From 346c1e71da89fda6afd665e560ba9853963dc17a Mon Sep 17 00:00:00 2001 From: Christophe Guyeux Date: Fri, 10 Oct 2014 15:53:27 +0200 Subject: [PATCH] fin de la preuve --- paper.tex | 13 ++++++++----- 1 file changed, 8 insertions(+), 5 deletions(-) diff --git a/paper.tex b/paper.tex index 012e7f1..d114bdd 100644 --- a/paper.tex +++ b/paper.tex @@ -774,13 +774,16 @@ Let $\operatorname{span}(S) = \left \{ {\sum_{i=1}^k \lambda_i v_i \Big| k \in \ $\min_{\alpha \in \mathbb{R}^s} ||b-R\alpha ||_2 = \min_{\alpha \in \mathbb{R}^s} ||b-AS\alpha ||_2$ $\begin{array}{ll} -& = \min_{x \in span\left(S_{k-s}, S_{k-s+1}, \hdots, S_{k-1} \right)} ||b-AS\alpha ||_2\\ -& = \min_{x \in span\left(x_{k-s}, x_{k-s}+1, \hdots, x_{k-1} \right)} ||b-AS\alpha ||_2\\ -& \leqslant \min_{x \in span\left( x_{k-1} \right)} ||b-Ax ||_2\\ -& \leqslant \min_{\lambda \in \mathbb{R}} ||b-\lambda Ax_{k-1} ||_2\\ -& \leqslant ||b-Ax_{k-1}||_2 . +& = \min_{x \in span\left(S_{k-s+1}, S_{k-s+2}, \hdots, S_{k} \right)} ||b-AS\alpha ||_2\\ +& = \min_{x \in span\left(x_{k-s+1}, x_{k-s}+2, \hdots, x_{k} \right)} ||b-AS\alpha ||_2\\ +& \leqslant \min_{x \in span\left( x_{k} \right)} ||b-Ax ||_2\\ +& \leqslant \min_{\lambda \in \mathbb{R}} ||b-\lambda Ax_{k} ||_2\\ +& \leqslant ||b-Ax_{k}||_2\\ +& = ||r_k||_2\\ +& \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||, \end{array}$ \end{itemize} +which concludes the induction and the proof. \end{proof} We can remark that, at each iterate, the residue of the TSIRM algorithm is lower -- 2.39.5