X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/e22868ab5dffa57e6db9bb8d6c9c21ae84411e2a..e9ef8e49b713cba35e5f44d77ad7c9cb1385c804:/paper.tex

diff --git a/paper.tex b/paper.tex
index e512018..0290628 100644
--- a/paper.tex
+++ b/paper.tex
@@ -748,10 +748,15 @@ the convergence of GMRES($m$) for all $m$ under that assumption regarding $A$.
 
 We can now claim that,
 \begin{proposition}
-If $A$ is a positive real matrix, then the TSIRM algorithm is convergent.
+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 
 \end{proof}
 
 %%%*********************************************************
@@ -1055,4 +1060,3 @@ Curie and Juqueen respectively based in France and Germany.
 
 % that's all folks
 \end{document}
-