Relecture

[GMRES2stage.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 3a51e45878d50c437d139e3d11009159adf232d1..a4545fd4733e8da1759f78dd354195c238bf34c7 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -627,14 +627,14 @@ inner solver. The current approximation of the Krylov method is then stored insi
 $S$, which is composed by the $s$ last solutions that have been computed during 
 the inner iterations phase.
 
 $S$, which is composed by the $s$ last solutions that have been computed during 
 the inner iterations phase.
 

-At each $s$ iterations, the minimization step is applied in order to
+At each $s$ iterations, another kind of minimization step is applied in order to

 compute a new  solution $x$. For that, the previous  residuals of $Ax=b$ are computed by
 the inner iterations with $(b-AS)$. The minimization of the residuals is obtained by  
 \begin{equation}
    \underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2
 \label{eq:01}
 \end{equation}
 compute a new  solution $x$. For that, the previous  residuals of $Ax=b$ are computed by
 the inner iterations with $(b-AS)$. The minimization of the residuals is obtained by  
 \begin{equation}
    \underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2
 \label{eq:01}
 \end{equation}

-with $R=AS$. Then the new solution $x$ is computed with $x=S\alpha$.
+with $R=AS$. The new solution $x$ is then computed with $x=S\alpha$.

 
 
 In  practice, $R$  is a  dense rectangular  matrix belonging in  $\mathbb{R}^{n\times s}$,
 
 
 In  practice, $R$  is a  dense rectangular  matrix belonging in  $\mathbb{R}^{n\times s}$,


@@ -664,8 +664,8 @@ appropriate than a single direct method in a parallel context.
 \label{algo:01}
 \end{algorithm}
 
 \label{algo:01}
 \end{algorithm}
 

-Algorithm~\ref{algo:01}  summarizes  the principle  of  our  method.  The  outer
-iteration is  inside the for  loop. Line~\ref{algo:solve}, the Krylov  method is
+Algorithm~\ref{algo:01}  summarizes  the principle  of  the proposed  method.  The  outer
+iteration is  inside the \emph{for}  loop. Line~\ref{algo:solve}, the Krylov  method is

 called for a  maximum of $max\_iter_{kryl}$ iterations.  In practice, we  suggest to set this parameter
 equals to  the restart  number of the  GMRES-like method. Moreover,  a tolerance
 threshold must be specified for the  solver. In practice, this threshold must be
 called for a  maximum of $max\_iter_{kryl}$ iterations.  In practice, we  suggest to set this parameter
 equals to  the restart  number of the  GMRES-like method. Moreover,  a tolerance
 threshold must be specified for the  solver. In practice, this threshold must be