From 2bf774646b9314d29be9cbe5d5e6771c391ae7c7 Mon Sep 17 00:00:00 2001
From: couturie <couturie@extinction>
Date: Thu, 14 Jan 2016 17:09:53 +0100
Subject: [PATCH] new

---
 paper.tex | 20 +++++++++++++++-----
 1 file changed, 15 insertions(+), 5 deletions(-)

diff --git a/paper.tex b/paper.tex
index 93a0366..81a5e46 100644
--- a/paper.tex
+++ b/paper.tex
@@ -661,19 +661,29 @@ EA: 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,. . . .,n
 \end{equation}
 
-contain 4 steps, start from the initial approximations of all the roots of the polynomial,the second step initialize the solution vector $Z$ using the Guggenheimer~\cite{Gugg86} method to assure the distinction of the initial vector roots,
+This methods 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.
 
- than in step 3 we apply the the iterative function based on the Newton's method~\cite{newt70} and Weiestrass operator~\cite{Weierstrass03}, wich will make it possible to converge to the roots solution, provided that all the root are different.
 
-
- At the end of each application of the iterative function, a stop condition is verified consists in stopping the iterative process when the whole of the modules of the roots are lower than a fixed value $\xi$. 
+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 $\xi$.
 
 \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
 \end{equation}
 \subsection{Improving Ehrlich-Aberth method}
-With high degree polynomial, the Ehrlich-Aberth method suffer from overflow because the limited number in the mantissa of floating points representations, which makes the computation of $p(z)$ wrong when z is large.
+With high degree polynomials, the Ehrlich-Aberth method suffers from
+floating point overflows due to the mantissa of floating points
+representations. This induces errors in the computation of $p(z)$ when
+$z$ is large.
  
 Experimentally, it is very difficult to solve polynomials with Ehrlich-Aberth method and have roots which except the circle of unit, represented by the radius $r$ evaluated as: 
 
-- 
2.39.5