Ehrlich-Aberth method and find the roots of very high degrees polynomials. More
details are given in Section ~\ref{sec2}.
\subsection{Convergence Condition}
-The convergence condition determines the termination of the algorithm. It consists in stopping from running the iterative function when the roots are sufficiently stable. We consider that the method converges sufficiently when:
+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}
\section{Improving the Ehrlich-Aberth Method for high degree polynomials with exp.log formulation}
\label{sec2}
-The Ehrlich-Aberth method implementation suffers of overflow problems. This
+With high degree polynomial, the Ehrlich-Aberth method implementation,
+as well as the Durand-Kerner implement, suffers from overflow problems. This
situation occurs, for instance, in the case where a polynomial
having positive coefficients and a large degree is computed at a
point $\xi$ where $|\xi| > 1$, where $|x|$ stands for the modolus of a complex $x$. Indeed, the limited number in the
manipulate lower absolute values and the roots for large polynomial's degrees can be looked for successfully~\cite{Karimall98}.
Applying this solution for the Ehrlich-Aberth method we obtain the
-iteration function with logarithm:
+iteration function with exponential and logarithm:
%%$$ \exp \bigl( \ln(p(z)_{k})-ln(\ln(p(z)_{k}^{'}))- \ln(1- \exp(\ln(p(z)_{k})-ln(\ln(p(z)_{k}^{'})+\ln\sum_{i\neq j}^{n}\frac{1}{z_{k}-z_{j}})$$
\begin{equation}
\label{Log_H2}
