X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/84e15020344b77e5497c4a516cc20b472b2914cd..7d6625e2b26c8e17c8cf8fafc8fbed964dda513a:/paper.tex

diff --git a/paper.tex b/paper.tex
index b08750d..ceffa3d 100644
--- a/paper.tex
+++ b/paper.tex
@@ -364,7 +364,6 @@
 \algnewcommand\Output{\item[\algorithmicoutput]}
 
 \newtheorem{proposition}{Proposition}
-\newtheorem{proof}{Proof}
 
 \begin{document}
 %
@@ -381,7 +380,7 @@
 % 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\\
@@ -742,10 +741,11 @@ 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 and GMRES($m$) is used as solver, then the TSIRM algorithm is convergent.
@@ -756,9 +756,17 @@ Let $r_k = b-Ax_k$, where $x_k$ is the approximation of the solution after the
 $k$-th iterate of TSIRM.
 We will prove that $r_k \rightarrow 0$ when $k \rightarrow +\infty$.
 
-Each step of the TSIRM algorithm 
+Each step of the TSIRM algorithm \\
+$\min_{\alpha \in \mathbb{R}^s} ||b-R\alpha ||_2 = \min_{\alpha \in \mathbb{R}^s} ||b-AS\alpha ||_2$
+
+$\begin{array}{ll}
+& = \min_{x \in Vect\left(x_0, x_1, \hdots, x_{k-1} \right)} ||b-AS\alpha ||_2\\
+& \leqslant \min_{x \in Vect\left( S_{k-1} \right)} ||b-Ax ||_2\\
+& \leqslant ||b-Ax_{k-1}||
+\end{array}$
 \end{proof}
 
+
 %%%*********************************************************
 %%%*********************************************************
 \section{Experiments using PETSc}
@@ -917,10 +925,10 @@ corresponds to 30*12, there are $max\_iter_{ls}$ which corresponds to 15.
 
 
 In  Figure~\ref{fig:01}, the number  of iterations  per second  corresponding to
-Table~\ref{tab:01}  is  displayed.   It  can  be  noticed  that  the  number  of
-iterations per second of FMGRES is  constant whereas it decrease with TSIRM with
-both preconditioner. This  can be explained by the fact that  when the number of
-core increases the time for the minimization step also increases but, generally,
+Table~\ref{tab:03}  is  displayed.   It  can  be  noticed  that  the  number  of
+iterations per second of FMGRES is  constant whereas it decreases with TSIRM with
+both preconditioners. This  can be explained by the fact that  when the number of
+cores increases the time for the least-squares minimization step also increases but, generally,
 when  the number  of cores  increases,  the number  of iterations  to reach  the
 threshold also increases,  and, in that case, TSIRM is  more efficient to reduce
 the number of iterations. So, the overall benefit of using TSIRM is interesting.
@@ -1060,3 +1068,4 @@ Curie and Juqueen respectively based in France and Germany.
 
 % that's all folks
 \end{document}
+