relecture section III A

diff --git a/paper.tex b/paper.tex
index 489b65bd076395d470d24a63b729a90d12c21282..a79078793cff82c66646820763a5abd84bd76fdc 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -663,24 +663,24 @@ z^{k+1}_{i}=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=1,\ldots,n
 \end{equation}
 
 {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=1,\ldots,n
 \end{equation}
 

-This method contains 4 steps. The first step consists of the initial
-approximations of all the roots of the polynomial. The second step
-initializes the solution vector $Z$ using the Guggenheimer
-method~\cite{Gugg86} to ensure the distinction of the initial vector
-roots. In step 3, the iterative function based on the Newton's
-method~\cite{newt70} and Weiestrass operator~\cite{Weierstrass03} is
-applied. With this step the computation of roots will converge,
-provided that all roots are different.
+This method contains 4 steps. The first step consists of the initial approximations of all the roots of the polynomial.\LZK{Pas compris??}
+The second step initializes the solution vector $Z$ using the Guggenheimer method~\cite{Gugg86} to ensure the distinction of the initial vector roots.\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"?} 
+In step 3, the iterative function based on the Newton's method~\cite{newt70} and Weiestrass operator~\cite{Weierstrass03} is applied. With this step the computation of roots will converge, provided that all roots are different.\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?}
+\LZK{Elle est où la 4th step??}
+\LZK{Conclusion: Méthode mal présentée et j'ai presque rien compris!}

 
 
 In order to stop the iterative function, a stop condition is
 applied. This condition checks that all the root modules are lower
 
 
 In order to stop the iterative function, a stop condition is
 applied. This condition checks that all the root modules are lower

-than a fixed value $\xi$.
+than a fixed value $\epsilon$.

 
 \begin{equation}
 \label{eq:Aberth-Conv-Cond}
 
 \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<\xi
+\forall i\in[1,n],~\vert\frac{z_i^k-z_i^{k-1}}{z_i^k}\vert<\epsilon

 \end{equation}
 \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.}
+

 \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
 \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