fMerge branch 'master' of ssh://bilbo.iut-bm.univ-fcomte.fr/GMRES2stage

[GMRES2stage.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index cba14daca4d4070c5dceb5345981a74323ddb1a9..1d4cac09f311bc2942f5bc0278957528aa0c19a3 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -364,6 +364,7 @@
 \algnewcommand\Output{\item[\algorithmicoutput]}
 
 \newtheorem{proposition}{Proposition}
 \algnewcommand\Output{\item[\algorithmicoutput]}
 
 \newtheorem{proposition}{Proposition}

+\newtheorem{proof}{Proof}

 
 \begin{document}
 %
 
 \begin{document}
 %


@@ -380,7 +381,7 @@
 % use a multiple column layout for up to two different
 % affiliations
 
 % 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\\
 \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\\


@@ -741,12 +742,23 @@ 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}
 \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}
 
 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}
+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 $k \rightarrow +\infty$.

 
 
 
 

+\end{proof}
+

 %%%*********************************************************
 %%%*********************************************************
 \section{Experiments using PETSc}
 %%%*********************************************************
 %%%*********************************************************
 \section{Experiments using PETSc}


@@ -1048,5 +1060,3 @@ Curie and Juqueen respectively based in France and Germany.
 
 % that's all folks
 \end{document}
 
 % that's all folks
 \end{document}

-
-