From 64f1ca2222ea5046f340e606f8978e8a9a4dcd1e Mon Sep 17 00:00:00 2001 From: Christophe Guyeux Date: Sat, 11 Oct 2014 10:55:26 +0200 Subject: [PATCH] TRuc --- paper.tex | 6 ++++++ 1 file changed, 6 insertions(+) diff --git a/paper.tex b/paper.tex index 896ac71..4d0e239 100644 --- 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 $S_i$. +$\|r_n\| \leq \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{n/2} \|r_0\|,$ +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 -- 2.39.5