterative method
+where $M_l$ are nonsingular matrices. Then the linear system is solved by iteration based
+on the multisplittings as follows
\begin{equation}
x^{k+1}=\displaystyle\sum^L_{l=1} E_l M^{-1}_l (N_l x^k + b),~k=1,2,3,\ldots
\label{eq02}
\end{equation}
-where $E_l$ is a non-negative and diagonal weighting matrix such that $\sum^L_{l=1}E_l=I$ ($I$ is the identity matrix).
+where $E_l$ are non-negative and diagonal weighting matrices such that $\sum^L_{l=1}E_l=I$ ($I$ is an identity matrix).
Thus the convergence of such a method is dependent on the condition
\begin{equation}
\rho(\displaystyle\sum^L_{l=1}E_l M^{-1}_l N_l)<1.
The advantage of the multisplitting method is that at each iteration $k$ there are $L$ different linear
systems
\begin{equation}
-y_l=M^{-1}_l N_l x_l^{k-1} + M^{-1}_l b,~l\in\{1,\ldots,L\},
+y_l^k=M^{-1}_l N_l x_l^{k-1} + M^{-1}_l b,~l\in\{1,\ldots,L\},
\label{eq04}
\end{equation}
-to be solved independently by a direct or an iterative method, where $y_l$ is the solution of the local system.
+to be solved independently by a direct or an iterative method, where $y_l^k$ is the solution of the local system.
A multisplitting method using an iterative method for solving the $L$ linear systems is called an inner-outer
iterative method or a two-stage method. The solution of the global linear system at the iteration $k$ is computed
as follows
\begin{equation}
-x^k = \displaystyle\sum^L_{l=1} E_l y_l,
+x^k = \displaystyle\sum^L_{l=1} E_l y_l^k,
\label{eq05}
\end{equation}
-In the case where the diagonal weighting matrices $E_l$ have only zero and one factors (i.e. $y_l$ are disjoint vectors),
+In the case where the diagonal weighting matrices $E_l$ have only zero and one factors (i.e. $y_l^k$ are disjoint vectors),
the multisplitting method is non-overlapping and corresponds to the block Jacobi method.
%%%%%%%%%%%%%%%%%%%%%%%
%% END
where for all $l\in\{1,\ldots,L\}$ $A_l$ is a rectangular block of size $n_l\times n$
and $X_l$ and $B_l$ are sub-vectors of size $n_l$, such that $\sum_ln_l=n$. In this
case, we use a row-by-row splitting without overlapping in such a way that successive
-rows of the sparse matrix $A$ and both vectors $x$ and $b$ are assigned to a cluster.
+rows of the sparse matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster.
So, the multisplitting format of the linear system is defined as follows:
\begin{equation}
\forall l\in\{1,\ldots,L\} \mbox{,~} \displaystyle\sum_{i=1}^{l-1}A_{li}X_i + A_{ll}X_l + \displaystyle\sum_{i=l+1}^{L}A_{li}X_i = B_l,
