From: Christophe Guyeux Date: Fri, 10 Oct 2014 14:28:04 +0000 (+0200) Subject: Relecture X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/commitdiff_plain/34a7850bea7d50191ce0ab301b26d2c6d11ef2ff?ds=inline Relecture --- diff --git a/paper.tex b/paper.tex index 3a51e45..a4545fd 100644 --- a/paper.tex +++ b/paper.tex @@ -627,14 +627,14 @@ inner solver. The current approximation of the Krylov method is then stored insi $S$, which is composed by the $s$ last solutions that have been computed during the inner iterations phase. -At each $s$ iterations, the minimization step is applied in order to +At each $s$ iterations, another kind of minimization step is applied in order to compute a new solution $x$. For that, the previous residuals of $Ax=b$ are computed by the inner iterations with $(b-AS)$. The minimization of the residuals is obtained by \begin{equation} \underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2 \label{eq:01} \end{equation} -with $R=AS$. Then the new solution $x$ is computed with $x=S\alpha$. +with $R=AS$. The new solution $x$ is then computed with $x=S\alpha$. In practice, $R$ is a dense rectangular matrix belonging in $\mathbb{R}^{n\times s}$, @@ -664,8 +664,8 @@ appropriate than a single direct method in a parallel context. \label{algo:01} \end{algorithm} -Algorithm~\ref{algo:01} summarizes the principle of our method. The outer -iteration is inside the for loop. Line~\ref{algo:solve}, the Krylov method is +Algorithm~\ref{algo:01} summarizes the principle of the proposed method. The outer +iteration is inside the \emph{for} loop. Line~\ref{algo:solve}, the Krylov method is called for a maximum of $max\_iter_{kryl}$ iterations. In practice, we suggest to set this parameter equals to the restart number of the GMRES-like method. Moreover, a tolerance threshold must be specified for the solver. In practice, this threshold must be