X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/kahina_paper1.git/blobdiff_plain/1c032eb8576d3d2ae1b6f4a297c7437af7c26c02..d2eaa5b6b9f4ba47dbfa2aeaae51427945accf72:/paper.tex?ds=sidebyside

diff --git a/paper.tex b/paper.tex
index 9e84151..04e1316 100644
--- a/paper.tex
+++ b/paper.tex
@@ -147,9 +147,12 @@ approximation of all the roots, starting with the Durand-Kerner (DK)
 method:
 %%\begin{center}
 \begin{equation}
- Z_{i}=Z_{i}-\frac{P(Z_{i})}{\prod_{i\neq j}(z_{i}-z_{j})}
+ Z_i^{k+1}=Z_{i}^k-\frac{P(Z_i^k)}{\prod_{i\neq j}(Z_i^k-Z_j^k)}
 \end{equation}
 %%\end{center}
+where $Z_i^k$ is the $i^{th}$ root of the polynomial $P$ at the
+iteration $k$.
+
 
 This formula is mentioned for the first time by
 Weiestrass~\cite{Weierstrass03} as part of the fundamental theorem
@@ -161,9 +164,11 @@ in the following form by Ehrlich~\cite{Ehrlich67} and
 Aberth~\cite{Aberth73} uses a different iteration formula given as fellows :
 %%\begin{center}
 \begin{equation}
- Z_{i}=Z_{i}-\frac{1}{{\frac {P'(Z_{i})} {P(Z_{i})}}-{\sum_{i\neq j}(z_{i}-z_{j})}}.
+ Z_i^{k+1}=Z_i^k-\frac{1}{{\frac {P'(Z_i^k)} {P(Z_i^k)}}-{\sum_{i\neq j}(Z_i^k-Z_j^k)}}.
 \end{equation}
 %%\end{center}
+where $P'(Z)$ is the polynomial derivative of $P$ evaluated in the
+point $Z$.
 
 Aberth, Ehrlich and Farmer-Loizou~\cite{Loizon83} have proved that
 the Ehrlich-Aberth method (EA) has a cubic order of convergence for simple roots whereas the Durand-Kerner has a quadratic order of convergence.
@@ -712,9 +717,20 @@ This figure show the execution time of the both algorithm EA and DK with sparse
 \label{fig:01}
 \end{figure}
 
-\bibliography{mybibfile}
+\subsubsection{The execution time of Ehrlich-Aberth algorithm on OpenMP(1 core, 4 cores) vs. on a Tesla GPU}
+
+\begin{figure}[H]
+\centering
+  \includegraphics[width=0.8\textwidth]{figures/openMP-GPU}
+\caption{The execution time of Ehrlich-Aberth algorithm on OpenMP(1core, 4cores) and GPU(Tesla k40)}
+\label{fig:01}
+\end{figure}
+
+
 
 
 \section{Conclusion and perspective}
 
+\bibliography{mybibfile}
+
 \end{document}
