From: Christophe Guyeux <guyeux@gmail.com>
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?ds=sidebyside

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