Correction d'une erreur dans la preuve

[GMRES2stage.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 436909ae599c7cc81316f5d2bfd8bdce47fed693..6add44ab51fafcf3ae6057f90a8ca01cf878f7d5 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -742,28 +742,35 @@ 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}
 
 
 We can now claim that,
 \begin{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 and GMRES($m$) is used as solver, then the TSIRM algorithm is convergent. Furthermore, we still have 
+If $A$ is a positive real 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 still have:

 \begin{equation}
 \begin{equation}

-||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_0|| ,
+||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0|| ,

 \end{equation}
 where $\alpha$ and $\beta$ are defined as in Proposition~\ref{prop:saad}.
 \end{proposition}
 
 \begin{proof}
 \end{equation}
 where $\alpha$ and $\beta$ are defined as in Proposition~\ref{prop:saad}.
 \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 $k \rightarrow +\infty$.
+We will prove by a mathematical induction that, for each $k \in \mathbb{N}^\ast$, 
+$||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{mk}{2}} ||r_0||.$
+
+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}.

 
 

-Each step of the TSIRM algorithm \\
+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:
+\begin{itemize}
+\item If $k \mod m \neq 0$, then the TSIRM algorithm consists in executing GMRES once. In that case, we obtain $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_{k-1}||\leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_0||$.

 
 

-Let $\operatorname{span}(S) = \left \{ {\sum_{i=1}^k \lambda_i v_i \Big| k \in \mathbb{N}, v_i \in S, \lambda _i \in \mathbb{R}} \right \}$ be the linear span of a set of vectors $S$. So,\\
+\item Else, let $\operatorname{span}(S) = \left \{ {\sum_{i=1}^k \lambda_i v_i \Big| k \in \mathbb{N}, v_i \in S, \lambda _i \in \mathbb{R}} \right \}$ be the linear span of a set of real vectors $S$. So,\\

 $\min_{\alpha \in \mathbb{R}^s} ||b-R\alpha ||_2 = \min_{\alpha \in \mathbb{R}^s} ||b-AS\alpha ||_2$
 
 $\begin{array}{ll}
 $\min_{\alpha \in \mathbb{R}^s} ||b-R\alpha ||_2 = \min_{\alpha \in \mathbb{R}^s} ||b-AS\alpha ||_2$
 
 $\begin{array}{ll}


@@ -773,6 +780,7 @@ $\begin{array}{ll}
 & \leqslant \min_{\lambda \in \mathbb{R}} ||b-\lambda Ax_{k-1} ||_2\\
 & \leqslant ||b-Ax_{k-1}||_2 .
 \end{array}$
 & \leqslant \min_{\lambda \in \mathbb{R}} ||b-\lambda Ax_{k-1} ||_2\\
 & \leqslant ||b-Ax_{k-1}||_2 .
 \end{array}$

+\end{itemize}

 \end{proof}
 
 We can remark that, at each iterate, the residue of the TSIRM algorithm is lower 
 \end{proof}
 
 We can remark that, at each iterate, the residue of the TSIRM algorithm is lower