\end{equation}
This solution is applied when the root except the circle unit, represented by the radius $R$ evaluated in C language as:
+
\begin{verbatim}
R = exp(log(DBL_MAX)/(2*n) );
\end{verbatim}
\caption{CUDA Algorithm to find roots with the Ehrlich-Aberth method}
\KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (error tolerance
- threshold), P(Polynomial to solve), Pu (the derivative of P), $n$
- (Polynomial's degrees), $\Delta z_{max}$ (maximum value of stop condition)}
+ threshold), P(Polynomial to solve), Pu (the derivative of P), $n$ (Polynomial's degrees),$\Delta z_{max}$ (maximum value of stop condition)}
\KwOut {$Z$ (The solution root's vector), $ZPrec$ (the previous solution root's vector)}
\end{algorithm}
~\\
-After the initialization step, all data of the root finding problem to be solved must be copied from the CPU memory to the GPU global memory, because the GPUs only access data already present in their memories. Next, all the data-parallel arithmetic operations inside the main loop \verb=(while(...))= are executed as kernels by the GPU. The first kernel named \textit{save} in line 6 of Algorithm~\ref{alg2-cuda} consists in saving the vector of polynomial's root found at the previous time-step in GPU memory, in order to check the convergence of the roots after each iteration (line 8, Algorithm~\ref{alg2-cuda}).
+After the initialization step, all data of the root finding problem to be solved must be copied from the CPU memory to the GPU global memory, because the GPUs only access data already present in their memories. Next, all the data-parallel arithmetic operations inside the main loop \verb=(do ... while(...))= are executed as kernels by the GPU. The first kernel named \textit{save} in line 6 of Algorithm~\ref{alg2-cuda} consists in saving the vector of polynomial's root found at the previous time-step in GPU memory, in order to check the convergence of the roots after each iteration (line 8, Algorithm~\ref{alg2-cuda}).
The second kernel executes the iterative function $H$ and updates
$d\_Z$, according to Algorithm~\ref{alg3-update}. We notice that the
\begin{algorithm}[H]
\label{alg3-update}
%\LinesNumbered
-\caption{Kernel\_update}
+\caption{Kernel update}
\eIf{$(\left|d\_Z\right|<= R)$}{
-$kernel\_update((d\_Z,d\_Pcoef,d\_Pdegres,d\_Pucoef,d\_Pudegres)$\;}
+$kernel\_update((d\_Z,d\_P,d\_Pu)$\;}
{
-$kernel\_update\_ExpoLog((d\_Z,d\_Pcoef,d\_Pdegres,d\_Pucoef,d\_Pudegres))$\;
+$kernel\_update\_ExpoLog((d\_Z,d\_P,\_Pu))$\;
}
\end{algorithm}
-The first form executes formula \ref{eq:SimplePolynome} if the modulus
+The first form executes formula the EA function Eq.~\ref{Eq:Hi} if the modulus
of the current complex is less than the a certain value called the
radius i.e. ($ |z^{k}_{i}|<= R$), else the kernel executes the EA.EL
function Eq.~\ref{Log_H2}
\label{fig:01}
\end{figure}
%%Figure 1 %%show a comparison of execution time between the parallel and sequential version of the Ehrlich-Aberth algorithm with sparse polynomial exceed 100000,
-In Figure~\ref{fig:01}, we report the execution times of the
-Ehrlich-Aberth method on one core of a Quad-Core Xeon E5620 CPU, on
-four cores on the same machine with \textit{OpenMP} and on a Nvidia
-Tesla K40c GPU. We chose different sparse polynomials with degrees
-ranging from 100,000 to 1,000,000. We can see that the implementation
-on the GPU is 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
-CPU (4 cores) implementation exceed 5,000 s for 250,000 degrees
-polynomials, in counterpart the GPU implementation for the same
-polynomials not reach 100 s, more than again, with an execution time
-under to 2,500 s CPU (4 cores) implementation can resolve With the GPU
-we can solve very high degrees polynomials very quickly up to degree
-of 1,000,000. We can also notice that the GPU implementation are
-almost 47 faster then those implementation on the CPU (4
-cores). However the CPU(4 cores) implementation are almost 4 faster
-then his implementation on CPU (1 core). Furthermore, the number of
-iterations and the convergence precision are similar with the CPU
-and the GPU implementation.
-
-%%This reduction
-%of time allows us to compute roots of polynomial of more important
-%degree at the same time than with a CPU.
+In Figure~\ref{fig:01}, we report respectively the execution time of the Ehrlich-Aberth method implemented initially on one core of the Quad-Core Xeon E5620 CPU than on four cores of the same machine with \textit{OpenMP} platform and the execution time of the same method implemented on one Nvidia Tesla K40c GPU, with sparse polynomial degrees ranging from 100,000 to 1,000,000. We can see that the method implemented on the GPU are faster than those implemented on the CPU (4 cores). This is due to the GPU ability to compute the data-parallel functions faster than its CPU counterpart. However, the execution time for the CPU(4 cores) implementation exceed 5,000 s for 250,000 degrees polynomials, in counterpart the GPU implementation for the same polynomials not reach 100 s, more than again, with an execution time under to 2,500 s CPU (4 cores) implementation can resolve polynomials degrees of only 200,000, whereas GPU implementation can resolve polynomials more than 1,000,000 degrees. We can also notice that the GPU implementation are almost 47 faster then those implementation on the CPU(4 cores). However the CPU(4 cores) implementation are almost 4 faster then his implementation on CPU (1 core). Furthermore, we verify that the number of iterations and the convergence precision is the same for the both CPU and GPU implementation. %This reduction of time allows us to compute roots of polynomial of more important degree at the same time than with a CPU.
%We notice that the convergence precision is a round $10^{-7}$ for the both implementation on CPU and GPU. Consequently, we can conclude that Ehrlich-Aberth on GPU are faster and accurately then CPU implementation.
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{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.
+In this experiment we report the performance of 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}
-The figure 3, show a comparison between the execution time of the Ehrlich-Aberth algorithm applying exp.log solution and the execution time of the Ehrlich-Aberth algorithm without applying exp.log solution, with full and sparse polynomials degrees. We can see that the execution time for the both algorithms are the same while the full polynomials degrees are less than 4000 and full polynomials are less than 150,000. After,we show clearly that the classical version of Ehrlich-Aberth algorithm (without applying log.exp) stop to converge and can not solving any polynomial sparse or full. In counterpart, the new version of Ehrlich-Aberth algorithm (applying log.exp solution) can solve very high and large full polynomial exceed 100,000 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 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 .
+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,
+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
+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
+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 .
+
-\subsection{A comparative study between Ehrlich-Aberth algorithm and Durand-kerner algorithm}
-In this part, we are interesting to compare the simultaneous methods, Ehrlich-Aberth and Durand-Kerner in parallel computer using GPU. We took into account the execution time, the number of iteration and the polynomial's size. for the both sparse and full polynomials.
+\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.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\textwidth]{figures/EA_DK}
-\caption{The execution time of Ehrlich-Aberth versus Durand-Kerner algorithm on GPU}
+\caption{Execution times of the Durand-Kerner and the Ehrlich-Aberth methods on GPU}
\label{fig:04}
\end{figure}
-This figure show the execution time of the both algorithm EA and DK with sparse polynomial degrees ranging from 1000 to 1000000. We can see that the Ehrlich-Aberth algorithm are faster than Durand-Kerner algorithm, with an average of 25 times as fast. Then, when degrees of polynomial exceed 500000 the execution time with EA is of the order 100 whereas DK passes in the order 1000. %with double precision not exceed $10^{-5}$.
+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
+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.
+
+%with double precision not exceed $10^{-5}$.
\begin{figure}[htbp]
\centering
