new

[kahina_paper1.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 384ff8bd6b4f41018c378b2ac329975295029649..788db9ac129fd34557039e46f743d4b9f398221c 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -151,7 +151,7 @@ method:
  DK: z_i^{k+1}=z_{i}^{k}-\frac{P(z_i^{k})}{\prod_{i\neq j}(z_i^{k}-z_j^{k})},   i = 1, . . . , n,
 \end{equation}
 %%\end{center}
  DK: z_i^{k+1}=z_{i}^{k}-\frac{P(z_i^{k})}{\prod_{i\neq j}(z_i^{k}-z_j^{k})},   i = 1, . . . , n,
 \end{equation}
 %%\end{center}

-where $z_i^k$ is the $i^{th}$ root of the polynomial $P$ at the
+where $z_i^k$ is the $i^{th}$ root of the polynomial $p$ at the

 iteration $k$.
 
 
 iteration $k$.
 
 


@@ -169,7 +169,7 @@ Aberth~\cite{Aberth73} uses a different iteration formula given as:
  EA: z_i^{k+1}=z_i^{k}-\frac{1}{{\frac {P'(z_i^{k})} {P(z_i^{k})}}-{\sum_{i\neq j}\frac{1}{(z_i^{k}-z_j^{k})}}}, i = 1, . . . , n,
 \end{equation}
 %%\end{center}
  EA: z_i^{k+1}=z_i^{k}-\frac{1}{{\frac {P'(z_i^{k})} {P(z_i^{k})}}-{\sum_{i\neq j}\frac{1}{(z_i^{k}-z_j^{k})}}}, i = 1, . . . , n,
 \end{equation}
 %%\end{center}

-where $P'(z)$ is the polynomial derivative of $P$ evaluated in the
+where $p'(z)$ is the polynomial derivative of $p$ evaluated in the

 point $z$.
 
 Aberth, Ehrlich and Farmer-Loizou~\cite{Loizou83} have proved that
 point $z$.
 
 Aberth, Ehrlich and Farmer-Loizou~\cite{Loizou83} have proved that


@@ -191,13 +191,13 @@ chain, for polynomials of degree up to 8. The third method often
 diverges, but the first two methods have speed-up equal to 5.5. Later,
 Freeman and Bane~\cite{Freemanall90}  considered asynchronous
 algorithms, in which each processor continues to update its
 diverges, but the first two methods have speed-up equal to 5.5. Later,
 Freeman and Bane~\cite{Freemanall90}  considered asynchronous
 algorithms, in which each processor continues to update its

-approximations even though the latest values of other $z_i((k))$
+approximations even though the latest values of other $z_i^{k}$

 have not been received from the other processors, in contrast with synchronous algorithms where it would wait those values before making a new iteration.
 Couturier and al.~\cite{Raphaelall01} proposed two methods of parallelization for
 a shared memory architecture and for distributed memory one. They were able to
 have not been received from the other processors, in contrast with synchronous algorithms where it would wait those values before making a new iteration.
 Couturier and al.~\cite{Raphaelall01} proposed two methods of parallelization for
 a shared memory architecture and for distributed memory one. They were able to

-compute the roots of sparse polynomials of degree 10000 in 430 seconds with only 8
+compute the roots of sparse polynomials of degree 10,000 in 430 seconds with only 8

 personal computers and 2 communications per iteration. Comparing to the sequential implementation
 personal computers and 2 communications per iteration. Comparing to the sequential implementation

-where it takes up to 3300 seconds to obtain the same results, the authors show an interesting speedup.
+where it takes up to 3,300 seconds to obtain the same results, the authors show an interesting speedup.

 
 Very few works had been performed since this last work until the appearing of
 the Compute Unified Device Architecture (CUDA)~\cite{CUDA10}, a
 
 Very few works had been performed since this last work until the appearing of
 the Compute Unified Device Architecture (CUDA)~\cite{CUDA10}, a


@@ -211,7 +211,7 @@ Ghidouche and al~\cite{Kahinall14} proposed an implementation of the
 Durand-Kerner method on GPU. Their main
 result showed that a parallel CUDA implementation is about 10 times faster than
 the sequential implementation on a single CPU for  sparse
 Durand-Kerner method on GPU. Their main
 result showed that a parallel CUDA implementation is about 10 times faster than
 the sequential implementation on a single CPU for  sparse

-polynomials of degree 48000. 
+polynomials of degree 48,000. 

 
 
 In this paper, we focus on the implementation of the Ehrlich-Aberth
 
 
 In this paper, we focus on the implementation of the Ehrlich-Aberth


@@ -225,15 +225,14 @@ 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
 \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
+and experimental study results. Finally, Section~\ref{sec7} concludes

 this paper and gives some hints for future research directions in this
 topic.
 
 \section{Ehrlich-Aberth method}
 \label{sec1}
 A cubically convergent iteration method for finding zeros of
 this paper and gives some hints for future research directions in this
 topic.
 
 \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
-following we present the main stages of our implementation the Ehrlich-Aberth method.
+polynomials was proposed by O. Aberth~\cite{Aberth73}. The Ehrlich-Aberth method contain 4 main steps, presented in the following. 

 %The Aberth method is a purely algebraic derivation. 
 %To illustrate the derivation, we let $w_{i}(z)$ be the product of linear factors 
 
 %The Aberth method is a purely algebraic derivation. 
 %To illustrate the derivation, we let $w_{i}(z)$ be the product of linear factors 
 


@@ -259,7 +258,7 @@ following we present the main stages of our implementation the Ehrlich-Aberth me
 
 
 \subsection{Polynomials Initialization}
 
 
 \subsection{Polynomials Initialization}

-The initialization of a polynomial p(z) is done by setting each of the $n$ complex coefficients $a_{i}$:
+The initialization of a polynomial $p(z)$ is done by setting each of the $n$ complex coefficients $a_{i}$:

 
 \begin{equation}
 \label{eq:SimplePolynome}
 
 \begin{equation}
 \label{eq:SimplePolynome}


@@ -267,9 +266,9 @@ The initialization of a polynomial p(z) is done by setting each of the $n$ compl
 \end{equation}
 
 
 \end{equation}
 
 

-\subsection{Vector $z^{(0)}$ Initialization}
+\subsection{Vector $Z^{(0)}$ Initialization}

 \label{sec:vec_initialization}
 \label{sec:vec_initialization}

-As for any iterative method, we need to choose $n$ initial guess points $z^{(0)}_{i}, i = 1, . . . , n.$
+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.
 In~\cite{Aberth73} the Ehrlich-Aberth iteration is started by selecting $n$
 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.
 In~\cite{Aberth73} the Ehrlich-Aberth iteration is started by selecting $n$


@@ -300,7 +299,7 @@ Here we give a second form of the iterative function used by Ehrlich-Aberth meth
 
 \begin{equation}
 \label{Eq:Hi}
 
 \begin{equation}
 \label{Eq:Hi}

-EA2: z^{k+1}=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}
+EA2: z^{k+1}_{i}=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}

 {1-\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}\sum_{j=1,j\neq i}^{j=n}{\frac{1}{(z_{i}^{k}-z_{j}^{k})}}}, i=0,. . . .,n
 \end{equation}
 It can be noticed that this equation is equivalent to Eq.~\ref{Eq:EA},
 {1-\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}\sum_{j=1,j\neq i}^{j=n}{\frac{1}{(z_{i}^{k}-z_{j}^{k})}}}, i=0,. . . .,n
 \end{equation}
 It can be noticed that this equation is equivalent to Eq.~\ref{Eq:EA},


@@ -316,14 +315,14 @@ The convergence condition determines the termination of the algorithm. It consis
 \end{equation}
 
 
 \end{equation}
 
 

-\section{Improving the Ehrlich-Aberth Method for high degree polynomials with exp.log formulation}
+\section{Improving the Ehrlich-Aberth Method for high degree polynomials with exp-log formulation}

 \label{sec2}
 With high degree polynomial, the Ehrlich-Aberth method implementation,
 as well as the Durand-Kerner implement, suffers from overflow problems. This
 situation occurs, for instance, in the case where a polynomial
 having positive coefficients and a large degree is computed at a
 \label{sec2}
 With high degree polynomial, the Ehrlich-Aberth method implementation,
 as well as the Durand-Kerner implement, suffers from overflow problems. This
 situation occurs, for instance, in the case where a polynomial
 having positive coefficients and a large degree is computed at a

-point $\xi$ where $|\xi| > 1$, where $|x|$ stands for the modolus of a complex $x$. Indeed, the limited number in the
-mantissa of floating points representations makes the computation of p(z) wrong when z
+point $\xi$ where $|\xi| > 1$, where $|z|$ stands for the modolus of a complex $z$. Indeed, the limited number in the
+mantissa of floating points representations makes the computation of $p(z)$ wrong when z

 is large. For example $(10^{50}) +1+ (- 10^{50})$ will give the wrong result
 of $0$ instead of $1$. Consequently, we can not compute the roots
 for large degrees. This problem was early discussed in
 is large. For example $(10^{50}) +1+ (- 10^{50})$ will give the wrong result
 of $0$ instead of $1$. Consequently, we can not compute the roots
 for large degrees. This problem was early discussed in


@@ -351,7 +350,7 @@ iteration function with exponential and logarithm:
 %%$$ \exp \bigl(  \ln(p(z)_{k})-ln(\ln(p(z)_{k}^{'}))- \ln(1- \exp(\ln(p(z)_{k})-ln(\ln(p(z)_{k}^{'})+\ln\sum_{i\neq j}^{n}\frac{1}{z_{k}-z_{j}})$$
 \begin{equation}
 \label{Log_H2}
 %%$$ \exp \bigl(  \ln(p(z)_{k})-ln(\ln(p(z)_{k}^{'}))- \ln(1- \exp(\ln(p(z)_{k})-ln(\ln(p(z)_{k}^{'})+\ln\sum_{i\neq j}^{n}\frac{1}{z_{k}-z_{j}})$$
 \begin{equation}
 \label{Log_H2}

-EA.EL: z^{k+1}=z_{i}^{k}-\exp \left(\ln \left(
+EA.EL: z^{k+1}_{i}=z_{i}^{k}-\exp \left(\ln \left(

 p(z_{i}^{k})\right)-\ln\left(p'(z^{k}_{i})\right)- \ln
 \left(1-Q(z^{k}_{i})\right)\right),
 \end{equation}
 p(z_{i}^{k})\right)-\ln\left(p'(z^{k}_{i})\right)- \ln
 \left(1-Q(z^{k}_{i})\right)\right),
 \end{equation}


@@ -361,10 +360,11 @@ where:
 \begin{equation}
 \label{Log_H1}
 Q(z^{k}_{i})=\exp\left( \ln (p(z^{k}_{i}))-\ln(p'(z^{k}_{i}))+\ln \left(
 \begin{equation}
 \label{Log_H1}
 Q(z^{k}_{i})=\exp\left( \ln (p(z^{k}_{i}))-\ln(p'(z^{k}_{i}))+\ln \left(

-\sum_{k\neq j}^{n}\frac{1}{z^{k}_{i}-z^{k}_{j}}\right)\right).
+\sum_{i\neq j}^{n}\frac{1}{z^{k}_{i}-z^{k}_{j}}\right)\right)i=1,...,n,

 \end{equation}
 
 This solution is applied when the root except the circle unit, represented by the radius $R$ evaluated in C language as:
 \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} 
 \begin{verbatim}
 R = exp(log(DBL_MAX)/(2*n) );
 \end{verbatim} 


@@ -582,51 +582,58 @@ Algorithm~\ref{alg2-cuda} shows a sketch of the Ehrlich-Aberth algorithm using C
 \caption{CUDA Algorithm to find roots with the Ehrlich-Aberth method}
 
 \KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (error tolerance
 \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 (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 {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}

 
 \BlankLine
 
 Initialization of the of P\;
 Initialization of the of Pu\;
 Initialization of the solution vector $Z^{0}$\;
 
 \BlankLine
 
 Initialization of the of P\;
 Initialization of the of Pu\;
 Initialization of the solution vector $Z^{0}$\;

-Allocate and copy initial data to the GPU global memory ($d\_Z,d\_ZPrec,d\_P,d\_Pu$)\;
+Allocate and copy initial data to the GPU global memory\;

 k=0\;
 \While {$\Delta z_{max} > \epsilon$}{
  Let $\Delta z_{max}=0$\;
 k=0\;
 \While {$\Delta z_{max} > \epsilon$}{
  Let $\Delta z_{max}=0$\;

-$ kernel\_save(d\_ZPrec,d\_Z)$\;
+$ kernel\_save(ZPrec,Z)$\;

 k=k+1\;
 k=k+1\;

-$ kernel\_update(d\_Z,d\_P,d\_Pu)$\;
-$kernel\_testConverge(\Delta z_{max},d\_Z,d\_ZPrec)$\;
+$ kernel\_update(Z,P,Pu)$\;
+$kernel\_testConverge(\Delta z_{max},Z,ZPrec)$\;

 
 }
 Copy results from GPU memory to CPU memory\;
 \end{algorithm}
 ~\\ 
 
 
 }
 Copy results from GPU memory to CPU memory\;
 \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
+must be copied from the CPU memory to the GPU global memory. 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}).

 
 The second kernel executes the iterative function $H$ and updates
 
 The second kernel executes the iterative function $H$ and updates

-$d\_Z$, according to Algorithm~\ref{alg3-update}. We notice that the
-update kernel is called in two forms, separated with the value of
+Z, according to Algorithm~\ref{alg3-update}. We notice that the
+update kernel is called in two forms, according to the value

 \emph{R} which determines the radius beyond which we apply the
 exponential logarithm algorithm. 
 
 \begin{algorithm}[H]
 \label{alg3-update}
 %\LinesNumbered
 \emph{R} which determines the radius beyond which we apply the
 exponential logarithm algorithm. 
 
 \begin{algorithm}[H]
 \label{alg3-update}
 %\LinesNumbered

-\caption{Kernel\_update}
+\caption{Kernel update}

 
 

-\eIf{$(\left|d\_Z\right|<= R)$}{
-$kernel\_update((d\_Z,d\_Pcoef,d\_Pdegres,d\_Pucoef,d\_Pudegres)$\;}
+\eIf{$(\left|Z\right|<= R)$}{
+$kernel\_update(Z,P,Pu)$\;}

 {
 {

-$kernel\_update\_ExpoLog((d\_Z,d\_Pcoef,d\_Pdegres,d\_Pucoef,d\_Pudegres))$\;
+$kernel\_update\_ExpoLog(Z,P,Pu)$\;

 }
 \end{algorithm}
 
 }
 \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$), else the kernel executes the EA.EL
 function Eq.~\ref{Log_H2}
 of the current complex is less than the a certain value called the
 radius i.e. ($ |z^{k}_{i}|<= R$), else the kernel executes the EA.EL
 function Eq.~\ref{Log_H2}


@@ -691,7 +698,7 @@ E5620@2.40GHz and a GPU K40 (with 6 Go of ram) is used.
 %First, performances of the Ehrlich-Aberth method  of root finding polynomials
 %implemented on CPUs and on GPUs are studied.
 
 %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 times, 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 CPU and on GPU.

 
 All experimental results obtained from the simulations are made in
 double precision data, the convergence threshold of the methods is set
 
 All experimental results obtained from the simulations are made in
 double precision data, the convergence threshold of the methods is set


@@ -705,8 +712,6 @@ 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}
 
 \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}[htbp]
 \centering
   \includegraphics[width=0.8\textwidth]{figures/openMP-GPU}
 \begin{figure}[htbp]
 \centering
   \includegraphics[width=0.8\textwidth]{figures/openMP-GPU}


@@ -714,37 +719,34 @@ of the methods are given in Section~\ref{sec:vec_initialization}.
   on a CPU with OpenMP (1 core, 4 cores) and on a Tesla GPU}
 \label{fig:01}
 \end{figure}
   on a CPU with OpenMP (1 core, 4 cores) and on a Tesla GPU}
 \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, 
+%%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.
 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.

+However, the execution time for the
+CPU (4 cores) implementation exceed 5,000s for 250,000 degrees
+polynomials. In counterpart, the GPU implementation for the same
+polynomials do not take more 100s. With the GPU
+we can solve high degrees polynomials very quickly up to degree
+ of 1,000,000. We can also notice that the GPU implementation are
+ almost 40 faster then those implementation 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 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.
+%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)}
  
  %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 50,000 and 10 different polynomials of size 500,000 degrees.
+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). 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 1,024, so we varied the number of threads per block from 8 to 1,024. 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}[htbp]
 \centering
 
 \begin{figure}[htbp]
 \centering


@@ -755,9 +757,9 @@ For that, we notice that the maximum number of threads per block for the Nvidia
 
 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.
 
 
 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{Influence of exp-log solution to compute high degree polynomials}

 
 

-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 the 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}
 \begin{figure}[htbp]
 \centering
   \includegraphics[width=0.8\textwidth]{figures/sparse_full_explog}


@@ -765,44 +767,83 @@ In this experiment we report the performance of log.exp solution describe in ~\r
 \label{fig:03}
 \end{figure}
 
 \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 method using the exp-log solution and the
+execution time of the Ehrlich-Aberth method 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 4,000 and sparse polynomials less than 150,000. We
+also clearly show that the classical version (without exp-log) 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 exp-log 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 exp-log 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 times, the number of iterations and the polynomials size for the both sparse and full polynomials.  

 
 \begin{figure}[htbp]
 \centering
   \includegraphics[width=0.8\textwidth]{figures/EA_DK}
 
 \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}
 
 \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 500,000 the execution times with DK are very long.
+
+%with double precision not exceed $10^{-5}$.

 
 \begin{figure}[htbp]
 \centering
   \includegraphics[width=0.8\textwidth]{figures/EA_DK_nbr}
 
 \begin{figure}[htbp]
 \centering
   \includegraphics[width=0.8\textwidth]{figures/EA_DK_nbr}

-\caption{The iteration number of Ehrlich-Aberth versus Durand-Kerner algorithm}
+\caption{The number of iterations to converge for the Ehrlich-Aberth
+  and the Durand-Kerner methods}

 \label{fig:05}
 \end{figure}
 
 \label{fig:05}
 \end{figure}
 

-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.
+Figure~\ref{fig:05} 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 the derivative of P (the polynomial to
+resolve) in the iterative function (Eq.~\ref{Eq:Hi}) executed by EA
+allows the algorithm to converge more quickly. In counterpart, the
+DK operator (Eq.~\ref{DK}) needs low operation, consequently low
+execution time per iteration, but it needs more iterations to converge.

 
 
 
 

- \section{Conclusion and perspective}
+ \section{Conclusion and perspectives}

 \label{sec7}
 \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.
+In this paper we have presented the parallel implementation
+Ehrlich-Aberth method on GPU for the problem of finding roots
+polynomial. Moreover, we have improved the classical Ehrlich-Aberth
+method which suffers from overflow problems, the exp-log solution
+applied to the iterative function allows to solve high degree
+polynomials.
+
+We have performed many experiments with the Ehrlich-Aberth method in
+GPU. These experiments highlight that this method is very efficient in
+GPU compared to all the other implementations. The improvement with
+the exponential logarithm solution allows us to solve sparse and full
+high degree polynomials up to 1,000,000 degree. Hence, it may be
+possible to consider to use polynomial root finding methods in other
+numerical applications on GPU.
+
+
+In future works, we plan to investigate the possibility of using
+several multiple GPUs simultaneously, either with multi-GPU machine or
+with cluster of GPUs.