X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/34a7850bea7d50191ce0ab301b26d2c6d11ef2ff..c5faa7bc352e0a123a457fb0a763a834124ae4ed:/paper.tex

diff --git a/paper.tex b/paper.tex
index a4545fd..896ac71 100644
--- a/paper.tex
+++ b/paper.tex
@@ -626,6 +626,7 @@ 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 $n \times s$ matrix
 $S$, which is composed by the $s$ last solutions that have been computed during 
 the inner iterations phase.
+In the remainder, the $i$-th column vector of $S$ will be denoted by $S_i$. 
 
 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
@@ -667,11 +668,12 @@ appropriate than a single direct method in a parallel context.
 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
+equal to  the restart  number in the  GMRES-like method. Moreover,  a tolerance
 threshold must be specified for the  solver. In practice, this threshold must be
-much  smaller  than the  convergence  threshold  of  the TSIRM  algorithm  (\emph{i.e.}
+much  smaller  than the  convergence  threshold  of  the TSIRM  algorithm  (\emph{i.e.},
 $\epsilon_{tsirm}$).  Line~\ref{algo:store}, $S_{k \mod s}=x^k$ consists in copying the
-solution  $x_k$  into the  column  $k \mod s$ of  the  matrix  $S$, where $S$ is a matrix of size $n\times s$ whose column vector $i$ is denoted by $S_i$. After  the
+solution  $x_k$  into the  column  $k \mod s$ of $S$.
+After  the
 minimization, the matrix $S$ is reused with the new values of the residuals.  To
 solve the minimization problem, an  iterative method is used. Two parameters are
 required for that: the maximum number of iterations and the threshold to stop the
@@ -736,7 +738,7 @@ these operations are easy to implement in PETSc or similar environment.
 
 \section{Convergence results}
 \label{sec:04}
-Let us recall the following result, see~\cite{Saad86}.
+Let us recall the following result, see~\cite{Saad86} for further readings.
 \begin{proposition}
 \label{prop:saad}
 Suppose that $A$ is a positive real matrix with symmetric part $M$. Then the residual norm provided at the $m$-th step of GMRES satisfies: