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diff --git a/paper.tex b/paper.tex
index 76b626e2b46a919c49ea888d44f23733381c28dd..276c50a9b3eb46e64687e1ceacbf25a4beeca9b6 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -308,7 +308,7 @@ but we prefer the latter one because we can use it to improve the
 Ehrlich-Aberth method and find the roots of very high degrees polynomials. More
 details are given in Section ~\ref{sec2}.
 \subsection{Convergence Condition}
 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}
 
 \begin{equation}
 \label{eq:Aberth-Conv-Cond}


@@ -319,7 +319,8 @@ The convergence condition determines the termination of the algorithm. It consis
 
 \section{Improving the Ehrlich-Aberth Method for high degree polynomials with exp.log formulation}
 \label{sec2}
 
 \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
 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