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@@ -229,7 +229,7 @@ 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.
 

-\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
 \label{sec1}
 A cubically convergent iteration method for finding zeros of
 polynomials was proposed by O. Aberth~\cite{Aberth73}. In the


@@ -496,48 +496,54 @@ polynomials of 48000.
 
 
 
 
 
 

-\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.
 
 \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.


@@ -553,18 +559,18 @@ 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 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. 
 
 
 %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. 
 


@@ -575,7 +581,8 @@ Algorithm~\ref{alg2-cuda} shows a sketch of the Ehrlich-Aberth algorithm using C
 %\LinesNumbered
 \caption{CUDA Algorithm to find roots with the Ehrlich-Aberth method}
 
 %\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)}
 
 
 \KwOut {$Z$ (The solution root's vector), $ZPrec$ (the previous solution root's vector)}