\end{equation}
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}
H(i,z^{k+1})=\frac{p(z^{(k)}_{i})}{p'(z^{(k)}_{i})-p(z^{(k)}_{i})\sum^{n}_{j=1 j\neq i}\frac{1}{z^{(k)}_{i}-z^{(k)}_{j}}}, i=1,...,n.
\end{equation}
-With the the Gauss-seidel iteration, we have:
+With the Gauss-seidel iteration, we have:
\begin{equation}
\label{eq:Aberth-H-GS}
H(i,z^{k+1})=\frac{p(z^{(k)}_{i})}{p'(z^{(k)}_{i})-p(z^{(k)}_{i})(\sum^{i-1}_{j=1}\frac{1}{z^{(k)}_{i}-z^{(k+1)}_{j}}+\sum^{n}_{j=i+1}\frac{1}{z^{(k)}_{i}-z^{(k)}_{j}})}, i=1,...,n.
\subsection{Comparative study}
-We initially carried out the convergence of Aberth algorithm with various sizes of polynomial, in second we evaluate the influence of the size of the threads per block....
-
-\subsubsection{Aberth algorithm on CPU and GPU}
-
-%\begin{table}[!ht]
-% \centering
-% \begin{tabular} {|R{2cm}|L{2.5cm}|L{2.5cm}|L{1.5cm}|L{1.5cm}|}
-% \hline Polynomial's degrees & $T_{exe}$ on CPU & $T_{exe}$ on GPU & CPU iteration & GPU iteration\\
-% \hline 5000 & 1.90 & 0.40 & 18 & 17\\
-% \hline 10000 & 172.723 & 0.59 & 21 & 24\\
-% \hline 20000 & 172.723 & 1.52 & 21 & 25\\
-% \hline 30000 & 172.723 & 2.77 & 21 & 33\\
-% \hline 50000 & 172.723 & 3.92 & 21 & 18\\
-% \hline 500000 & $>$1h & 497.109 & & 24\\
-% \hline 1000000 & $>$1h & 1,524.51& & 24\\
-% \hline
-% \end{tabular}
-% \caption{the convergence of Aberth algorithm}
-% \label{tab:theConvergenceOfAberthAlgorithm}
-%\end{table}
-
+In this section, we discuss the performance Ehrlish-Aberth method of root finding polynomials implemented on CPUs and on GPUs.
+
+We performed a set of experiments on the sequential and the parallel algorithms, for both sparse and full polynomials and different sizes. We took into account the execution time,the polynomial size and the number of threads per block performed by sum or each experiment on CPUs and on GPUs.
+
+All experimental results obtained from the simulations are made in double precision data, for a convergence tolerance of the methods set to $10^{-7}$. Since we were more interested in the comparison of the performance behaviors of Ehrlish-Aberth and Durand-Kerner methods on CPUs versus on GPUs. The initialization values of the vector solution of the Ehrlich-Aberth method are given in section 2.2.
+\subsubsection{The execution time in seconds of Ehrlisch-Aberth algorithm on CPU core vs. on a Tesla GPU}
+
+
\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{figures/Compar_EA_algorithm_CPU_GPU}
-\caption{Aberth algorithm on CPU and GPU}
+\caption{The execution time in seconds of Ehrlisch-Aberth algorithm on CPU core vs. on a Tesla GPU}
\label{fig:01}
\end{figure}
+Figure 1 %%show a comparison of execution time between the parallel and sequential version of the Ehrlisch-Aberth algorithm with sparse polynomial exceed 100000,
+We report the execution times of the Ehrlisch-Aberth method implemented on one core of the Quad-Core Xeon E5620 CPU and those of the same methods implemented on one Nvidia Tesla K40c GPU, with sparse polynomial degrees ranging from 100,000 to 1,000,000. We can see that the methods implemented on the GPU are faster than those implemented on the CPU. This is due to the GPU ability to compute the data-parallel functions faster than its CPU counterpart. However, the execution time for the sequential implementation exceed 16,000 s for 450,000 degrees polynomials, in counterpart the GPU implementation for the same polynomials need only 350 s, more than again, with 1,000,000 polynomials degrees GPU implementation not reach 2,300 s degrees. While CPU implementation need more than 10 hours. We can also notice that the GPU implementation are almost 47 faster then those implementation on the CPU. Furthermore, we verify that the number of iterations is the same. This reduction of time allows us to compute roots of polynomial of more important degree at the same time than with a CPU.
+
-\subsubsection{The impact of the thread's number into the convergence of Aberth algorithm}
-
-%\begin{table}[!h]
-% \centering
-% \begin{tabular} {|R{2.5cm}|L{2.5cm}|L{2.5cm}|}
-% \hline Thread's numbers & Execution time &Number of iteration\\
-% \hline 1024 & 523 & 27\\
-% \hline 512 & 449.426 & 24\\
-% \hline 256 & 440.805 & 24\\
-% \hline 128 & 456.175 & 22\\
-% \hline 64 & 472.862 & 23\\
-% \hline 32 & 830.152 & 24\\
-% \hline 8 & 2632.78 & 23 \\
-% \hline
-% \end{tabular}
-% \caption{The impact of the thread's number into the convergence of Aberth algorithm}
-% \label{tab:Theimpactofthethread'snumberintotheconvergenceofAberthalgorithm}
-%
-%\end{table}
+\subsubsection{Influence of the number of threads on the execution times of different polynomials (sparse and full)}
+It is also interesting to see the influence of the number of threads per block on the execution time. For that, we notice that the maximum number of threads per block for the Nvidia Tesla K40c GPU is 1024, so we varied the number of threads per block from 8 to 1024. We took into account the execution time for both sparse and full polynomials of size 50000 and 500000 degrees.
\begin{figure}[H]
\centering
\label{fig:01}
\end{figure}
+The figure 2 show that, the best execution time for both sparse and full polynomial are given while the threads number varies between 64 and 256 threads per bloc. We notice that with small polynomials the number of threads per block is 64, Whereas, the large polynomials the number of threads per block is 256. However,In the following experiments we specify that the number of thread by block is 256.
+
\subsubsection{The impact of exp-log solution to compute very high degrees of polynomial}
+
+In this experiment we report the performance of log.exp solution describe in ~\ref{sec2} to compute very high degrees polynomials.
\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{figures/log_exp}
\label{fig:01}
\end{figure}
+The figure 3, show a comparison between the execution time of the Ehrlisch-Aberth algorithm applying log-exp solution and the execution time of the Ehrlisch-Aberth algorithm without applying log-exp solution, with full polynomials degrees. We can see that the execution time for the both algorithms are the same while the polynomials degrees are less than 4500.After,we show clearly that the Ehrlisch-Aberth algorithm without applying log.exp stop to converge consequently,can not solving polynomial exceed 4500, in counterpart, applying log.exp solution the Ehrlisch-Aberth algorithm can solving very high and large full polynomial exceed 500,000 degrees.
+
+in fact, when the modulus of the roots are up than R given in (~\ref{eq:radiusR}),this exceed the the limited number in the mantissa of floating points representations who justify the divergence of the Ehrlisch-Aberth algorithm without log.exp. However, applying log.exp solution given in equation~\ref{alg3-update} took into account the limit of floating using the iterative function given in~\ref{eq:Aberth-H-GS}.
+
+
+
+%we report the performances of the exp.log for the Ehrlisch-Aberth algorithm for solving very high degree of polynomial.
+
+
\subsubsection{A comparative study between Aberth and Durand-kerner algorithm}
\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{figures/EA_DK}
-\caption{Ehrlisch-Aberth and Durand-Kerner algorithm on GPU}
+\caption{The execution time of Ehrlisch-Aberth versus Durand-Kerner algorithm}
\label{fig:01}
\end{figure}
+\begin{figure}[H]
+\centering
+ \includegraphics[width=0.8\textwidth]{figures/EA_DK_nbr}
+\caption{The iteration number of Ehrlisch-Aberth versus Durand-Kerner algorithm}
+\label{fig:01}
+\end{figure}
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