this paper and gives some hints for future research directions in this
topic.
-\section{The Sequential Ehrlich-Aberth method}
+\section{Ehrlich-Aberth method}
\label{sec1}
A cubically convergent iteration method for finding zeros of
polynomials was proposed by O. Aberth~\cite{Aberth73}. In the
\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}
-\subsection{Sequential Ehrlich-Aberth algorithm}
-The main steps of Ehrlich-Aberth method are shown in Algorithm.~\ref{alg1-seq} :
-%\LinesNumbered
-\begin{algorithm}[H]
-\label{alg1-seq}
+%% \subsection{Sequential Ehrlich-Aberth algorithm}
+%% The main steps of Ehrlich-Aberth method are shown in Algorithm.~\ref{alg1-seq} :
+%% %\LinesNumbered
+%% \begin{algorithm}[H]
+%% \label{alg1-seq}
-\caption{A sequential algorithm to find roots with the Ehrlich-Aberth method}
+%% \caption{A sequential 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) $\Delta z_{max}$ (maximum value
- of stop condition), k (number of iteration), n (Polynomial's degrees)}
-\KwOut {$Z$ (The solution root's vector), $ZPrec$ (the previous solution root's vector)}
+%% \KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (error tolerance
+%% threshold), $P$ (Polynomial to solve),$Pu$ (the derivative of P) $\Delta z_{max}$ (maximum value
+%% of stop condition), k (number of iteration), n (Polynomial's degrees)}
+%% \KwOut {$Z$ (The solution root's vector), $ZPrec$ (the previous solution root's vector)}
-\BlankLine
+%% \BlankLine
-Initialization of $P$\;
-Initialization of $Pu$\;
-Initialization of the solution vector $Z^{0}$\;
-$\Delta z_{max}=0$\;
- k=0\;
+%% Initialization of $P$\;
+%% Initialization of $Pu$\;
+%% Initialization of the solution vector $Z^{0}$\;
+%% $\Delta z_{max}=0$\;
+%% k=0\;
-\While {$\Delta z_{max} > \varepsilon$}{
- Let $\Delta z_{max}=0$\;
-\For{$j \gets 0 $ \KwTo $n$}{
-$ZPrec\left[j\right]=Z\left[j\right]$;// save Z at the iteration k.\
+%% \While {$\Delta z_{max} > \varepsilon$}{
+%% Let $\Delta z_{max}=0$\;
+%% \For{$j \gets 0 $ \KwTo $n$}{
+%% $ZPrec\left[j\right]=Z\left[j\right]$;// save Z at the iteration k.\
-$Z\left[j\right]=H\left(j, Z, P, Pu\right)$;//update Z with the iterative function.\
-}
-k=k+1\;
+%% $Z\left[j\right]=H\left(j, Z, P, Pu\right)$;//update Z with the iterative function.\
+%% }
+%% k=k+1\;
-\For{$i \gets 0 $ \KwTo $n-1$}{
-$c= testConverge(\Delta z_{max},ZPrec\left[j\right],Z\left[j\right])$\;
-\If{$c > \Delta z_{max}$ }{
-$\Delta z_{max}$=c\;}
-}
+%% \For{$i \gets 0 $ \KwTo $n-1$}{
+%% $c= testConverge(\Delta z_{max},ZPrec\left[j\right],Z\left[j\right])$\;
+%% \If{$c > \Delta z_{max}$ }{
+%% $\Delta z_{max}$=c\;}
+%% }
-}
-\end{algorithm}
+%% }
+%% \end{algorithm}
-~\\
-In this sequential algorithm, one CPU thread executes all the steps. Let us look to the $3^{rd}$ step i.e. the execution of the iterative function, 2 sub-steps are needed. The first sub-step \textit{save}s the solution vector of the previous iteration, the second sub-step \textit{update}s or computes the new values of the roots vector.
-There exists two ways to execute the iterative function that we call a Jacobi one and a Gauss-Seidel one. With the Jacobi iteration, at iteration $k+1$ we need all the previous values $z^{(k)}_{i}$ to compute the new values $z^{(k+1)}_{i}$, that is :
+%% ~\\
+%% In this sequential algorithm, one CPU thread executes all the steps. Let us look to the $3^{rd}$ step i.e. the execution of the iterative function, 2 sub-steps are needed. The first sub-step \textit{save}s the solution vector of the previous iteration, the second sub-step \textit{update}s or computes the new values of the roots vector.
+
+\subsection{Parallel implementation with CUDA }
+
+In order to implement the Ehrlich-Aberth method in CUDA, it is
+possible to use the Jacobi scheme or the Gauss Seidel one. With the
+Jacobi iteration, at iteration $k+1$ we need all the previous values
+$z^{(k)}_{i}$ to compute the new values $z^{(k+1)}_{i}$, that is :
\begin{equation}
EAJ: z^{k+1}_{i}=\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.
\textit{Z}, we expect the Gauss-Seidel iteration to converge more
quickly because, just as any Jacobi algorithm (for solving linear systems of equations), it uses the most fresh computed roots $z^{k+1}_{i}$.
-The $4^{th}$ step of the algorithm checks the convergence condition using Eq.~\ref{eq:Aberth-Conv-Cond}.
-Both steps 3 and 4 use 1 thread to compute all the $n$ roots on CPU, which is very harmful for performance in case of the large degree polynomials.
+%The $4^{th}$ step of the algorithm checks the convergence condition using Eq.~\ref{eq:Aberth-Conv-Cond}.
+%Both steps 3 and 4 use 1 thread to compute all the $n$ roots on CPU, which is very harmful for performance in case of the large degree polynomials.
-\subsection{Parallel implementation with CUDA }
-On the CPU, both steps 3 and 4 contain the loop \verb=for= and a single thread executes all the instructions in the loop $n$ times. In this subsection, we explain how the GPU architecture can compute this loop and reduce the execution time.
-In the GPU, the scheduler assigns the execution of this loop to a
-group of threads organised as a grid of blocks with block containing a
-number of threads. All threads within a block are executed
-concurrently in parallel. The instructions run on the GPU are grouped
-in special function called kernels. With CUDA, a programmer must
-describe the kernel execution context: the size of the Grid, the number of blocks and the number of threads per block.
+
+%On the CPU, both steps 3 and 4 contain the loop \verb=for= and a single thread executes all the instructions in the loop $n$ times. In this subsection, we explain how the GPU architecture can compute this loop and reduce the execution time.
+%In the GPU, the scheduler assigns the execution of this loop to a
+%group of threads organised as a grid of blocks with block containing a
+%number of threads. All threads within a block are executed
+%concurrently in parallel. The instructions run on the GPU are grouped
+%in special function called kernels. With CUDA, a programmer must
+%describe the kernel execution context: the size of the Grid, the number of blocks and the number of threads per block.
%In CUDA programming, all the instructions of the \verb=for= loop are executed by the GPU as a kernel. A kernel is a function written in CUDA and defined by the \verb=__global__= qualifier added before a usual \verb=C= function, which instructs the compiler to generate appropriate code to pass it to the CUDA runtime in order to be executed on the GPU.
%\LinesNumbered
\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)}
+\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)}
\KwOut {$Z$ (The solution root's vector), $ZPrec$ (the previous solution root's vector)}
\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}
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
