X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/7d1a785e0f85113eb0bc24d6b4421a95d200d9ce..a12cd1ab6b9ca67bc2943937cc73e59782d724e3:/paper.tex

diff --git a/paper.tex b/paper.tex
index 693edae..f60ef9a 100644
--- a/paper.tex
+++ b/paper.tex
@@ -716,7 +716,7 @@ $error$, which is defined by $||Ax_k-b||_2$.
 
 Let us summarize the most important parameters of TSIRM:
 \begin{itemize}
-\item $\epsilon_{tsirm}$: the threshold to stop the TSIRM method;
+\item $\epsilon_{tsirm}$: the threshold that stops the TSIRM method;
 \item $max\_iter_{kryl}$: the maximum number of iterations for the Krylov method;
 \item $s$: the number of outer iterations before applying the minimization step;
 \item $max\_iter_{ls}$: the maximum number of iterations for the iterative least-squares method;
@@ -727,9 +727,9 @@ Let us summarize the most important parameters of TSIRM:
 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
+our   code,   we   have   chosen  to   use   PETSc~\cite{petsc-web-page}.    In
+line~\ref{algo:matrix_mul}, the  matrix-matrix multiplication is  implemented and
+efficient since the  matrix $A$ is sparse and the  matrix $S$ contains few
 columns in  practice. As explained  previously, at least  two methods seem  to be
 interesting to solve the least-squares minimization, CGLS and LSQR.
 
@@ -764,7 +764,7 @@ the parallelization of CGLS which is  similar to LSQR.
 
 
 In each iteration  of CGLS, there is two  matrix-vector multiplications and some
-classical  operations:  dot  product,   norm,  multiplication  and  addition  on
+classical  operations:  dot  product,   norm,  multiplication,  and  addition  on
 vectors.  All  these  operations are  easy  to  implement  in PETSc  or  similar
 environment.  It should be noticed that LSQR follows the same principle, it is a
 little bit longer but it performs more or less the same operations.
@@ -846,8 +846,11 @@ $\begin{array}{ll}
 which concludes the induction and the proof.
 \end{proof}
 
-%We can remark that, at each iterate, the residue of the TSIRM algorithm is lower 
-%than the one of the GMRES method.
+Remark that a similar proposition can be formulated at each time
+the given solver satisfies an inequality of the form $||r_n|| \leqslant \mu^n ||r_0||$,
+with $|\mu|<1$. Furthermore, it is \emph{a priori} possible in some particular cases 
+regarding $A$, 
+that the proposed TSIRM converges while the GMRES($m$) does not.
 
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