From 317020501fc8da3f9e443457700821b26f66da55 Mon Sep 17 00:00:00 2001 From: Christophe Guyeux Date: Fri, 10 Oct 2014 14:58:22 +0200 Subject: [PATCH] maj de la prop --- paper.tex | 11 +++++++++-- 1 file changed, 9 insertions(+), 2 deletions(-) diff --git a/paper.tex b/paper.tex index 4a8bc4d..436909a 100644 --- a/paper.tex +++ b/paper.tex @@ -380,7 +380,7 @@ % 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\\ @@ -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{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|| , @@ -748,7 +749,11 @@ the convergence of GMRES($m$) for all $m$ under that assumption regarding $A$. 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} @@ -770,6 +775,8 @@ $\begin{array}{ll} \end{array}$ \end{proof} +We can remark that, at each iterate, the residue of the TSIRM algorithm is lower +than the one of the GMRES method. %%%********************************************************* %%%********************************************************* -- 2.39.5