TRuc

diff --git a/paper.tex b/paper.tex
index 896ac71233815d6aa5c7b3f00399c26d575d4248..4d0e239a02f1d21caede6978887dbfb368d7a932 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -628,6 +628,12 @@ $S$, 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 $S_i$. 
 
 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  
 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