X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/7de6cc5b2d794c02ef46116780ac061eff79ab7a..a1237a8d52aa2c3f927cdb0388c9f049243471f4:/paper.tex

diff --git a/paper.tex b/paper.tex
index 063cbf9..4e40730 100644
--- a/paper.tex
+++ b/paper.tex
@@ -760,8 +760,9 @@ Each step of the TSIRM algorithm \\
 $\min_{\alpha \in \mathbb{R}^s} ||b-R\alpha ||_2 = \min_{\alpha \in \mathbb{R}^s} ||b-AS\alpha ||_2$
 
 $\begin{array}{ll}
-& = \min_{x \in Vect\left(x_0, x_1, \hdots, x_{k-1} \right)} ||b-AS\alpha ||_2\\
-& \leqslant \min_{x \in Vect\left( S_{k-1} \right)} ||b-Ax ||_2\\
+& = \min_{x \in Vect\left(S_{k-s}, S_{k-s+1}, \hdots, S_{k-1} \right)} ||b-AS\alpha ||_2\\
+& = \min_{x \in Vect\left(x_{k-s}, x_{k-s}+1, \hdots, x_{k-1} \right)} ||b-AS\alpha ||_2\\
+& \leqslant \min_{x \in Vect\left( x_{k-1} \right)} ||b-Ax ||_2\\
 & \leqslant ||b-Ax_{k-1}||
 \end{array}$
 \end{proof}