maj de la prop

diff --git a/paper.tex b/paper.tex
index 4a8bc4dbfa834bd3f6d674d8b7e44f662e08fc8b..436909ae599c7cc81316f5d2bfd8bdce47fed693 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -380,7 +380,7 @@
 % 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\\


@@ -737,6 +737,7 @@ these operations are easy to implement in PETSc or similar environment.
 \label{sec:04}
 Let us recall the following result, see~\cite{Saad86}.
 \begin{proposition}
 \label{sec:04}
 Let us recall the following result, see~\cite{Saad86}.
 \begin{proposition}

+\label{prop:saad}

 Suppose that $A$ is a positive real matrix with symmetric part $M$. Then the residual norm provided at the $m$-th step of GMRES satisfies:
 \begin{equation}
 ||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_0|| ,
 Suppose that $A$ is a positive real matrix with symmetric part $M$. Then the residual norm provided at the $m$-th step of GMRES satisfies:
 \begin{equation}
 ||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_0|| ,


@@ -748,7 +749,11 @@ the convergence of GMRES($m$) for all $m$ under that assumption regarding $A$.
 
 We can now claim that,
 \begin{proposition}
 
 We can now claim that,
 \begin{proposition}

-If $A$ is a positive real matrix and GMRES($m$) is used as solver, 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. Furthermore, we still have 
+\begin{equation}
+||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_0|| ,
+\end{equation}
+where $\alpha$ and $\beta$ are defined as in Proposition~\ref{prop:saad}.

 \end{proposition}
 
 \begin{proof}
 \end{proposition}
 
 \begin{proof}


@@ -770,6 +775,8 @@ $\begin{array}{ll}
 \end{array}$
 \end{proof}
 
 \end{array}$
 \end{proof}
 

+We can remark that, at each iterate, the residue of the TSIRM algorithm is lower 
+than the one of the GMRES method.

 
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