X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/acead508c1f820b919aa203c78668e5e645cc9b8..7d6625e2b26c8e17c8cf8fafc8fbed964dda513a:/paper.tex

diff --git a/paper.tex b/paper.tex
index bf9e767..ceffa3d 100644
--- a/paper.tex
+++ b/paper.tex
@@ -746,6 +746,26 @@ 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.
+\end{proposition}
+
+\begin{proof}
+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 \\
+$\min_{\alpha \in \mathbb{R}^s} ||b-R\alpha ||_2 = \min_{\alpha \in \mathbb{R}^s} ||b-AS\alpha ||_2$
+
+$\begin{array}{ll}
+& = \min_{x \in Vect\left(x_0, x_1, \hdots, x_{k-1} \right)} ||b-AS\alpha ||_2\\
+& \leqslant \min_{x \in Vect\left( S_{k-1} \right)} ||b-Ax ||_2\\
+& \leqslant ||b-Ax_{k-1}||
+\end{array}$
+\end{proof}
+
 
 %%%*********************************************************
 %%%*********************************************************
@@ -905,7 +925,7 @@ 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
+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,
@@ -1049,4 +1069,3 @@ Curie and Juqueen respectively based in France and Germany.
 % that's all folks
 \end{document}
 
-