From d2f6dd23c329a1dc4056aa229e08119172290acb Mon Sep 17 00:00:00 2001 From: ziane Date: Tue, 21 Apr 2015 19:25:13 +0200 Subject: [PATCH 1/1] modifs sec 04 --- paper.tex | 24 +++++++++++++++++++++++- 1 file changed, 23 insertions(+), 1 deletion(-) diff --git a/paper.tex b/paper.tex index 8583e51..77dde18 100644 --- a/paper.tex +++ b/paper.tex @@ -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} -- 2.39.5