+%And let a rational function $R_{i}(z)$ be the correction term of the
+%Weistrass method~\cite{Weierstrass03}
+
+%\begin{equation}
+%R_{i}(z)=\frac{p(z)}{w_{i}(z)} , i=1,2,...,n.
+%\end{equation}
+
+%Differentiating the rational function $R_{i}(z)$ and applying the
+%Newton method, we have:
+
+%\begin{equation}
+%\frac{R_{i}(z)}{R_{i}^{'}(z)}= \frac{p(z)}{p^{'}(z)-p(z)\frac{w_{i}(z)}{w_{i}^{'}(z)}}= \frac{p(z)}{p^{'}(z)-p(z) \sum _{j=1,j \neq i}^{n}\frac{1}{z-x_{j}}}, i=1,2,...,n
+%\end{equation}
+%where R_{i}^{'}(z)is the rational function derivative of F evaluated in the point z
+%Substituting $x_{j}$ for $z_{j}$ we obtain the Aberth iteration method.%
+
+
+%\subsubsection{Polynomials Initialization}
+%The initialization of a polynomial $p(z)$ is done by setting each of the $n$ complex coefficients %$a_{i}$:
+
+%\begin{equation}
+%\label{eq:SimplePolynome}
+% p(z)=\sum{a_{i}z^{n-i}} , a_{n} \neq 0,a_{0}=1, a_{i}\subset C
+%\end{equation}
+
+
+%\subsubsection{Vector $Z^{(0)}$ Initialization}
+%\label{sec:vec_initialization}
+%As for any iterative method, we need to choose $n$ initial guess points $z^{0}_{i}, i = 1, . . . , %n.$
+%The initial guess is very important since the number of steps needed by the iterative method to %reach
+%a given approximation strongly depends on it.
+%In~\cite{Aberth73} the Ehrlich-Aberth iteration is started by selecting $n$
+%equi-distant points on a circle of center 0 and radius r, where r is
+%an upper bound to the moduli of the zeros. Later, Bini and al.~\cite{Bini96}
+%performed this choice by selecting complex numbers along different
+%circles which relies on the result of~\cite{Ostrowski41}.
+
+%\begin{equation}
+%\label{eq:radiusR}
+%%\begin{align}
+%\sigma_{0}=\frac{u+v}{2};u=\frac{\sum_{i=1}^{n}u_{i}}{n.max_{i=1}^{n}u_{i}};
+%v=\frac{\sum_{i=0}^{n-1}v_{i}}{n.min_{i=0}^{n-1}v_{i}};\\
+%%\end{align}
+%\end{equation}
+%Where:
+%\begin{equation}
+%u_{i}=2.|a_{i}|^{\frac{1}{i}};
+%v_{i}=\frac{|\frac{a_{n}}{a_{i}}|^{\frac{1}{n-i}}}{2}.
+%\end{equation}
+
+%\subsubsection{Iterative Function}
+%The operator used by the Aberth method corresponds to the
+%equation~\ref{Eq:EA1}, it enables the convergence towards
+%the polynomials zeros, provided all the roots are distinct.
+
+%Here we give a second form of the iterative function used by the Ehrlich-Aberth method:
+
+%\begin{equation}
+%\label{Eq:EA1}
+%EA: z^{k+1}_{i}=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}
+%{1-\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}\sum_{j=1,j\neq i}^{j=n}{\frac{1}{(z_{i}^{k}-z_{j}^{k})}}}, %i=1,. . . .,n
+%\end{equation}
+
+%\subsubsection{Convergence Condition}
+%The convergence condition determines the termination of the algorithm. It consists in stopping the %iterative function when the roots are sufficiently stable. We consider that the method converges %sufficiently when:
+
+%\begin{equation}
+%\label{eq:Aberth-Conv-Cond}
+%\forall i \in [1,n];\vert\frac{z_{i}^{k}-z_{i}^{k-1}}{z_{i}^{k}}\vert<\xi
+%\end{equation}
+
+
+%\begin{figure}[htbp]
+%\centering
+ % \includegraphics[angle=-90,width=0.5\textwidth]{EA-Algorithm}
+%\caption{The Ehrlich-Aberth algorithm on single GPU}
+%\label{fig:03}
+%\end{figure}
+
+%the Ehrlich-Aberth method is an iterative method, contain 4 steps, start from the initial approximations of all the roots of the polynomial,the second step initialize the solution vector $Z$ using the Guggenheimer method to assure the distinction of the initial vector roots, than in step 3 we apply the the iterative function based on the Newton's method and Weiestrass operator~\cite{,}, wich will make it possible to converge to the roots solution, provided that all the root are different.
+
+The Ehrlich-Aberth method is a simultaneous method~\cite{} using the following iteration
+\begin{equation}
+\label{Eq:EA1}
+z^{k+1}_i=z_i^k-\frac{\frac{p(z_i^k)}{p'(z_i^k)}}{1-\frac{p(z_i^k)}{p'(z_i^k)}\displaystyle\sum\limits_{\substack{j=1 \\ j\neq i}}^{j=n}{\frac{1}{(z_i^k-z_j^k)}}}, i=1,\ldots,n,
+\end{equation}
+to find the roots $Z$
+
+ contain 4 steps, start from the initial approximations of all the roots of the polynomial,the second step initialize the solution vector $Z$ using the Guggenheimer method to assure the distinction of the initial vector roots, than in step 3 we apply the the iterative function based on the Newton's method and Weiestrass operator~\cite{,}, wich will make it possible to converge to the roots solution, provided that all the root are different.
+
+
+
+ At the end of each application of the iterative function, a stop condition is verified consists in stopping the iterative process when the whole of the modules of the roots are lower than a fixed value $\xi$
+
+\begin{equation}
+\label{eq:Aberth-Conv-Cond}
+\forall i \in [1,n];\vert\frac{z_{i}^{k}-z_{i}^{k-1}}{z_{i}^{k}}\vert<\xi
+\end{equation}
+
+\subsection{EA parallel implementation on CUDA}
+We introduced three paradigms of parallel programming. Our objective consists in implementing a root finding polynomial algorithm on multiple GPUs. To this end, it is primordial to know how to manage CUDA contexts of different GPUs. A direct method for controlling the various GPUs is to use as many threads or processes as GPU devices. We can choose the GPU index based on the identifier of OpenMP thread or the rank of the MPI process. Both approaches will be investigated.
+
+
+
+
+Like any parallel code, a GPU parallel implementation first
+requires to determine the sequential tasks and the
+parallelizable parts of the sequential version of the
+program/algorithm. In our case, all the operations that are easy
+to execute in parallel must be made by the GPU to accelerate
+the execution of the application, like the step 3 and step 4. On the other hand, all the
+sequential operations and the operations that have data
+dependencies between threads or recursive computations must
+be executed by only one CUDA or CPU thread (step 1 and step 2). Initially, we specify the organization of parallel threads, by specifying the dimension of the grid Dimgrid, the number of blocks per grid DimBlock and the number of threads per block.
+
+The code is organzed by what is named kernels, portions o code that are run on GPU devices. For step 3, there are two kernels, the
+first named \textit{save} is used to save vector $Z^{K-1}$ and the seconde one is named
+\textit{update} and is used to update the $Z^{K}$ vector. For step 4, a kernel
+tests the convergence of the method. In order to
+compute the function H, we have two possibilities: either to use
+the Jacobi mode, or the Gauss-Seidel mode of iterating which uses the
+most recent computed roots. It is well known that the Gauss-
+Seidel mode converges more quickly. So, we used the Gauss-Seidel mode of iteration. To
+parallelize the code, we created kernels and many functions to
+be executed on the GPU for all the operations dealing with the
+computation on complex numbers and the evaluation of the
+polynomials. As said previously, we managed both functions
+of evaluation of a polynomial: the normal method, based on
+the method of Horner and the method based on the logarithm
+of the polynomial. All these methods were rather long to
+implement, as the development of corresponding kernels with
+CUDA is longer than on a CPU host. This comes in particular
+from the fact that it is very difficult to debug CUDA running
+threads like threads on a CPU host. In the following paragraph
+Algorithm~\ref{alg1-cuda} shows the GPU parallel implementation of Ehrlich-Aberth method.
+
+\begin{enumerate}
+\begin{algorithm}[htpb]
+\label{alg1-cuda}
+%\LinesNumbered
+\caption{CUDA 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), $\Delta z_{max}$ (Maximum value of stop condition)}
+
+\KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}
+
+%\BlankLine
+
+\item Initialization of the of P\;
+\item Initialization of the of Pu\;
+\item Initialization of the solution vector $Z^{0}$\;
+\item Allocate and copy initial data to the GPU global memory\;
+\item k=0\;
+\While {$\Delta z_{max} > \epsilon$}{
+\item Let $\Delta z_{max}=0$\;
+\item $ kernel\_save(ZPrec,Z)$\;
+\item k=k+1\;
+\item $ kernel\_update(Z,P,Pu)$\;
+\item $kernel\_testConverge(\Delta z_{max},Z,ZPrec)$\;
+
+}
+\item Copy results from GPU memory to CPU memory\;
+\end{algorithm}
+\end{enumerate}
+~\\
+
+
+
+\section{The EA algorithm on Multiple GPUs}
+\label{sec4}
+\subsection{M-GPU : an OpenMP-CUDA approach}
+Our OpenMP-CUDA implementation of EA algorithm is based on the hybrid OpenMP and CUDA programming model. It works as follows.
+Based on the metadata, a shared memory is used to make data evenly shared among OpenMP threads. The shared data are the solution vector $Z$, the polynomial to solve $P$, and the error vector $\Delta z$. Let (T\_omp) the number of OpenMP threads be equal to the number of GPUs, each OpenMP thread binds to one GPU, and controls a part of the shared memory, that is a part of the vector Z , that is $(n/num\_gpu)$ roots where $n$ is the polynomial's degree and $num\_gpu$ the total number of available GPUs. Each OpenMP thread copies its data from host memory to GPU’s device memory. Then every GPU will have a grid of computation organized according to the device performance and the size of data on which it runs the computation kernels. %In principle a grid is set by two parameter DimGrid, the number of block per grid, DimBloc: the number of threads per block. The following schema shows the architecture of (CUDA,OpenMP).
+
+%\begin{figure}[htbp]
+%\centering
+ % \includegraphics[angle=-90,width=0.5\textwidth]{OpenMP-CUDA}
+%\caption{The OpenMP-CUDA architecture}
+%\label{fig:03}
+%\end{figure}
+%Each thread OpenMP compute the kernels on GPUs,than after each iteration they copy out the data from GPU memory to CPU shared memory. The kernels are re-runs is up to the roots converge sufficiently. Here are below the corresponding algorithm:
+
+$num\_gpus$ OpenMP threads are created using \verb=omp_set_num_threads();=function (step $3$, Algorithm \ref{alg2-cuda-openmp}), the shared memory is created using \verb=#pragma omp parallel shared()= OpenMP function (line $5$, Algorithm\ref{alg2-cuda-openmp}), then each OpenMP thread allocates memory and copies initial data from CPU memory to GPU global memory, executes the kernels on GPU, but computes only his portion of roots indicated with variable \textit{index} initialized in (line 5, Algorithm \ref{alg2-cuda-openmp}), used as input data in the $kernel\_update$ (line 10, Algorithm \ref{alg2-cuda-openmp}). After each iteration, all OpenMP threads synchronize using \verb=#pragma omp barrier;= to gather all the correct values of $\Delta z$, thus allowing the computation the maximum stop condition on vector $\Delta z$ (line 12, Algorithm \ref{alg2-cuda-openmp}). Finally, threads copy the results from GPU memories to CPU memory. The OpenMP threads execute kernels until the roots sufficiently converge.
+\begin{enumerate}
+\begin{algorithm}[htpb]
+\label{alg2-cuda-openmp}
+%\LinesNumbered
+\caption{CUDA-OpenMP 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 degree), $\Delta z$ ( Vector of errors for stop condition), $num_gpus$ (number of OpenMP threads/ Number of GPUs), $Size$ (number of roots)}
+
+\KwOut {$Z$ ( Root's vector), $ZPrec$ (Previous root's vector)}
+
+\BlankLine
+
+\item Initialization of P\;
+\item Initialization of Pu\;
+\item Initialization of the solution vector $Z^{0}$\;
+\verb=omp_set_num_threads(num_gpus);=
+\verb=#pragma omp parallel shared(Z,$\Delta$ z,P);=
+\verb=cudaGetDevice(gpu_id);=
+\item Allocate and copy initial data from CPU memory to the 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 $kernel\_testConverge(\Delta z[gpu\_id],Z,ZPrec)$\;
+%\verb=#pragma omp barrier;=
+\item error= Max($\Delta z$)\;
+}
+
+\item Copy results from GPU memories to CPU memory\;
+\end{algorithm}
+\end{enumerate}
+~\\
+
+
+
+\subsection{Multi-GPU : an MPI-CUDA approach}
+%\begin{figure}[htbp]
+%\centering
+ % \includegraphics[angle=-90,width=0.2\textwidth]{MPI-CUDA}
+%\caption{The MPI-CUDA architecture }
+%\label{fig:03}
+%\end{figure}
+Our parallel implementation of EA to find root of polynomials using a CUDA-MPI approach is a data parallel approach. It splits input data of the polynomial to solve among MPI processes. In Algorithm \ref{alg2-cuda-mpi}, input data are the polynomial to solve $P$, the solution vector $Z$, the previous solution vector $ZPrev$, and the value of errors of stop condition $\Delta z$. Let $p$ denote the number of MPI processes on and $n$ the degree of the polynomial to be solved. The algorithm performs a simple data partitioning by creating $p$ portions, of at most $\lceil n/p \rceil$ roots to find per MPI process, for each $Z$ and $ZPrec$. Consequently, each MPI process of rank $k$ will have its own solution vector $Z_{k}$ and $ZPrec$, the error related to the stop condition $\Delta z_{k}$, enabling each MPI process to compute $\lceil n/p \rceil$ roots.
+
+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}
+~\\
+
+\section{Experiments}
+\label{sec5}
+We study two categories of polynomials: sparse polynomials and full polynomials.\\
+{\it A sparse polynomial} is a polynomial for which only some coefficients are not null. In this paper, we consider sparse polynomials for which the roots are distributed on 2 distinct circles:
+\begin{equation}
+ \forall \alpha_{1} \alpha_{2} \in C,\forall n_{1},n_{2} \in N^{*}; P(z)= (z^{n_{1}}-\alpha_{1})(z^{n_{2}}-\alpha_{2})
+\end{equation}\noindent
+{\it A full polynomial} is, in contrast, a polynomial for which all the coefficients are not null. A full polynomial is defined by:
+%%\begin{equation}
+ %%\forall \alpha_{i} \in C,\forall n_{i}\in N^{*}; P(z)= \sum^{n}_{i=1}(z^{n^{i}}.a_{i})
+%%\end{equation}
+
+\begin{equation}
+ {\Large \forall a_{i} \in C, i\in N; p(x)=\sum^{n}_{i=0} a_{i}.x^{i}}
+\end{equation}
+For our tests, a CPU Intel(R) Xeon(R) CPU E5620@2.40GHz and a GPU K40 (with 6 Go of ram) are used.
+%SIDER : Une meilleure présentation de l'architecture est à faire ici.
+
+In order to evaluate both the M-GPU and Multi-GPU approaches, we performed a set of experiments on a single GPU and multiple GPUs using OpenMP or MPI by EA algorithm, for both sparse and full polynomials of different sizes.
+All experimental results obtained are made in double precision data whereas the convergence threshold of the EA method is set to $10^{-7}$.
+%Since we were more interested in the comparison of the
+%performance behaviors of Ehrlich-Aberth and Durand-Kerner methods on
+%CPUs versus on GPUs.
+The initialization values of the vector solution
+of the methods are given in %Section~\ref{sec:vec_initialization}.
+
+\subsection{Evaluating the M-GPU (CUDA-OpenMP) approach}
+
+We report here the results of the set of experiments with the M-GPU approach for full and sparse polynomials of different degrees, and we compare it with a Single GPU execution.
+\subsubsection{Execution time of the EA method for solving sparse polynomials on multiple GPUs using the M-GPU approach}
+
+In this experiments we report the execution time of the EA algorithm, on single GPU and Multi-GPU with (2,3,4) GPUs, for different sparse polynomial degrees ranging from 100,000 to 1,400,000.
