10-10-2014 12

[GMRES2stage.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index bf9e767c64f4e0b6780e51b6506acfdc91dc9403..3b19b2de33824a775e7438a9dc202f344890924b 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -745,7 +745,22 @@ where $\alpha = \lambda_min(M)^2$ and $\beta = \lambda_max(A^T A)$, which proves
 the convergence of GMRES($m$) for all $m$ under that assumption regarding $A$.
 \end{proposition}
 
 the convergence of GMRES($m$) for all $m$ under that assumption regarding $A$.
 \end{proposition}
 

+<<<<<<< HEAD

 
 

+=======
+We can now claim that,
+\begin{proposition}
+If $A$ is a positive real matrix and GMRES($m$) is used as solver, then the TSIRM algorithm is convergent.
+\end{proposition}
+
+\begin{proof}
+Let $r_k = b-Ax_k$, where $x_k$ is the approximation of the solution after the
+$k$-th iterate of TSIRM.
+We will prove that $r_k \rightarrow 0$ when $k \rightarrow +\infty$.
+
+Each step of the TSIRM algorithm 
+\end{proof}
+>>>>>>> 84e15020344b77e5497c4a516cc20b472b2914cd

 
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@@ -905,7 +920,7 @@ corresponds to 30*12, there are $max\_iter_{ls}$ which corresponds to 15.
 
 
 In  Figure~\ref{fig:01}, the number  of iterations  per second  corresponding to
 
 
 In  Figure~\ref{fig:01}, the number  of iterations  per second  corresponding to

-Table~\ref{tab:01}  is  displayed.   It  can  be  noticed  that  the  number  of
+Table~\ref{tab:03}  is  displayed.   It  can  be  noticed  that  the  number  of

 iterations per second of FMGRES is  constant whereas it decreases with TSIRM with
 both preconditioners. This  can be explained by the fact that  when the number of
 cores increases the time for the least-squares minimization step also increases but, generally,
 iterations per second of FMGRES is  constant whereas it decreases with TSIRM with
 both preconditioners. This  can be explained by the fact that  when the number of
 cores increases the time for the least-squares minimization step also increases but, generally,