X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/kahina_paper2.git/blobdiff_plain/4b3ba00ad3edb6f2c001e4a4e842ed4a2a8bb566..87920647de834836776958ae2479e1e91567ab76:/paper.tex?ds=sidebyside

diff --git a/paper.tex b/paper.tex
index 489b65b..a790787 100644
--- 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}
 
-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 diffÃ©rence 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 mÃ©thode de Newton et de l'opÃ©rateur de  Weiestrass?}
+\LZK{Elle est oÃ¹ la 4th step??}
+\LZK{Conclusion: MÃ©thode mal prÃ©sentÃ©e 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
-than a fixed value $\xi$.
+than a fixed value $\epsilon$.
 
 \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}
+
+\LZK{On ne dit pas plutÃ´t "the relative errors" Ã  la place de "root modules"? Raph nous confirmera quelle critÃ¨re d'arrÃªt a utilisÃ©.}
+
 \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
