\end{abstract}
% no keywords
-\LZK{Faut pas mettre des keywords?}
+\LZK{Faut pas mettre des keywords?\KG{Oui d'après ça: "no keywords" qui se trouve dans leur fichier source!!, mais c'est Bizzard!!!}}
\IEEEpeerreviewmaketitle
though the latest values of other approximations $z^{k}_{i}$ have not
been received from the other processors, in contrast with synchronous
algorithms where it would wait those values before making a new
-iteration. Couturier et al.~\cite{Raphaelall01} proposed two methods
+iteration. Couturier and al.~\cite{Raphaelall01} proposed two methods
of parallelization for a shared memory architecture with OpenMP and
for a distributed memory one with MPI. They are able to compute the
-roots of sparse polynomials of degree 10,000 in 116 seconds with
-OpenMP and 135 seconds with MPI only by using 8 personal computers and
-2 communications per iteration. The authors showed an interesting
-speedup comparing to the sequential implementation which takes up-to
-3,300 seconds to obtain same results.
+roots of sparse polynomials of degree 10,000. The authors showed an interesting
+speedup that is 20 times as fast as the sequential implementation.
+%which takes up-to 3,300 seconds to obtain same results.
\RC{si on donne des temps faut donner le proc, comme c'est vieux à mon avis faut supprimer ca, votre avis?}
\LZK{Supprimons ces détails et mettons une référence s'il y en a une}
-
+\KG{Je viens de supprimer les détails, la référence existe déja, a reverifier}
Very few work had been performed since then until the appearing of the Compute Unified Device Architecture (CUDA)~\cite{CUDA15}, a parallel computing platform and a programming model invented by NVIDIA. The computing power of GPUs (Graphics Processing Units) has exceeded that of traditional processors CPUs. However, CUDA adopts a totally new computing architecture to use the hardware resources provided by the GPU in order to offer a stronger computing ability to the massive data computing. Ghidouche et al.~\cite{Kahinall14} proposed an implementation of the Durand-Kerner method on a single GPU. Their main results showed that a parallel CUDA implementation is about 10 times faster than the sequential implementation on a single CPU for sparse polynomials of degree 48,000.
In this paper we propose the parallelization of Ehrlich-Aberth method which has a good convergence and it is suitable to be implemented in parallel computers. We use two parallel programming paradigms OpenMP and MPI on CUDA multi-GPU platforms. Our CUDA-MPI and CUDA-OpenMP codes are the first implementations of Ehrlich-Aberth method with multiple GPUs for finding roots of polynomials. Our major contributions include:
\item The parallel implementation of Ehrlich-Aberth algorithm on a multi-GPU platform with a distributed memory using MPI API, such that each GPU is attached and managed by a MPI process. The GPUs exchange their data by message-passing communications.
\end{itemize}
This latter approach is more used on clusters to solve very complex problems that are too large for traditional supercomputers, which are very expensive to build and run.
-\LZK{Pas d'autres contributions possibles? J'ai supprimé les deux premiers points proposés précédemment.}
+\LZK{Pas d'autres contributions possibles? J'ai supprimé les deux premiers points proposés précédemment.
+\AS{La résolution du problème pour des polynomes pleins de degré 6M est une contribution aussi à mon avis}}
The paper is organized as follows. In Section~\ref{sec2} we present three different parallel programming models OpenMP, MPI and CUDA. In Section~\ref{sec3} we present the implementation of the Ehrlich-Aberth algorithm on a single GPU. In Section~\ref{sec4} we present the parallel implementations of the Ehrlich-Aberth algorithm on multiple GPUs using the OpenMP and MPI approaches. In section~\ref{sec5} we present our experiments and discuss them. Finally, Section~\ref{sec6} concludes this paper and gives some hints for future research directions in this topic.
The code is organized as kernels which are parts of code that are run
on GPU devices. Algorithm~\ref{alg1-cuda} describes the CUDA
implementation of the Ehrlich-Aberth on a GPU. This algorithms starts
-by initializinf the polynomial and its derivative (line 1). The
+by initializing the polynomial and its derivative (line 1). The
initialization of the roots is performed (line 2). Data are transfered
from the CPU to the GPU (after allocation of the required memory on
the GPU) (line 3). Then at each iteration, if the error is greater
