Avancées dans la preuve

[GMRES2stage.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 3b19b2de33824a775e7438a9dc202f344890924b..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}
 
-<<<<<<< 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.
@@ -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 $k \rightarrow +\infty$.
 
-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}
->>>>>>> 84e15020344b77e5497c4a516cc20b472b2914cd
+
 
 %%%*********************************************************
 %%%*********************************************************
@@ -1064,4 +1069,3 @@ Curie and Juqueen respectively based in France and Germany.
 % that's all folks
 \end{document}
 
-