+This method contains 4 steps. The first step consists in the
+initializing the polynomial.\LZK{Pas compris?? \RC{changé}}.
+The second step initializes the solution vector $Z$ using the
+Guggenheimer method~\cite{Gugg86} to ensure that initial roots are all
+distinct from each other. \LZK{Quelle est la différence entre la 1st
+ step et la 2nd step? Que veut dire " to ensure the distinction of
+ the initial vector roots"? \RC{reformulé}}
+In step 3, the iterative function based on the Newton's
+method~\cite{newt70} and Weiestrass operator~\cite{Weierstrass03} is
+applied. In our case, the Ehrlich-Aberth is applied as in (\ref{Eq:EA1}).
+Iterations of the EA method will converge to the roots of the
+considered polynomial.\LZK{On ne peut pas expliquer un peu plus
+ comment? Donner des formules comment elle se base sur la méthode de
+ Newton et de l'opérateur de Weiestrass? \RC{amélioré}}
+\LZK{Elle est où la 4th step??}
+\LZK{Conclusion: Méthode mal présentée et j'ai presque rien compris!
+ \RC{après} }
+
+
+In order to stop the iterative function, a stop condition is applied,
+this is the 4th step. This condition checks that all the root modules
+are lower than a fixed value $\epsilon$.
+
+\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<\epsilon
+\end{equation}
+
+\LZK{On ne dit pas plutôt "the relative errors" à la place de "root
+ modules"? Raph nous confirmera quelle critère d'arrêt a
+ utilisé. \RC{normalement c'est bon, l'erreur est calculée avec le
+ module de chaque racine}}
+
+\subsection{Improving Ehrlich-Aberth method}
+With high degree polynomials, the Ehrlich-Aberth method suffers from floating point overflows due to the mantissa of floating points representations. This induces errors in the computation of $p(z)$ when $z$ is large.
+
+%Experimentally, it is very difficult to solve polynomials with the Ehrlich-Aberth method and have roots which except the circle of unit, represented by the radius $r$ evaluated as:
+
+%\begin{equation}
+%\label{R.EL}
+%R = exp(log(DBL\_MAX)/(2*n) );
+%\end{equation}
+
+
+
+% where \verb=DBL_MAX= stands for the maximum representable \verb=double= value.
+
+In order to solve this problem, we propose to modify the iterative
+function by using the logarithm and the exponential of a complex and
+we propose a new version of the Ehrlich-Aberth method. This method
+allows us to exceed the computation of the polynomials of degree
+100,000 and to reach a degree up to more than 1,000,000. The reformulation of the iteration~(\ref{Eq:EA1}) of the Ehrlich-Aberth method with exponential and logarithm is defined as follows, for $i=1,\dots,n$:
+
+\begin{equation}
+\label{Log_H2}
+z^{k+1}_i = z_i^k - \exp(\ln(p(z_i^k)) - \ln(p'(z^k_i)) - \ln(1-Q(z^k_i))),
+\end{equation}
+
+where:
+
+\begin{equation}
+\label{Log_H1}
+Q(z^k_i) = \exp(\ln(p(z^k_i)) - \ln(p'(z^k_i)) + \ln(\sum_{i\neq j}^n\frac{1}{z^k_i-z^k_j})).
+\end{equation}
+
+
+%We propose to use the logarithm and the exponential of a complex in order to compute the power at a high exponent.
+Using the logarithm and the exponential operators, we can replace any
+multiplications and divisions with additions and
+subtractions. Consequently, computations manipulate lower values in absolute
+values~\cite{Karimall98}. \LZK{Je n'ai pas compris cette dernière
+ phrase? \RC{changé : on veut dire on manipule des valeurs plus petites en valeur absolues}}
+
+%This problem was discussed earlier in~\cite{Karimall98} for the Durand-Kerner method. The authors
+%propose to use the logarithm and the exponential of a complex in order to compute the power at a high exponent. Using the logarithm and the exponential operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations manipulate lower absolute values and the roots for large polynomial degrees can be looked for successfully~\cite{Karimall98}.
+
+\subsection{The Ehrlich-Aberth 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. \LZK{Répétition! Le même texte est déjà écrit comme
+ intro dans la section II. Sinon ici on parle seulement de
+ l'implémentation cuda sans mpi et openmp! \RC{Je suis d'accord à
+ revoir après, quand les 2 parties suivantes seront plus stables}}
