new

diff --git a/paper.tex b/paper.tex
index 93a0366dee45f3631c166795417e7c079cdcfd09..81a5e46f2c5f5f8112d19dcd9f873c60bdd2f9a9 100644 (file)
--- 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}
 
 {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}
 
 \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: 
 
  
 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: