\subsection{Influence of the number of threads on the execution times of different polynomials (sparse and full)}
To optimize the performances of an algorithm on a GPU, it is necessary to maximize the use of cores GPU (maximize the number of threads executed in parallel) and to optimize the use of the various memoirs GPU. In fact, it is interesting to see the influence of the number of threads per block on the execution time of Ehrlich-Aberth algorithm.
-For that, we notice that the maximum number of threads per block for the Nvidia Tesla K40 GPU is 1024, so we varied the number of threads per block from 8 to 1,024. We took into account the execution time for both sparse and full of 10 different polynomials of size 50,000 and 10 different polynomials of size 500,000 degrees.
+For that, we notice that the maximum number of threads per block for the Nvidia Tesla K40 GPU is 1,024, so we varied the number of threads per block from 8 to 1,024. We took into account the execution time for both sparse and full of 10 different polynomials of size 50,000 and 10 different polynomials of size 500,000 degrees.
\begin{figure}[htbp]
\centering
\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\textwidth]{figures/sparse_full_explog}
-\caption{The impact of exp.log solution to compute very high degrees of polynomial.}
+\caption{The impact of exp-log solution to compute very high degrees of polynomial.}
\label{fig:03}
\end{figure}
with full and sparse polynomials degrees. We can see that the
execution times for both algorithms are the same with full polynomials
degrees less than 4,000 and sparse polynomials less than 150,000. We
-also clearly show that the classical version (without log-exp) of
+also clearly show that the classical version (without exp-log) of
Ehrlich-Aberth algorithm do not converge after these degree with
sparse and full polynomials. In counterpart, the new version of
Ehrlich-Aberth algorithm with the exp-log solution can solve very
high degree polynomials.
-%in fact, when the modulus of the roots are up than \textit{R} given in ~\ref{R},this exceed the limited number in the mantissa of floating points representations and can not compute the iterative function given in ~\ref{eq:Aberth-H-GS} to obtain the root solution, who justify the divergence of the classical Ehrlich-Aberth algorithm. However, applying log.exp solution given in ~\ref{sec2} took into account the limit of floating using the iterative function in(Eq.~\ref{Log_H1},Eq.~\ref{Log_H2} and allows to solve a very large polynomials degrees .
+%in fact, when the modulus of the roots are up than \textit{R} given in ~\ref{R},this exceed the limited number in the mantissa of floating points representations and can not compute the iterative function given in ~\ref{eq:Aberth-H-GS} to obtain the root solution, who justify the divergence of the classical Ehrlich-Aberth algorithm. However, applying exp-log solution given in ~\ref{sec2} took into account the limit of floating using the iterative function in(Eq.~\ref{Log_H1},Eq.~\ref{Log_H2} and allows to solve a very large polynomials degrees .
\label{fig:04}
\end{figure}
-\begin{figure}[htbp]
-\centering
- \includegraphics[width=0.8\textwidth]{figures/EA_DK1}
-\caption{Execution times of the Durand-Kerner and the Ehrlich-Aberth methods on GPU}
-\label{fig:0}
-\end{figure}
-
Figure~\ref{fig:04} shows the execution times of both methods with
sparse polynomial degrees ranging from 1,000 to 1,000,000. We can see
that the Ehrlich-Aberth algorithm is faster than Durand-Kerner
execution time per iteration, but it needs more iterations to converge.
- \section{Conclusion and perspective}
+ \section{Conclusion and perspectives}
\label{sec7}
In this paper we have presented the parallel implementation
Ehrlich-Aberth method on GPU for the problem of finding roots
