X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/2013b77d04aa67e242c885349c663c432142782a..7de6cc5b2d794c02ef46116780ac061eff79ab7a:/paper.tex

diff --git a/paper.tex b/paper.tex
index c0e8b16..063cbf9 100644
--- a/paper.tex
+++ b/paper.tex
@@ -669,8 +669,8 @@ called for a  maximum of $max\_iter_{kryl}$ iterations.  In practice, we  sugges
 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
 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  $S$. After  the
+$\epsilon_{tsirm}$).  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
 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
@@ -686,13 +686,13 @@ Let us summarize the most important parameters of TSIRM:
 \end{itemize}
 
 
-The  parallelisation  of  TSIRM  relies   on  the  parallelization  of  all  its
+The  parallelization  of  TSIRM  relies   on  the  parallelization  of  all  its
 parts. More  precisely, except  the least-squares step,  all the other  parts are
 obvious to  achieve out in parallel. In  order to develop a  parallel version of
 our   code,   we   have   chosen  to   use   PETSc~\cite{petsc-web-page}.    For
 line~\ref{algo:matrix_mul} the  matrix-matrix multiplication is  implemented and
 efficient since the  matrix $A$ is sparse and since the  matrix $S$ contains few
-colums in  practice. As explained  previously, at least  two methods seem  to be
+columns in  practice. As explained  previously, at least  two methods seem  to be
 interesting to solve the least-squares minimization, CGLS and LSQR.
 
 In the following  we remind the CGLS algorithm. The LSQR  method follows more or