+
+A cubically convergent iteration method to find zeros of
+polynomials was proposed by O. Aberth~\cite{Aberth73}. The
+Ehrlich-Aberth method contains 4 main steps, presented in what
+follows.
+
+%The Aberth method is a purely algebraic derivation.
+%To illustrate the derivation, we let $w_{i}(z)$ be the product of linear factors
+
+%\begin{equation}
+%w_{i}(z)=\prod_{j=1,j \neq i}^{n} (z-x_{j})
+%\end{equation}
+
+%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-spaced 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 is corresponding to the
+following equation~\ref{Eq:EA} which will enable the convergence towards
+polynomial solutions, 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:EA}
+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 $ε$
+
+
