\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}
+
%%%*********************************************************
%%%*********************************************************
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.
% that's all folks
\end{document}
-
