X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/kahina_paper1.git/blobdiff_plain/5870d4e72db0c954d03231f1d807de7aee47ff74..e74bf528b696246b88bb8a5542efc542fd09788a:/paper.tex?ds=inline

diff --git a/paper.tex b/paper.tex
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--- a/paper.tex
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@@ -232,7 +232,8 @@ topic.
 \section{The Sequential Ehrlich-Aberth method}
 \label{sec1}
 A cubically convergent iteration method for finding zeros of
-polynomials was proposed by O. Aberth~\cite{Aberth73}. In the fellowing we present the main stages of the running of the Ehrlich-Aberth method.
+polynomials was proposed by O. Aberth~\cite{Aberth73}. In the
+following we present the main stages of our implementation the Ehrlich-Aberth method.
 %The Aberth method is a purely algebraic derivation. 
 %To illustrate the derivation, we let $w_{i}(z)$ be the product of linear factors 
 
@@ -268,7 +269,7 @@ The initialization of a polynomial p(z) is done by setting each of the $n$ compl
 
 \subsection{Vector $z^{(0)}$ Initialization}
 
-Like for any iterative method, we need to choose $n$ initial guess points $z^{(0)}_{i}, i = 1, . . . , n.$
+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 $n$
@@ -302,7 +303,10 @@ Here we give a second form of the iterative function used by Ehrlich-Aberth meth
 EA2: z^{k+1}=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=0,. . . .,n
 \end{equation}
-we notice that the function iterative in Eq.~\ref{Eq:Hi} it the same those presented in Eq.~\ref{Eq:EA}, but we prefer used the last one seen the advantage of its use to improve the Ehrlich-Aberth method and resolve very high degrees polynomials. More detail in the section ~\ref{sec2}.    
+It can be noticed that this equation is equivalent to Eq.~\ref{Eq:EA},
+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}
 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: