X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/5890d7827b348b48bcfa9f454f2a10f591deef14..9300a8195f1fd0aba96819dbd29244ca8bc90a33:/paper.tex

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@@ -743,33 +743,45 @@ these operations are easy to implement in PETSc or similar environment.
 
 \section{Convergence results}
 \label{sec:04}
-Let us recall the following result, see~\cite{Saad86} for further readings.
-\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|| ,
-\end{equation}
-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}
-If $A$ is a positive real matrix and GMRES($m$) is used as solver, then the TSIRM algorithm is convergent. Furthermore, 
+\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
 $k$-th residue of TSIRM, then
-we still have:
+we have the following boundaries:
+\begin{itemize}
+\item when $A$ is positive:
 \begin{equation}
 ||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}.
+where $M$ is the symmetric part of $A$, $\alpha = \lambda_{min}(M)^2$ and $\beta = \lambda_{max}(A^T A)$;
+\item when $A$ is positive definite:
+\begin{equation}
+\|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}
+\end{itemize}
+%In the general case, where A is not positive definite, we have
+%$\|r_n\| \le \inf_{p \in P_n} \|p(A)\| \le \kappa_2(V) \inf_{p \in P_n} \max_{\lambda \in \sigma(A)} |p(\lambda)| \|r_0\|, .$
 \end{proposition}
 
 \begin{proof}
-We will prove by a mathematical induction that, for each $k \in \mathbb{N}^\ast$, 
-$||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{mk}{2}} ||r_0||.$
+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\|, .
+\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}.
+
+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}.
 
