From: Christophe Guyeux <guyeux@gmail.com> Date: Fri, 10 Oct 2014 12:43:50 +0000 (+0200) Subject: Typos dans la preuve X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/commitdiff_plain/257eb3828b7e512fff530ca5609ec15a27e3fa56 Typos dans la preuve --- diff --git a/paper.tex b/paper.tex index 64a88a8..517cb8e 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}