From 6ee06aaa241c211c1fbd5d1efd3697c9e1e20b41 Mon Sep 17 00:00:00 2001
From: Christophe Guyeux <guyeux@gmail.com>
Date: Fri, 10 Oct 2014 13:23:10 +0200
Subject: [PATCH 1/1] Reprise du boulot
---
paper.tex | 12 +++++++++---
1 file changed, 9 insertions(+), 3 deletions(-)
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}
-
--
2.39.5