new

[book_gpu.git] / BookGPU / Chapters / chapter6 / PartieAsync.tex

diff --git a/BookGPU/Chapters/chapter6/PartieAsync.tex b/BookGPU/Chapters/chapter6/PartieAsync.tex
index 44eadd5609ee4032d59adf459dfb4654dcc60360..4a2b6d2aee7e1bae827e28527f39cd5e039b42d7 100644 (file)
--- a/BookGPU/Chapters/chapter6/PartieAsync.tex
+++ b/BookGPU/Chapters/chapter6/PartieAsync.tex
@@ -15,34 +15,59 @@ Formally, if we denote by $f=(f_1,...,f_n)$ the function representing the
 iterative process and by $x^t=(x_1^t,...,x_n^t)$ the values of the $n$ elements of
 the system at iteration $t$, we pass from a synchronous iterative scheme of the
 form:
 iterative process and by $x^t=(x_1^t,...,x_n^t)$ the values of the $n$ elements of
 the system at iteration $t$, we pass from a synchronous iterative scheme of the
 form:

+%% \begin{algorithm}[H]
+%%   \caption{Synchronous iterative scheme}\label{algo:ch6p2sync}
+%%   \begin{Algo}
+%%     $x^{0}=(x_{1}^{0},...,x_{n}^{0})$\\
+%%     \textbf{for} $t=0,1,...$\\
+%%     \>\textbf{for} $i=1,...,n$\\
+%%     \>\>$x_{i}^{t+1}=f_{i}(x_{1}^t,...,x_i^t,...,x_{n}^t)$\\
+%%     \>\textbf{endfor}\\
+%%     \textbf{endfor}
+%%   \end{Algo}
+%% \end{algorithm}

 \begin{algorithm}[H]
   \caption{Synchronous iterative scheme}\label{algo:ch6p2sync}
 \begin{algorithm}[H]
   \caption{Synchronous iterative scheme}\label{algo:ch6p2sync}

-  \begin{Algo}
-    $x^{0}=(x_{1}^{0},...,x_{n}^{0})$\\
-    \textbf{for} $t=0,1,...$\\
-    \>\textbf{for} $i=1,...,n$\\
-    \>\>$x_{i}^{t+1}=f_{i}(x_{1}^t,...,x_i^t,...,x_{n}^t)$\\
-    \>\textbf{endfor}\\
-    \textbf{endfor}
-  \end{Algo}
+  $x^{0}=(x_{1}^{0},...,x_{n}^{0})$\;
+  \For{ $t=0,1,...$} {
+    \For{ $i=1,...,n$}{
+      $x_{i}^{t+1}=f_{i}(x_{1}^t,...,x_i^t,...,x_{n}^t)$\;
+    }
+  }

 \end{algorithm}
 \end{algorithm}

+
+

 \noindent
 \noindent

-to an asynchronous iterative scheme of the form:
+to an asynchronous iterative scheme of the form:\\
+%% \begin{algorithm}[H]
+%%   \caption{Asynchronous iterative scheme}\label{algo:ch6p2async}
+%%   \begin{Algo}
+%%     $x^{0}=(x_{1}^{0},...,x_{n}^{0})$\\
+%%     \textbf{for} $t=0,1,...$\\
+%%     \>\textbf{for} $i=1,...,n$\\
+%%     \>\>$x_{i}^{t+1}=\left\{
+%%       \begin{array}[h]{ll}
+%%         x_i^t & \text{if } i \text{ is \emph{not} updated at iteration } i\\
+%%         f_i(x_1^{s_1^i(t)},...,x_n^{s_n^i(t)}) & \text{if } i \text{ is updated at iteration } i
+%%       \end{array}
+%%     \right.$\\
+%%     \>\textbf{endfor}\\
+%%     \textbf{endfor}
+%%   \end{Algo}
+%% \end{algorithm}

 \begin{algorithm}[H]
   \caption{Asynchronous iterative scheme}\label{algo:ch6p2async}
 \begin{algorithm}[H]
   \caption{Asynchronous iterative scheme}\label{algo:ch6p2async}

-  \begin{Algo}
-    $x^{0}=(x_{1}^{0},...,x_{n}^{0})$\\
-    \textbf{for} $t=0,1,...$\\
-    \>\textbf{for} $i=1,...,n$\\
-    \>\>$x_{i}^{t+1}=\left\{
+    $x^{0}=(x_{1}^{0},...,x_{n}^{0})$\;
+    \For {$t=0,1,...$} {
+      \For{ $i=1,...,n$} {
+        $x_{i}^{t+1}=\left\{

       \begin{array}[h]{ll}
         x_i^t & \text{if } i \text{ is \emph{not} updated at iteration } i\\
         f_i(x_1^{s_1^i(t)},...,x_n^{s_n^i(t)}) & \text{if } i \text{ is updated at iteration } i
       \end{array}
       \begin{array}[h]{ll}
         x_i^t & \text{if } i \text{ is \emph{not} updated at iteration } i\\
         f_i(x_1^{s_1^i(t)},...,x_n^{s_n^i(t)}) & \text{if } i \text{ is updated at iteration } i
       \end{array}

-    \right.$\\
-    \>\textbf{endfor}\\
-    \textbf{endfor}
-  \end{Algo}
+    \right.$
+      }
+    }

 \end{algorithm}
 where $s_j^i(t)$ is the iteration number of the production of the value $x_j$ of
 element $j$  that is  used on element  $i$ at iteration  $t$ (see for example~\cite{BT89,
 \end{algorithm}
 where $s_j^i(t)$ is the iteration number of the production of the value $x_j$ of
 element $j$  that is  used on element  $i$ at iteration  $t$ (see for example~\cite{BT89,