+%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[...,...], 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 $ε$
+
+
+\subsection{EA parallel implementation on CUDA}
+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}
+~\\
+
+
+
