X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/0152824d3e001a7084c17325a1171e9efe4c51ec..6ee06aaa241c211c1fbd5d1efd3697c9e1e20b41:/paper.tex

diff --git a/paper.tex b/paper.tex
index fe7fa39..a2a3e9d 100644
--- a/paper.tex
+++ b/paper.tex
@@ -364,6 +364,7 @@
 \algnewcommand\Output{\item[\algorithmicoutput]}
 
 \newtheorem{proposition}{Proposition}
+\newtheorem{proof}{Proof}
 
 \begin{document}
 %
@@ -380,7 +381,7 @@
 % use a multiple column layout for up to two different
 % affiliations
 
-\author{\IEEEauthorblockN{Rapha\"el Couturier\IEEEauthorrefmark{1}, Lilia Ziane Khodja \IEEEauthorrefmark{2}, and Christophe Guyeux\IEEEauthorrefmark{1}}
+\author{\IEEEauthorblockN{Rapha\"el Couturier\IEEEauthorrefmark{1}, Lilia Ziane Khodja\IEEEauthorrefmark{2}, and Christophe Guyeux\IEEEauthorrefmark{1}}
 \IEEEauthorblockA{\IEEEauthorrefmark{1} Femto-ST Institute, University of Franche Comte, France\\
 Email: \{raphael.couturier,christophe.guyeux\}@univ-fcomte.fr}
 \IEEEauthorblockA{\IEEEauthorrefmark{2} INRIA Bordeaux Sud-Ouest, France\\
@@ -739,11 +740,17 @@ Suppose that $A$ is a positive real matrix with symmetric part $M$. Then the res
 \begin{equation}
 ||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_0|| ,
 \end{equation}
-where $\alpha = \lambda_min(M)^2$ and $\beta = \lambda_max(A^T A)$, which proves 
+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}
 
+We can now claim that,
+\begin{proposition}
+If $A$ is a positive real matrix, then the TSIRM algorithm is convergent.
+\end{proposition}
 
+\begin{proof}
+\end{proof}
 
 %%%*********************************************************
 %%%*********************************************************
@@ -1048,4 +1055,3 @@ Curie and Juqueen respectively based in France and Germany.
 % that's all folks
 \end{document}
 
-