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-The key idea of the multisplitting method for solving a large system of linear equations
-$Ax=b$ consists in partitioning the matrix $A$ in $L$ several ways
+The key idea of the multisplitting method for solving a large system
+of linear equations $Ax=b$ consists in partitioning the matrix $A$ in
+$L$ several ways
\begin{equation}
A = M_l - N_l,~l\in\{1,\ldots,L\},
\label{eq01}
\end{equation}
-where $M_l$ are nonsingular matrices. Then the linear system is solved by iteration based
-on the multisplittings as follows
+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 $E_l$ 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
+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.
\label{eq03}
\end{equation}
-The advantage of the multisplitting method is that at each iteration $k$ there are $L$ different linear
-systems
+The advantage of the multisplitting method is that at each iteration
+$k$ there are $L$ different linear sub-systems
\begin{equation}
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 $y_l^k$ 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
+to be solved independently by a direct or an iterative method, where
+$y_l^k$ is the solution of the local sub-system. A multisplitting
+method using an iterative method for solving the $L$ linear
+sub-systems is called an inner-outer iterative method or a two-stage
+method. The results $y_l^k$ obtained from the different
+splittings~(\ref{eq04}) are combined to compute the solution $x^k$ of
+the linear system by using the diagonal weighting matrices
\begin{equation}
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. $y_l^k$ are disjoint vectors),
-the multisplitting method is non-overlapping and corresponds to the block Jacobi method.
+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.
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