Avancées dans la preuve

[GMRES2stage.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index f1becbd3f27a9396a408bf0e5459428194ed1eea..ceffa3d7903def8c6c51d0395e382a9487e67e2a 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -745,9 +745,7 @@ 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}
 
 the convergence of GMRES($m$) for all $m$ under that assumption regarding $A$.
 \end{proposition}
 

-<<<<<<< HEAD

 
 

-=======

 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.
 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.


@@ -758,9 +756,16 @@ 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 $k \rightarrow +\infty$.
 
 $k$-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}
 \end{proof}

->>>>>>> 84e15020344b77e5497c4a516cc20b472b2914cd
+

 
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@@ -920,7 +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
 
 
 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,
 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,


@@ -1064,4 +1069,3 @@ Curie and Juqueen respectively based in France and Germany.
 % that's all folks
 \end{document}
 
 % that's all folks
 \end{document}
 

-