X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/a94b4bdcd2dbd9bbfb4d24fc8add30c2f46e7a59..34a7850bea7d50191ce0ab301b26d2c6d11ef2ff:/paper.tex

diff --git a/paper.tex b/paper.tex
index dd1ff68..a4545fd 100644
--- a/paper.tex
+++ b/paper.tex
@@ -623,17 +623,18 @@ solved using only $m$
 iterations of an iterative method, this latter being initialized with the 
 last obtained approximation. 
 GMRES method~\cite{Saad86}, or any of its variants, can potentially be used as
-inner solver. The current approximation of the Krylov method is then stored inside a matrix
-$S$ composed by the successive solutions that are computed during inner iterations.
+inner solver. The current approximation of the Krylov method is then stored inside a $n \times s$ matrix
+$S$, 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}
-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  $\mathbb{R}^{n\times s}$,
@@ -663,8 +664,8 @@ appropriate than a single direct method in a parallel context.
 \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