From: Christophe Guyeux Date: Sat, 11 Oct 2014 08:55:44 +0000 (+0200) Subject: dfmqslk X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/commitdiff_plain/296102e5b791e8d44dcc357426e5590f466a54e9 dfmqslk Merge branch 'master' of ssh://info.iut-bm.univ-fcomte.fr/GMRES2stage --- 296102e5b791e8d44dcc357426e5590f466a54e9 diff --cc paper.tex index 4d0e239,126ff34..8764073 --- a/paper.tex +++ b/paper.tex @@@ -618,22 -622,15 +622,21 @@@ It can be summarized as follows: th inner solver is a Krylov based one. In order to accelerate its convergence, the outer solver periodically applies a least-squares minimization on the residuals computed by the inner one. %Tsolver which does not required to be changed. - At each outer iteration, the sparse linear system $Ax=b$ is partially - solved using only $m$ - iterations of an iterative method, this latter being initialized with the - last obtained approximation. - GMRES method~\cite{Saad86}, or any of its variants, can potentially be used as - inner solver. The current approximation of the Krylov method is then stored inside a $n \times s$ matrix - $S$, which is composed by the $s$ last solutions that have been computed during - the inner iterations phase. - In the remainder, the $i$-th column vector of $S$ will be denoted by $S_i$. + At each outer iteration, the sparse linear system $Ax=b$ is partially solved + using only $m$ iterations of an iterative method, this latter being initialized + with the last obtained approximation. GMRES method~\cite{Saad86}, or any of its + variants, can potentially be used as inner solver. The current approximation of + the Krylov method is then stored inside a $n \times s$ matrix $S$, which is + composed by the $s$ last solutions that have been computed during the inner + iterations phase. In the remainder, the $i$-th column vector of $S$ will be + denoted by $S_i$. +$\|r_n\| \leq \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{n/2} \|r_0\|,$ +In the general case, where A is not positive definite, we have + +$\|r_n\| \le \inf_{p \in P_n} \|p(A)\| \le \kappa_2(V) \inf_{p \in P_n} \max_{\lambda \in \sigma(A)} |p(\lambda)| \|r_0\|, \,$ + + 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