Merge

[GMRES2stage.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 063cbf9acbeb195e2500a1bbacaa905fa9327143..4e407309a44605787351dec9043baa5b54742bc7 100644 (file)
--- 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}
 $\min_{\alpha \in \mathbb{R}^s} ||b-R\alpha ||_2 = \min_{\alpha \in \mathbb{R}^s} ||b-AS\alpha ||_2$
 
 $\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}
 & \leqslant ||b-Ax_{k-1}||
 \end{array}$
 \end{proof}