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{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}
+
+
\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{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{ $i=1,...,n$} {
+ $x_{i}^{t+1}=\left\{
\begin{array}[h]{ll}
x_i^t & \text{if } i \text{ is \emph{not} updated at iteration } i\\
f_i(x_1^{s_1^i(t)},...,x_n^{s_n^i(t)}) & \text{if } i \text{ is updated at iteration } i
\end{array}
- \right.$\\
- \>\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,
