07-01-2014 V1

[Krylov_multi.git] / krylov_multi.tex

diff --git a/krylov_multi.tex b/krylov_multi.tex
index 3295c82e9c5b16cdeb4287c5192d00e088bfb3e8..e61890d318e3fd40e2237f2ff4cc310399fefaa4 100644 (file)
--- a/krylov_multi.tex
+++ b/krylov_multi.tex
@@ -56,41 +56,48 @@ On ne peut pas parler de tout...\\
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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}
 \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}
 \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}
 
 \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}
 \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}    
 \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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