\end{equation}
-\section{Improving the Ehrlich-Aberth Method for high degree polynomials with exp.log formulation}
+\section{Improving the Ehrlich-Aberth Method for high degree polynomials with exp-log formulation}
\label{sec2}
With high degree polynomial, the Ehrlich-Aberth method implementation,
as well as the Durand-Kerner implement, suffers from overflow problems. This
%First, performances of the Ehrlich-Aberth method of root finding polynomials
%implemented on CPUs and on GPUs are studied.
-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 times, the polynomial size and the number of threads per block performed by sum or each experiment 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 times, the polynomial size and the number of threads per block performed by sum or each experiment on CPU and on GPU.
All experimental results obtained from the simulations are made in
double precision data, the convergence threshold of the methods is set
\subsection{Comparison of execution times of the Ehrlich-Aberth method
on a CPU with OpenMP (1 core and 4 cores) vs. on a Tesla GPU}
-
-
\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\textwidth]{figures/openMP-GPU}
\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 1024. 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
The figure 2 show that, the best execution time for both sparse and full polynomial are given when the threads number varies between 64 and 256 threads per bloc. We notice that with small polynomials the best number of threads per block is 64, Whereas, the large polynomials the best number of threads per block is 256. However,In the following experiments we specify that the number of thread by block is 256.
-\subsection{The impact of exp.log solution to compute very high degrees of polynomial}
+\subsection{Influence of exp-log solution to compute high degree polynomials}
-In this experiment we report the performance of exp-log solution described in Section~\ref{sec2} to compute very high degrees polynomials.
+In this experiment we report the performance of the exp-log solution described in Section~\ref{sec2} to compute very high degrees polynomials.
\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}
Figure~\ref{fig:03} shows a comparison between the execution time of
-the Ehrlich-Aberth algorithm using the exp.log solution and the
-execution time of the Ehrlich-Aberth algorithm without this solution,
+the Ehrlich-Aberth method using the exp-log solution and the
+execution time of the Ehrlich-Aberth method without this solution,
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 4000 and sparse polynomials less than 150,000. We
-also clearly show that the classical version (without log.exp) of
+degrees less than 4,000 and sparse polynomials less than 150,000. We
+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 log.exp solution can solve very
+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 .
\subsection{Comparison of the Durand-Kerner and the Ehrlich-Aberth methods}
In this part, we compare the Durand-Kerner and the Ehrlich-Aberth
-methods on GPU. We took into account the execution time, the number of iteration and the polynomial's size for the both sparse and full polynomials.
+methods on GPU. We took into account the execution times, the number of iterations and the polynomials size for the both sparse and full polynomials.
\begin{figure}[htbp]
\centering
sparse polynomial degrees ranging from 1,000 to 1,000,000. We can see
that the Ehrlich-Aberth algorithm is faster than Durand-Kerner
algorithm, with an average of 25 times faster. Then, when degrees of
-polynomial exceed 500000 the execution time with EA is of the order
-100 whereas DK passes in the order 1000.
+polynomial exceed 500,000 the execution times with DK are very long.
%with double precision not exceed $10^{-5}$.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\textwidth]{figures/EA_DK_nbr}
-\caption{The iteration number of Ehrlich-Aberth versus Durand-Kerner algorithm}
+\caption{The number of iterations to converge for the Ehrlich-Aberth
+ and the Durand-Kerner methods}
\label{fig:05}
\end{figure}
-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.
+Figure~\ref{fig:05} 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 the derivative of P (the polynomial to
+resolve) in the iterative function (Eq.~\ref{Eq:Hi}) executed by EA
+allows the algorithm to converge more quickly. In counterpart, the
+DK operator (Eq.~\ref{DK}) needs low operation, consequently low
+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 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.
+In this paper we have presented the parallel implementation
+Ehrlich-Aberth method on GPU for the problem of finding roots
+polynomial. Moreover, we have improved the classical Ehrlich-Aberth
+method which suffers from overflow problems, the exp-log solution
+applied to the iterative function allows to solve high degree
+polynomials.
+
+We have performed many experiments with the Ehrlich-Aberth method in
+GPU. These experiments highlight that this method is very efficient in
+GPU compared to all the other implementations. The improvement with
+the exponential logarithm solution allows us to solve sparse and full
+high degree polynomials up to 1,000,000 degree. Hence, it may be
+possible to consider to use polynomial root finding methods in other
+numerical applications on GPU.
+
+
+In future works, we plan to investigate the possibility of using
+several multiple GPUs simultaneously, either with multi-GPU machine or
+with cluster of GPUs.
