X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/kahina_paper1.git/blobdiff_plain/c42326ed2bebe65f1b51b355637cf4d89d70bbce..1df92e2629fef95f9b236c8d952d94c08f5f34a0:/paper.tex

diff --git a/paper.tex b/paper.tex
index a2102e7..1d70450 100644
--- a/paper.tex
+++ b/paper.tex
@@ -268,7 +268,7 @@ The initialization of a polynomial $p(z)$ is done by setting each of the $n$ com
 
 \subsection{Vector $Z^{(0)}$ Initialization}
 \label{sec:vec_initialization}
-As 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 $n$
@@ -299,7 +299,7 @@ Here we give a second form of the iterative function used by Ehrlich-Aberth meth
 
 \begin{equation}
 \label{Eq:Hi}
-EA2: z^{k+1}=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}
+EA2: 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=0,. . . .,n
 \end{equation}
 It can be noticed that this equation is equivalent to Eq.~\ref{Eq:EA},
@@ -321,8 +321,8 @@ With high degree polynomial, the Ehrlich-Aberth method implementation,
 as well as the Durand-Kerner implement, suffers from overflow problems. This
 situation occurs, for instance, in the case where a polynomial
 having positive coefficients and a large degree is computed at a
-point $\xi$ where $|\xi| > 1$, where $|x|$ stands for the modolus of a complex $x$. Indeed, the limited number in the
-mantissa of floating points representations makes the computation of p(z) wrong when z
+point $\xi$ where $|\xi| > 1$, where $|z|$ stands for the modolus of a complex $z$. Indeed, the limited number in the
+mantissa of floating points representations makes the computation of $p(z)$ wrong when z
 is large. For example $(10^{50}) +1+ (- 10^{50})$ will give the wrong result
 of $0$ instead of $1$. Consequently, we can not compute the roots
 for large degrees. This problem was early discussed in