From 6ee06aaa241c211c1fbd5d1efd3697c9e1e20b41 Mon Sep 17 00:00:00 2001 From: Christophe Guyeux Date: Fri, 10 Oct 2014 13:23:10 +0200 Subject: [PATCH] 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