X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/kahina_paper1.git/blobdiff_plain/fdc11565a5688a74effe8d08b78b4a1853387f3a..69096b5c21f0e3416c85888d5652b8d11cf4ca4a:/paper.tex?ds=inline

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@@ -744,16 +744,21 @@ This figure show the execution time of the both algorithm EA and DK with sparse
 \label{fig:05}
 \end{figure}
 
-%\subsubsection{The execution time of Ehrlich-Aberth algorithm on OpenMP(1 core, 4 cores) vs. on a Tesla GPU}
-
+This figure show the evaluation of the number of iteration according to degree of polynomial from both EA and DK algorithms, we can see that the iteration number of DK is of order 100 while EA is of order 10. Indeed the computing of derivative of P (the polynomial to resolve) in the iterative function(Eq.~\ref{Eq:Hi}) executed by EA, offers him a possibility to converge more quickly. In counterpart the DK operator(Eq.~\ref{DK}) need low operation, consequently low execution time per iteration,but it need lot of iteration to converge.
 
+ \section{Conclusion and perspective}
+\label{sec7}
+In this paper we have presented the parallel implementation Ehrlich-Aberth method on GPU and on CPU (openMP) for the problem of finding roots polynomial. Moreover, we have improved the classical Ehrlich-Aberth method witch suffer of overflow problems, the exp.log solution applying to the iterative function to resolve high degree polynomial.
 
+Then, we have described the parallel implementation of the Ehrlich-Aberth algorithm on GPU. 
+We have performed some experiments on Ehrlich-Aberth algorithm in CPU and GPU from the both sparse and full polynomial. These experiments lead us to conclude that the iterative methods using data-parallel operations are more efficient on the GPU than on the CPU. Moreover, the experiment showed that Ehrlich-Aberth algorithm on GPU converge from the both sparse and full polynomials with precision of $10^{-7}$ and the execution time very faster than the CPU version. 
+The experiences showed that the improvement brought to Ehrlich-Aberth allows to resolve very large degree polynomial exceed 100,000.
+Finally, we have compared Ehrlich-Aberth algorithm to Durand-Kerner algorithm, we have conclude that Ehrlich-Aberth converges more quickly than Durand-Kerner in execution time, it is due in fact that Ehrlich-Aberth has cubic one convergence While Durand-Kerner is quadratic. In counterpart, the execution time per iteration are very low for Durand-Kerner algorithm compare to the Ehrlich-Aberth algorithm, consequently, it need lot of iterations to converge. We have to notice that Durand-Kerner does not converge for full polynomial which exceed 5000 degrees while Ehrlich-Aberth was able to solve full polynomial of degree 500,000.
 
+In future work, we plan to perform some experiments using several GPU with a cluster of GPU. So it is interesting to implement algorithms using at least two forms of parallelism on GPU and CPU.
 
 
-\section{Conclusion and perspective}
 
-\label{sec7}
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