%% roots sufficiently converge.
-\begin{algorithm}[htpb]
+\begin{algorithm}[h]
\label{alg2-cuda-openmp}
\LinesNumbered
\SetAlgoNoLine
Initialization of Pu\;
Initialization of the solution vector $Z^{0}$\;
omp\_set\_num\_threads(num\_gpus)\;
-\#pragma omp parallel shared(Z,$\Delta$ z,P)\;
+\#pragma omp parallel shared(Z,$\Delta z$,P)\;
\Indp
{
gpu\_id=cudaGetDevice()\;
copy $Z_{loc}$ from GPU to Z in CPU
}
\Indm}
+\RC{Est ce qu'on fait apparaitre le pragma? J'hésite...}
\end{algorithm}
Since a GPU works only on data already allocated in its memory, all local input data, $Z_{k}$, $ZPrec$ and $\Delta z_{k}$, must be transferred from CPU memories to the corresponding GPU memories. Afterwards, the same EA algorithm (Algorithm \ref{alg1-cuda}) is run by all processes but on different polynomial subset of roots $ p(x)_{k}=\sum_{i=1}^{n} a_{i}x^{i}, k=1,...,p$. Each MPI process executes the loop \verb=(While(...)...do)= containing the CUDA kernels but each MPI process computes only its own portion of the roots according to the rule ``''owner computes``''. The local range of roots is indicated with the \textit{index} variable initialized at (line 5, Algorithm \ref{alg2-cuda-mpi}), and passed as an input variable to $kernel\_update$ (line 10, Algorithm \ref{alg2-cuda-mpi}). After each iteration, MPI processes synchronize (\verb=MPI_Allreduce= function) by a reduction on $\Delta z_{k}$ in order to compute the maximum error related to the stop condition. Finally, processes copy the values of new computed roots from GPU memories to CPU memories, then communicate their results to other processes with \verb=MPI_Alltoall= broadcast. If the stop condition is not verified ($error > \epsilon$) then processes stay withing the loop \verb= while(...)...do= until all the roots sufficiently converge.
-%% \begin{enumerate}
-%% \begin{algorithm}[htpb]
-%% \label{alg2-cuda-mpi}
-%% %\LinesNumbered
-%% \caption{CUDA-MPI Algorithm to find roots with the Ehrlich-Aberth method}
-
-%% \KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (Error tolerance
-%% threshold), P (Polynomial to solve), Pu (Derivative of P), $n$ (Polynomial degrees), $\Delta z$ ( error of stop condition), $num_gpus$ (number of MPI processes/ number of GPUs), Size (number of roots)}
-
-%% \KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}
-
-%% \BlankLine
-%% \item Initialization of P\;
-%% \item Initialization of Pu\;
-%% \item Initialization of the solution vector $Z^{0}$\;
-%% \item Allocate and copy initial data from CPU memories to GPU global memories\;
-%% \item $index= Size/num_gpus$\;
-%% \item k=0\;
-%% \While {$error > \epsilon$}{
-%% \item Let $\Delta z=0$\;
-%% \item $kernel\_save(ZPrec,Z)$\;
-%% \item k=k+1\;
-%% \item $kernel\_update(Z,P,Pu,index)$\;
-%% \item $kernel\_testConverge(\Delta z,Z,ZPrec)$\;
-%% \item ComputeMaxError($\Delta z$,error)\;
-%% \item Copy results from GPU memories to CPU memories\;
-%% \item Send $Z[id]$ to all processes\;
-%% \item Receive $Z[j]$ from every other process j\;
-%% }
-%% \end{algorithm}
-%% \end{enumerate}
-%% ~\\
-
-%% \RC{ENCORE ENCORE PIRE}
+\begin{algorithm}[htpb]
+\label{alg2-cuda-mpi}
+%\LinesNumbered
+\caption{CUDA-MPI Algorithm to find roots with the Ehrlich-Aberth method}
+
+\KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (Error tolerance
+ threshold), P (Polynomial to solve), Pu (Derivative of P), $n$ (Polynomial degrees), $\Delta z$ ( error of stop condition), $num_gpus$ (number of MPI processes/ number of GPUs), Size (number of roots)}
+
+\KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}
+
+\BlankLine
+Initialization of P\;
+Initialization of Pu\;
+Initialization of the solution vector $Z^{0}$\;
+Distribution of Z\;
+Allocate memory to GPU\;
+\While {$error > \epsilon$}{
+copy Z from CPU to GPU\;
+$ZPrec_{loc}=kernel\_save(Z_{loc})$\;
+$Z_{loc}=kernel\_update(Z,P,Pu)$\;
+$\Delta z=kernel\_testConv(Z_{loc},ZPrec_{loc})$\;
+$error=MPI\_Reduce(\Delta z)$\;
+$Copy Z_{loc} from GPU to CPU$\;
+$Z=MPI\_AlltoAll(Z_{loc})$\;
+}
+\end{algorithm}
+
\section{Experiments}
\label{sec5}
