08-01-2013

[Krylov_multi.git] / krylov_multi.tex

diff --git a/krylov_multi.tex b/krylov_multi.tex
index d81125b19fbbc0e1c4edccb70eb1e669d90ed6f1..3295c82e9c5b16cdeb4287c5192d00e088bfb3e8 100644 (file)
--- a/krylov_multi.tex
+++ b/krylov_multi.tex
@@ -62,12 +62,13 @@ $Ax=b$ consists in partitioning the matrix $A$ in $L$ several ways
 A = M_l - N_l,~l\in\{1,\ldots,L\},
 \label{eq01}
 \end{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
+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 $E_l$ is a non-negative and diagonal weighting matrix such that $\sum^L_{l=1}E_l=I$ ($I$ is the identity matrix).
+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.
 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.


@@ -77,18 +78,18 @@ Thus the convergence of such a method is dependent on the condition
 The advantage of the multisplitting method is that at each iteration $k$ there are $L$ different linear
 systems
 \begin{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\},
+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}
 \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.
+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
 \begin{equation}
 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,
+x^k = \displaystyle\sum^L_{l=1} E_l y_l^k,

 \label{eq05}
 \end{equation}    
 \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),
+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.  
 %%%%%%%%%%%%%%%%%%%%%%%
 %% END
 the multisplitting method is non-overlapping and corresponds to the block Jacobi method.  
 %%%%%%%%%%%%%%%%%%%%%%%
 %% END


@@ -146,7 +147,7 @@ b & = & [B_{1}, \ldots, B_{L}]
 where for all $l\in\{1,\ldots,L\}$ $A_l$ is a rectangular block of size $n_l\times n$
 and $X_l$ and $B_l$ are sub-vectors of size $n_l$, such that $\sum_ln_l=n$. In this
 case, we use a row-by-row splitting without overlapping in such a way that successive
 where for all $l\in\{1,\ldots,L\}$ $A_l$ is a rectangular block of size $n_l\times n$
 and $X_l$ and $B_l$ are sub-vectors of size $n_l$, such that $\sum_ln_l=n$. In this
 case, we use a row-by-row splitting without overlapping in such a way that successive

-rows of the sparse matrix $A$ and both vectors $x$ and $b$ are assigned to a cluster.
+rows of the sparse matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster.

 So, the multisplitting format of the linear system is defined as follows:
 \begin{equation}
 \forall l\in\{1,\ldots,L\} \mbox{,~} \displaystyle\sum_{i=1}^{l-1}A_{li}X_i + A_{ll}X_l + \displaystyle\sum_{i=l+1}^{L}A_{li}X_i = B_l, 
 So, the multisplitting format of the linear system is defined as follows:
 \begin{equation}
 \forall l\in\{1,\ldots,L\} \mbox{,~} \displaystyle\sum_{i=1}^{l-1}A_{li}X_i + A_{ll}X_l + \displaystyle\sum_{i=l+1}^{L}A_{li}X_i = B_l,