From: Christophe Guyeux <guyeux@gmail.com>
Date: Sat, 11 Oct 2014 09:25:27 +0000 (+0200)
Subject: Avancées dans la réécriture de la preuve
X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/commitdiff_plain/166f32f18ec198e8744c5092290f640921199099

Avancées dans la réécriture de la preuve
---

diff --git a/paper.tex b/paper.tex
index 812326f..82cc12a 100644
--- a/paper.tex
+++ b/paper.tex
@@ -748,8 +748,9 @@ these operations are easy to implement in PETSc or similar environment.
 We can now claim that,
 \begin{proposition}
 \label{prop:saad}
-If $A$ is either a definite positive or a positive matrix and GMRES($m$) is used as solver, then the TSIRM algorithm is convergent. Furthermore, 
-let $r_k$ be the
+If $A$ is either a definite positive or a positive matrix and GMRES($m$) is used as solver, then the TSIRM algorithm is convergent. 
+
+Furthermore, let $r_k$ be the
 $k$-th residue of TSIRM, then
 we have the following boundaries:
 \begin{itemize}
@@ -770,20 +771,20 @@ where $M$ is the symmetric part of $A$, $\alpha = \lambda_{min}(M)^2$ and $\beta
 \begin{proof}
 Let us first recall that the residue is under control when considering the GMRES algorithm on a positive definite matrix, and it is bounded as follows:
 \begin{equation*}
-\|r_n\| \le \inf_{p \in P_n} \|p(A)\| \le \kappa_2(V) \inf_{p \in P_n} \max_{\lambda \in \sigma(A)} |p(\lambda)| \|r_0\|, .
+\|r_k\| \leq \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_0\| .
 \end{equation*}
 Additionally, when $A$ 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|| ,
 \end{equation*}
 where $\alpha$ and $\beta$ are defined as in Proposition~\ref{prop:saad}, which proves 
-the convergence of GMRES($m$) for all $m$ under that assumption regarding $A$.
-Such well-known results can be found, \emph{e.g.}, in~\cite{Saad86}.
+the convergence of GMRES($m$) for all $m$ under such assumptions regarding $A$.
+These well-known results can be found, \emph{e.g.}, in~\cite{Saad86}.
 
 We will now prove by a mathematical induction that, for each $k \in \mathbb{N}^\ast$, 
 $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{mk}{2}} ||r_0||$ when $A$ is positive, and $\|r_k\| \leq \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_0\|$ when $A$ is positive definite.
 
-The base case is obvious, as for $k=1$, the TSIRM algorithm simply consists in applying GMRES($m$) once, leading to a new residual $r_1$ which follows the inductive hypothesis due to Proposition~\ref{prop:saad}.
+The base case is obvious, as for $k=1$, the TSIRM algorithm simply consists in applying GMRES($m$) once, leading to a new residual $r_1$ which follows the inductive hypothesis due to the results recalled above.
 
 Suppose now that the claim holds for all $m=1, 2, \hdots, k-1$, that is, $\forall m \in \{1,2,\hdots, k-1\}$, $||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$.
 We will show that the statement holds too for $r_k$. Two situations can occur: