X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/fd0a7d17543c653d442bccd0c7ee035764e83650..19a2f48d7a537fbcf6f320847d686e8f8b9efcb4:/paper.tex

diff --git a/paper.tex b/paper.tex
index e23ccc2..fc64064 100644
--- a/paper.tex
+++ b/paper.tex
@@ -613,11 +613,10 @@ $b\in\mathbb{R}^n$ is  the right-hand side.  The algorithm is implemented  as an
 inner-outer iteration  solver based  on iterative Krylov  methods. The  main key
 points of our solver are given in Algorithm~\ref{algo:01}.
 
-In order to accelerate the convergence, the outer iteration is implemented as an
-iterative  Krylov method  which minimizes  some  error functions  over a  Krylov
-subspace~\cite{saad96}. At  each iteration, the  sparse linear system  $Ax=b$ is
+In order to accelerate the convergence, the outer iteration applies a least-square minimization on the residuals computed by the inner   some  error functions  over a  Krylov
+subspace~\cite{Saad2003}. At  each iteration, the  sparse linear system  $Ax=b$ is
 solved   iteratively    with   an   iterative   method,    for   example   GMRES
-method~\cite{saad86} or  some of its variants,  and the Krylov  subspace that we
+method~\cite{Saad86} or  some of its variants,  and the Krylov  subspace that we
 used is spanned by a basis  $S$ composed of successive solutions issued from the
 inner iteration
 \begin{equation}
@@ -640,7 +639,7 @@ which is associated with the least-squares problem
 such  that $R=AS$  is a  dense rectangular  matrix in  $\mathbb{R}^{n\times s}$,
 $s\ll n$,  and $R^T$ denotes  the transpose of  matrix $R$. We use  an iterative
 method   to  solve   the  least-squares   problem~(\ref{eq:01})  such   as  CGLS
-~\cite{hestenes52}  or LSQR~\cite{paige82}  which  are more  appropriate than  a
+~\cite{Hestenes52}  or LSQR~\cite{Paige82}  which  are more  appropriate than  a
 direct method in the parallel context.
 
 \begin{algorithm}[t]
@@ -852,23 +851,23 @@ Curie and Juqueen respectively based in France and Germany.
 % http://www.ctan.org/tex-archive/biblio/bibtex/contrib/doc/
 % The IEEEtran BibTeX style support page is at:
 % http://www.michaelshell.org/tex/ieeetran/bibtex/
-%\bibliographystyle{IEEEtran}
+\bibliographystyle{IEEEtran}
 % argument is your BibTeX string definitions and bibliography database(s)
-%\bibliography{IEEEabrv,../bib/paper}
+\bibliography{biblio}
 %
 % <OR> manually copy in the resultant .bbl file
 % set second argument of \begin to the number of references
 % (used to reserve space for the reference number labels box)
-\begin{thebibliography}{1}
+%% \begin{thebibliography}{1}
 
-\bibitem{saad86} Y.~Saad and M.~H.~Schultz, \emph{GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems}, SIAM Journal on Scientific and Statistical Computing, 7(3):856--869, 1986.
+%% \bibitem{saad86} Y.~Saad and M.~H.~Schultz, \emph{GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems}, SIAM Journal on Scientific and Statistical Computing, 7(3):856--869, 1986.
 
-\bibitem{saad96} Y.~Saad, \emph{Iterative Methods for Sparse Linear Systems}, PWS Publishing, New York, 1996.
+%% \bibitem{saad96} Y.~Saad, \emph{Iterative Methods for Sparse Linear Systems}, PWS Publishing, New York, 1996.
 
-\bibitem{hestenes52} M.~R.~Hestenes and E.~Stiefel, \emph{Methods of conjugate gradients for solving linear system}, Journal of Research of National Bureau of Standards, B49:409--436, 1952.
+%% \bibitem{hestenes52} M.~R.~Hestenes and E.~Stiefel, \emph{Methods of conjugate gradients for solving linear system}, Journal of Research of National Bureau of Standards, B49:409--436, 1952.
 
-\bibitem{paige82} C.~C.~Paige and A.~M.~Saunders, \emph{LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares}, ACM Trans. Math. Softw. 8(1):43--71, 1982.
-\end{thebibliography}
+%% \bibitem{paige82} C.~C.~Paige and A.~M.~Saunders, \emph{LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares}, ACM Trans. Math. Softw. 8(1):43--71, 1982.
+%% \end{thebibliography}