Most of the numerical methods that deal with the polynomial root-finding problem are simultaneous methods, \textit{i.e.} the iterative methods to find simultaneous approximations of the $n$ polynomial roots. These methods start from the initial approximations of all $n$ polynomial roots and give a sequence of approximations that converge to the roots of the polynomial. Two examples of well-known simultaneous methods for root-finding problem of polynomials are Durand-Kerner method~\cite{Durand60,Kerner66} and Ehrlich-Aberth method~\cite{Ehrlich67,Aberth73}.
-The convergence time of simultaneous methods drastically increases with the increasing of the polynomial's degree. The great challenge with simultaneous methods is to parallelize them and to improve their convergence. Many authors have proposed parallel simultaneous methods~\cite{Freeman89,Loizou83,Freemanall90,cs01:nj,Couturier02}, using several paradigms of parallelization (synchronous or asynchronous computations, mechanism of shared or distributed memory, etc). However, they have treated only polynomials not exceeding degrees of 20,000.
+The convergence time of simultaneous methods drastically increases with the increasing of the polynomial's degree. The great challenge with simultaneous methods is to parallelize them and to improve their convergence. Many authors have proposed parallel simultaneous methods~\cite{Freeman89,Loizou83,Freemanall90,bini96,cs01:nj,Couturier02}, using several paradigms of parallelization (synchronous or asynchronous computations, mechanism of shared or distributed memory, etc). However, they have treated only polynomials not exceeding degrees of 20,000.
%The main problem of the simultaneous methods is that the necessary
%time needed for the convergence increases with the increasing of the
multiplications and divisions with additions and
subtractions. Consequently, computations manipulate lower values in
absolute values~\cite{Karimall98}. In practice, the exponential and
-logarithm mode is used a root excepts the circle unit, \LZK{Je n'ai pas compris cette phrase!} represented by the radius $R$ evaluated in C language as :
+logarithm mode is used when a root is outisde the circle unit represented by the radius $R$ evaluated in C language with:
\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.
+where \verb=DBL_MAX= stands for the maximum representable
+\verb=double= value and $n$ is the degree of the polynimal.
\subsection{The Ehrlich-Aberth parallel implementation on CUDA}
