X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/5890d7827b348b48bcfa9f454f2a10f591deef14..296102e5b791e8d44dcc357426e5590f466a54e9:/paper.tex

diff --git a/paper.tex b/paper.tex
index 126ff34..8764073 100644
--- a/paper.tex
+++ b/paper.tex
@@ -631,6 +631,12 @@ 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