%%\usepackage[utf8]{inputenc}
%%\usepackage[T1]{fontenc}
%%\usepackage[french]{babel}
+\usepackage{float}
\usepackage{amsmath,amsfonts,amssymb}
\usepackage[ruled,vlined]{algorithm2e}
%\usepackage[french,boxed,linesnumbered]{algorithm2e}
\sum_{k\neq j}^{n}\frac{1}{z_{k}-z_{j}}\right)\right).
\end{equation}
-This solution is applied when it is necessary ??? When ??? (SIDER)
+This solution is applied when the root except the circle unit, represented by the radius $R$ evaluated as:
+
+$$R = \exp( \log(DBL\_MAX) / (2*n) )$$ where $DBL\_MAX$ stands for the maximum representable double value.
\section{The implementation of simultaneous methods in a parallel computer}
\label{secStateofArt}
Moreover, they have fast rate of convergence (quadratic for the
Durand-Kerner and cubic for the Ehrlisch-Aberth). Various parallel
algorithms reported for these methods can be found
-in~\cite{Cosnard90, Freeman89,Freemanall90,,Jana99,Janall99}.
+in~\cite{Cosnard90, Freeman89,Freemanall90,Jana99,Janall99}.
Freeman and Bane~\cite{Freemanall90} presented two parallel
algorithms on a local memory MIMD computer with the compute-to
communication time ratio O(n). However, their algorithms require
\section {A CUDA parallel Ehrlisch-Aberth method}
+In the following, we describe the parallel implementation of Ehrlisch-Aberth method on GPU
+for solving high degree polynomials. First, the hardware and software of the GPUs are presented. Then, a CUDA parallel Ehrlisch-Aberth method are presented.
\subsection{Background on the GPU architecture}
A GPU is viewed as an accelerator for the data-parallel and
\begin{equation}
\forall \alpha_{1} \alpha_{2} \in C,\forall n_{1},n_{2} \in N^{*}; P(z)= (z^{n_{1}}-\alpha_{1})(z^{n_{2}}-\alpha_{2})
\end{equation}
-
This form makes it possible to associate roots having two
different modules and thus to work on a polynomial constitute
of four non zero terms.
% \label{tab:theConvergenceOfAberthAlgorithm}
%\end{table}
-\begin{figure}[htbp]
+\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{figures/Compar_EA_algorithm_CPU_GPU}
\caption{Aberth algorithm on CPU and GPU}
%\end{table}
-\begin{figure}[htbp]
+\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{figures/influence_nb_threads}
\caption{Influence of the number of threads on the execution times of different polynomials (sparse and full)}
\label{fig:01}
\end{figure}
-\begin{figure}[htbp]
+\subsubsection{The impact of exp-log solution to compute very high degrees of polynomial}
+\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{figures/log_exp}
\caption{The impact of exp-log solution to compute very high degrees of polynomial.}
\end{figure}
\subsubsection{A comparative study between Aberth and Durand-kerner algorithm}
-\begin{table}[htbp]
- \centering
- \begin{tabular} {|R{2cm}|L{2.5cm}|L{2.5cm}|L{1.5cm}|L{1.5cm}|}
- \hline Polynomial's degrees & Aberth $T_{exe}$ & D-Kerner $T_{exe}$ & Aberth iteration & D-Kerner iteration\\
- \hline 5000 & 0.40 & 3.42 & 17 & 138 \\
- \hline 50000 & 3.92 & 385.266 & 17 & 823\\
- \hline 500000 & 497.109 & 4677.36 & 24 & 214\\
- \hline
- \end{tabular}
- \caption{Aberth algorithm compare to Durand-Kerner algorithm}
- \label{tab:AberthAlgorithCompareToDurandKernerAlgorithm}
-\end{table}
+
+
+\begin{figure}[H]
+\centering
+ \includegraphics[width=0.8\textwidth]{figures/EA_DK}
+\caption{Ehrlisch-Aberth and Durand-Kerner algorithm on GPU}
+\label{fig:01}
+\end{figure}
\bibliography{mybibfile}
