method for high degree polynomials on GPU. We propose an adaptation of
the exponential logarithm in order to be able to solve sparse and full
polynomial of degree up to $1,000,000$. The paper is organized as
-follows. Initially, we recall the Ehrlich-Aberth method in Section
-\ref{sec1}. Improvements for the Ehrlich-Aberth method are proposed in
-Section \ref{sec2}. Related work to the implementation of simultaneous
-methods using a parallel approach is presented in Section
-\ref{secStateofArt}. In Section \ref{sec5} we propose a parallel
+follows. Initially, we recall the Ehrlich-Aberth method in
+Section~\ref{sec1}. Improvements for the Ehrlich-Aberth method are
+proposed in Section \ref{sec2}. Related work to the implementation of
+simultaneous methods using a parallel approach is presented in Section
+\ref{secStateofArt}. In Section~\ref{sec5} we propose a parallel
implementation of the Ehrlich-Aberth method on GPU and discuss
-it. Section \ref{sec6} presents and investigates our implementation
-and experimental study results. Finally, Section\ref{sec7} 6 concludes
+it. Section~\ref{sec6} presents and investigates our implementation
+and experimental study results. Finally, Section~\ref{sec7} 6 concludes
this paper and gives some hints for future research directions in this
topic.
\subsection{Vector $z^{(0)}$ Initialization}
-
+\label{sec:vec_initialization}
As for any iterative method, we need to choose $n$ initial guess points $z^{(0)}_{i}, i = 1, . . . , n.$
The initial guess is very important since the number of steps needed by the iterative method to reach
a given approximation strongly depends on it.
It can be noticed that this equation is equivalent to Eq.~\ref{Eq:EA},
but we prefer the latter one because we can use it to improve the
Ehrlich-Aberth method and find the roots of very high degrees polynomials. More
-details are given in Section ~\ref{sec2}.
+details are given in Section~\ref{sec2}.
\subsection{Convergence Condition}
The convergence condition determines the termination of the algorithm. It consists in stopping the iterative function when the roots are sufficiently stable. We consider that the method converges sufficiently when:
EAGS: z^{k+1}_{i}=\frac{p(z^{k}_{i})}{p'(z^{k}_{i})-p(z^{k}_{i})(\sum^{i-1}_{j=1}\frac{1}{z^{k}_{i}-z^{k+1}_{j}}+\sum^{n}_{j=i+1}\frac{1}{z^{k}_{i}-z^{k}_{j}})}, i=1,...,n.
\end{equation}
%%Here a finiched my revision %%
-Using Equation.~\ref{eq:Aberth-H-GS} to update the vector solution
+Using Eq.~\ref{eq:Aberth-H-GS} to update the vector solution
\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 Equation.~\ref{eq:Aberth-Conv-Cond}.
+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.
%performance behaviors of Ehrlich-Aberth and Durand-Kerner methods on
%CPUs versus on GPUs.
The initialization values of the vector solution
-of the methods are given in section 2.2.
+of the methods are given in Section~\ref{sec:vec_initialization}.
+
\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}[H]
+\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\textwidth]{figures/openMP-GPU}
\caption{Comparison of execution times of the Ehrlich-Aberth method
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.
-\begin{figure}[H]
+\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\textwidth]{figures/influence_nb_threads}
\caption{Influence of the number of threads on the execution times of different polynomials (sparse and full)}
\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.
-\begin{figure}[H]
+\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.}
\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.
-\begin{figure}[H]
+\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}
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 $10^{-5}$.
-\begin{figure}[H]
+\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\textwidth]{figures/EA_DK_nbr}
\caption{The iteration number of Ehrlich-Aberth versus Durand-Kerner algorithm}
