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@@ -218,14 +218,14 @@ In this paper, we focus on the implementation of the Ehrlich-Aberth
 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
 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
 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.
 
 this paper and gives some hints for future research directions in this
 topic.
 


@@ -268,7 +268,7 @@ The initialization of a polynomial p(z) is done by setting each of the $n$ compl
 
 
 \subsection{Vector $z^{(0)}$ Initialization}
 
 
 \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.
 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.


@@ -306,7 +306,7 @@ EA2: z^{k+1}=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}
 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
 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:
 
 \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:
 


@@ -548,11 +548,11 @@ With the Gauss-Seidel iteration, we have:
 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 %%
 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 $z^{k+1}_{i}$.
 
 \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.
 
 
 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.
 
 


@@ -627,17 +627,18 @@ The last kernel checks the convergence of the roots after each update
 of $Z^{(k)}$, according to formula Eq.~\ref{eq:Aberth-Conv-Cond}. We used the functions of the CUBLAS Library (CUDA Basic Linear Algebra Subroutines) to implement this kernel. 
 
 The kernel terminates its computations when all the roots have
 of $Z^{(k)}$, according to formula Eq.~\ref{eq:Aberth-Conv-Cond}. We used the functions of the CUBLAS Library (CUDA Basic Linear Algebra Subroutines) to implement this kernel. 
 
 The kernel terminates its computations when all the roots have

-converged. Many important remarks should be noticed. First, as blocks
-of threads are scheduled automatically by the GPU, we have absolutely
-no control on the order of the blocks. Consequently, our algorithm is
-executed more or less in an asynchronous iterations way, where blocks
-of roots are updated in a non deterministic way. As the Durand-Kerner
-method has been proved to convergence with asynchronous iterations, we
-think it is similar with the Ehrlich-Aberth method, but we did not try
-to prove this in that paper. Another consequence of that, is that
-several executions of our algorithm with the same polynomials do no
-give necessarily the same result with the same number of iterations
-(even if the variation is not very significant).
+converged. It should be noticed that, as blocks of threads are
+scheduled automatically by the GPU, we have absolutely no control on
+the order of the blocks. Consequently, our algorithm is executed more
+or less in an asynchronous iteration model, where blocks of roots are
+updated in a non deterministic way. As the Durand-Kerner method has
+been proved to converge with asynchronous iterations, we think it is
+similar with the Ehrlich-Aberth method, but we did not try to prove
+this in that paper. Another consequence of that, is that several
+executions of our algorithm with the same polynomial do no give
+necessarily the same result (but roots have the same accuracy) and the
+same number of iterations (even if the variation is not very
+significant).

 
 
 
 
 
 


@@ -647,14 +648,14 @@ give necessarily the same result with the same number of iterations
 \section{Experimental study}
 \label{sec6}
 %\subsection{Definition of the used polynomials }
 \section{Experimental study}
 \label{sec6}
 %\subsection{Definition of the used polynomials }

-We study two categories of polynomials : the sparse polynomials and the full polynomials.
-\paragraph{A sparse polynomial}:is a polynomial for which only some coefficients are not null. We use in the following polynomial for which the roots are distributed on 2 distinct circles :
+We study two categories of polynomials: sparse polynomials and the full polynomials.\\
+{\it A sparse polynomial} is a polynomial for which only some
+coefficients are not null. In this paper, we consider sparse polynomials for which the roots are distributed on 2 distinct circles:

 \begin{equation}
        \forall \alpha_{1} \alpha_{2} \in C,\forall n_{1},n_{2} \in N^{*}; P(z)= (z^{n_{1}}-\alpha_{1})(z^{n_{2}}-\alpha_{2})
 \begin{equation}
        \forall \alpha_{1} \alpha_{2} \in C,\forall n_{1},n_{2} \in N^{*}; P(z)= (z^{n_{1}}-\alpha_{1})(z^{n_{2}}-\alpha_{2})

-\end{equation}
-
-
-\paragraph{A full polynomial}:is in contrast, a polynomial for which all the coefficients are not null. the second form used to obtain a full polynomial is:
+\end{equation}\noindent
+{\it A full polynomial} is, in contrast, a polynomial for which
+all the coefficients are not null. A full polynomial is defined by:

 %%\begin{equation}
        %%\forall \alpha_{i} \in C,\forall n_{i}\in N^{*}; P(z)= \sum^{n}_{i=1}(z^{n^{i}}.a_{i})
 %%\end{equation}
 %%\begin{equation}
        %%\forall \alpha_{i} \in C,\forall n_{i}\in N^{*}; P(z)= \sum^{n}_{i=1}(z^{n^{i}}.a_{i})
 %%\end{equation}


@@ -662,45 +663,56 @@ We study two categories of polynomials : the sparse polynomials and the full pol
 \begin{equation}
      {\Large \forall a_{i} \in C, i\in N;  p(x)=\sum^{n}_{i=0} a_{i}.x^{i}} 
 \end{equation}
 \begin{equation}
      {\Large \forall a_{i} \in C, i\in N;  p(x)=\sum^{n}_{i=0} a_{i}.x^{i}} 
 \end{equation}

-With this form, we can have until \textit{n} non zero terms whereas the sparse ones have just two non zero terms.
+%With this form, we can have until \textit{n} non zero terms whereas the sparse ones have just two non zero terms.

 
 %\subsection{The study condition} 
 
 %\subsection{The study condition} 

-The our experiences results concern two parameters which are
-the polynomial degree and the execution time of our program
-to converge on the solution. The polynomial degree allows us
-to validate that our algorithm is powerful with high degree
-polynomials. The execution time remains the
-element-key which justifies our work of parallelization.
-       For our tests we used a CPU Intel(R) Xeon(R) CPU
-E5620@2.40GHz and a GPU K40 (with 6 Go of ram).
+%Two parameters are studied are
+%the polynomial degree and the execution time of our program
+%to converge on the solution. The polynomial degree allows us
+%to validate that our algorithm is powerful with high degree
+%polynomials. The execution time remains the
+%element-key which justifies our work of parallelization.
+For our tests, a CPU Intel(R) Xeon(R) CPU
+E5620@2.40GHz and a GPU K40 (with 6 Go of ram) is used. 

 
 
 %\subsection{Comparative study}
 
 
 %\subsection{Comparative study}

-In this section, we discuss the performance Ehrlich-Aberth method  of root finding polynomials implemented on CPUs and on GPUs.
+%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 time, 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 CPUs and on GPUs.

 
 

-All experimental results obtained from the simulations are made in double precision data, for a convergence tolerance of the methods set to $10^{-7}$. Since we were more interested in the comparison of the performance behaviors of Ehrlich-Aberth and Durand-Kerner methods on CPUs versus on GPUs. The initialization values of the vector solution of the Ehrlich-Aberth method are given in section 2.2. 
-\subsection{The execution time in seconds of Ehrlich-Aberth algorithm on CPU OpenMP (1 core, 4 cores) vs. on a Tesla GPU}
+All experimental results obtained from the simulations are made in
+double precision data, the convergence threshold of the methods is set
+to $10^{-7}$.
+%Since we were more interested in the comparison of the
+%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~\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}
 \centering
   \includegraphics[width=0.8\textwidth]{figures/openMP-GPU}

-\caption{The execution time in seconds of Ehrlich-Aberth algorithm on CPU OpenMP (1 core, 4 cores) vs. on a Tesla GPU}
+\caption{Comparison of execution times  of the Ehrlich-Aberth method
+  on a CPU with OpenMP (1 core, 4 cores) and on a Tesla GPU}

 \label{fig:01}
 \end{figure}
 \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, 
-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 2500 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.
+%%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 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.
 
 \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. 
  
  %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.
 
 \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 50000 and 10 different polynomials of size 500000 degrees.
+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)}
 \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)}


@@ -712,7 +724,7 @@ The figure 2 show that, the best execution time for both sparse and full polynom
 \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.   
 \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.}
 \centering
   \includegraphics[width=0.8\textwidth]{figures/sparse_full_explog}
 \caption{The impact of exp-log solution to compute very high degrees of  polynomial.}


@@ -728,7 +740,7 @@ in fact, when the modulus of the roots are up than \textit{R} given in ~\ref{R},
 \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{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}
 \centering
   \includegraphics[width=0.8\textwidth]{figures/EA_DK}
 \caption{The execution time of Ehrlich-Aberth versus Durand-Kerner algorithm on GPU}


@@ -737,23 +749,29 @@ In this part, we are interesting to compare the simultaneous methods, Ehrlich-Ab
 
 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 $10^{-5}$.
 
 
 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}
 \label{fig:05}
 \end{figure}
 
 \centering
   \includegraphics[width=0.8\textwidth]{figures/EA_DK_nbr}
 \caption{The iteration number of Ehrlich-Aberth versus Durand-Kerner algorithm}
 \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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