X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/d0f18e8e70d6e9c4404a5c5b5300f736ac4257c1..6ee06aaa241c211c1fbd5d1efd3697c9e1e20b41:/paper.tex

diff --git a/paper.tex b/paper.tex
index e3d19ec..a2a3e9d 100644
--- a/paper.tex
+++ b/paper.tex
@@ -364,6 +364,7 @@
 \algnewcommand\Output{\item[\algorithmicoutput]}
 
 \newtheorem{proposition}{Proposition}
+\newtheorem{proof}{Proof}
 
 \begin{document}
 %
@@ -380,7 +381,7 @@
 % use a multiple column layout for up to two different
 % affiliations
 
-\author{\IEEEauthorblockN{Rapha\"el Couturier\IEEEauthorrefmark{1}, Lilia Ziane Khodja \IEEEauthorrefmark{2}, and Christophe Guyeux\IEEEauthorrefmark{1}}
+\author{\IEEEauthorblockN{Rapha\"el Couturier\IEEEauthorrefmark{1}, Lilia Ziane Khodja\IEEEauthorrefmark{2}, and Christophe Guyeux\IEEEauthorrefmark{1}}
 \IEEEauthorblockA{\IEEEauthorrefmark{1} Femto-ST Institute, University of Franche Comte, France\\
 Email: \{raphael.couturier,christophe.guyeux\}@univ-fcomte.fr}
 \IEEEauthorblockA{\IEEEauthorrefmark{2} INRIA Bordeaux Sud-Ouest, France\\
@@ -425,16 +426,17 @@ Email: lilia.ziane@inria.fr}
 
 
 \begin{abstract}
-In  this article,  a  two-stage  iterative algorithm is proposed to improve  the
-convergence of Krylov based iterative methods,  typically those of GMRES variants. The
-principle of  the proposed approach  is to  build an external  iteration over  the Krylov
-method, and to  frequently store its current  residual   (at  each
-GMRES restart for instance). After a given number of outer iterations, a minimization
-step  is applied  on the  matrix composed by the  saved residuals,  in  order to
-compute a better solution and to make  new iterations if required.  It is proven that
-the proposal has  the same convergence properties than the  inner embedded method itself. 
-Experiments using up  to 16,394 cores also show that the proposed algorithm
-runs around 5 or 7 times faster than GMRES.
+In  this article, a  two-stage iterative  algorithm is  proposed to  improve the
+convergence  of  Krylov  based  iterative  methods,  typically  those  of  GMRES
+variants.  The  principle of  the  proposed approach  is  to  build an  external
+iteration over the  Krylov method, and to frequently  store its current residual
+(at each GMRES restart for instance).  After a given number of outer iterations,
+a least-squares minimization step is applied on the matrix composed by the saved
+residuals, in order  to compute a better solution and to  make new iterations if
+required.  It  is proven that the  proposal has the  same convergence properties
+than the  inner embedded  method itself.  Experiments  using up to  16,394 cores
+also  show that the  proposed algorithm  runs around  5 or  7 times  faster than
+GMRES.
 \end{abstract}
 
 \begin{IEEEkeywords}
@@ -738,11 +740,17 @@ Suppose that $A$ is a positive real matrix with symmetric part $M$. Then the res
 \begin{equation}
 ||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_0|| ,
 \end{equation}
-where $\alpha = \lambda_min(M)^2$ and $\beta = \lambda_max(A^T A)$, which proves 
+where $\alpha = \lambda_{min}(M)^2$ and $\beta = \lambda_{max}(A^T A)$, which proves 
 the convergence of GMRES($m$) for all $m$ under that assumption regarding $A$.
 \end{proposition}
 
+We can now claim that,
+\begin{proposition}
+If $A$ is a positive real matrix, then the TSIRM algorithm is convergent.
+\end{proposition}
 
+\begin{proof}
+\end{proof}
 
 %%%*********************************************************
 %%%*********************************************************
@@ -1047,4 +1055,3 @@ Curie and Juqueen respectively based in France and Germany.
 % that's all folks
 \end{document}
 
-