From b56c355fe82d00c83b2c729ebea92733f1f30686 Mon Sep 17 00:00:00 2001
From: lilia <lilia@mondomaine.fr>
Date: Tue, 7 Jan 2014 00:10:18 +0100
Subject: [PATCH] 07-01-2014

---
 krylov_multi.tex | 46 +++++++++++++++++++++++++++++++++++++++++++++-
 1 file changed, 45 insertions(+), 1 deletion(-)

diff --git a/krylov_multi.tex b/krylov_multi.tex
index 82cf45e..d81125b 100644
--- a/krylov_multi.tex
+++ b/krylov_multi.tex
@@ -48,7 +48,51 @@ thousands of cores are used.
 
 
 A completer...
-On ne peut pas parler de tout...
+On ne peut pas parler de tout...\\
+
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%
+%% BEGIN
+%%%%%%%%%%%%%%%%%%%%%%%
+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 $M_l$ is a nonsingular matrix, and then solving the linear system by the iterative method
+\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).
+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
+\begin{equation}
+y_l=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.
+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,
+\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),
+the multisplitting method is non-overlapping and corresponds to the block Jacobi method.  
+%%%%%%%%%%%%%%%%%%%%%%%
+%% END
+%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Related works}
 
-- 
2.39.5