X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/ba23dbc774b2cbdf134b422e9fe09e2f2fadf298..d2f6dd23c329a1dc4056aa229e08119172290acb:/paper.tex

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@@ -98,7 +98,29 @@ ABSTRACT
 
 \section{SimGrid}
 
-\section{Simulation of the multisplitting method}
+%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Two-stage splitting methods}
+\label{sec:04}
+\subsection{Multisplitting methods for sparse linear systems}
+\label{sec:04.01}
+Let us consider the following sparse linear system of $n$ equations in $\mathbb{R}$:
+\begin{equation}
+Ax=b,
+\label{eq:01}
+\end{equation}
+where $A$ is a sparse square and nonsingular matrix, $b$ is the right-hand side and $x$ is the solution of the system. The multisplitting methods solve the linear system~(\ref{eq:01}) iteratively as follows: 
+\begin{equation}
+x^{k+1}=\displaystyle\sum^L_{\ell=1} E_\ell M^{-1}_\ell (N_\ell x^k + b),~k=1,2,3,\ldots
+\label{eq:02}
+\end{equation}
+where a collection of $L$ triplets $(M_\ell, N_\ell, E_\ell)$ defines the multisplitting of matrix $A$, such that: the different splittings are defined as $A=M_\ell-N_\ell$ where $M_\ell$ are nonsingular matrices, and $\sum_\ell{E_\ell=I}$ are diagonal nonnegative weighting matrices and $I$ is the identity matrix.
+
+\subsection{Simulation of two-stage methods using SimGrid framework}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Experimental, Results and Comments}