MAJ

diff --git a/figures/EA_DK.pdf b/figures/EA_DK.pdf
index f992927f22752539856d3fbcaddb9c5f08199899..2bfe0281810c4546f86934142798541ded15de17 100644 (file)
Binary files a/figures/EA_DK.pdf and b/figures/EA_DK.pdf differ

diff --git a/figures/EA_DK.plot b/figures/EA_DK.plot
index 33c2d80ed21e6bd1177028e4a74dad1d20a19963..e9568f154e152b1aa5e50a8f4101b70414daf566 100644 (file)
--- a/figures/EA_DK.plot
+++ b/figures/EA_DK.plot
@@ -3,9 +3,14 @@ set encoding iso_8859_1
 set terminal x11
 set size 1,0.5
 set term postscript enhanced portrait "Helvetica" 12
 set terminal x11
 set size 1,0.5
 set term postscript enhanced portrait "Helvetica" 12

+#set decimalsign ","
+#set decimalsign locale "fr_FR"

 
 set ylabel "execution times (in s)" 
 
 set ylabel "execution times (in s)" 

-set xlabel "polynomial's degree" 
+set xlabel "polynomial's degree"
+#set format x "%6.4e"
+#set format x "%2.0e"
+#set format x "%2.0t{/Symbol \327}10^{%L}"

 #set logscale x
 #set logscale y
 
 #set logscale x
 #set logscale y
 

diff --git a/figures/EA_DK.txt b/figures/EA_DK.txt
index 6414285280e318b227be485d03d89ffd712d168d..a22a15155b05d7a987a9b5e4bd90a6939da92250 100644 (file)
--- a/figures/EA_DK.txt
+++ b/figures/EA_DK.txt
@@ -6,28 +6,28 @@
 100000         12.45           16              54.5215         30                      
 150000          28.67          17              156.63          33                      
 200000         40              23              330.456         43                      
 100000         12.45           16              54.5215         30                      
 150000          28.67          17              156.63          33                      
 200000         40              23              330.456         43                      

-250000         93.76           20              518.342         47                      
-300000         138.94          21              313.697         21                      
-350000         159.65          18                              28                      
+250000         93.76           20              #518.342        47                      
+300000         138.94          21              #313.697        21                      
+350000         159.65          18              #               28                      

 400000         258.91          22              580.649         20                      
 400000         258.91          22              580.649         20                      

-450000         339.47          23              
+450000         339.47          23              #

 500000         419.78          23              739.882         28
 500000         419.78          23              739.882         28

-550000         415.94          19
-600000         549.70          21
-650000         612.12          20
-700000         864.21          24
-750000         940.87          23
-800000         1247.16         26
-850000         1702.12         32
-900000         1803.17         30
-950000         2280.07         34
-1000000                2400.51         30
+550000         415.94          19              #
+600000         549.70          21              1419.76         27
+650000         612.12          20              #
+700000         864.21          24              2149.02         48
+750000         940.87          23              #
+800000         1247.16         26              2459.5          26
+850000         1702.12         32              #
+900000         1803.17         30              3001.47         28
+950000         2280.07         34              #
+1000000                2400.51         30              3321.03         27

 
 # Second data block (index 1)
 
 #DK            sparse                          full                                    
                times           nb iter         times       nb iter             
 
 # Second data block (index 1)
 
 #DK            sparse                          full                                    
                times           nb iter         times       nb iter             

-5000           3.42            138             638.572     9597
+5000           3.42            138             622.617     9483

 50000          385.266         823             9.27        19
 100000         447.364         408             7.73        15
 150000         1524.08         552             8.64        21
 50000          385.266         823             9.27        19
 100000         447.364         408             7.73        15
 150000         1524.08         552             8.64        21

diff --git a/paper.tex b/paper.tex
index bcf399a3b5c7b8109b9046412ec75e0961cf30c0..c49a7052603cf95dc5ec785bea154dabfa93bbff 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -147,7 +147,8 @@ approximation of all the roots, starting with the Durand-Kerner (DK)
 method:
 %%\begin{center}
 \begin{equation}
 method:
 %%\begin{center}
 \begin{equation}

- 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,
+\label{DK}
+ 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
 \end{equation}
 %%\end{center}
 where $z_i^k$ is the $i^{th}$ root of the polynomial $P$ at the


@@ -161,11 +162,11 @@ Docev~\cite{Docev62}, Durand~\cite{Durand60},
 Kerner~\cite{Kerner66}. Another method discovered by
 Borsch-Supan~\cite{ Borch-Supan63} and also described and brought
 in the following form by Ehrlich~\cite{Ehrlich67} and
 Kerner~\cite{Kerner66}. Another method discovered by
 Borsch-Supan~\cite{ Borch-Supan63} and also described and brought
 in the following form by Ehrlich~\cite{Ehrlich67} and

-Aberth~\cite{Aberth73} uses a different iteration formula given as fellows :
+Aberth~\cite{Aberth73} uses a different iteration formula given as:

 %%\begin{center}
 \begin{equation}
 \label{Eq:EA}
 %%\begin{center}
 \begin{equation}
 \label{Eq: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,
+ 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
 \end{equation}
 %%\end{center}
 where $P'(z)$ is the polynomial derivative of $P$ evaluated in the


@@ -279,21 +280,21 @@ u_{i}=2.|a_{i}|^{\frac{1}{i}};
 v_{i}=\frac{|\frac{a_{n}}{a_{i}}|^{\frac{1}{n-i}}}{2}.
 \end{equation}
 
 v_{i}=\frac{|\frac{a_{n}}{a_{i}}|^{\frac{1}{n-i}}}{2}.
 \end{equation}
 

-\subsection{Iterative Function $H_{i}(z^{k})$}
-The operator used by the Aberth method is corresponding to the
-following equation~\ref{Eq:EA} which will enable the convergence towards
-polynomial solutions, provided all the roots are distinct.
+\subsection{Iterative Function}
+%The operator used by the Aberth method is corresponding to the
+%following equation~\ref{Eq:EA} which will enable the convergence towards
+%polynomial solutions, provided all the roots are distinct.
+
+Here we give a second form of the iterative function used by Ehrlich-Aberth method: 

 
 \begin{equation}
 \label{Eq:Hi}
 
 \begin{equation}
 \label{Eq:Hi}

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

-we notice that the function iterative $H_{i}$ in Eq.~\ref{Eq:Hi} it the same those presented in Eq.~\ref{Eq:EA}, but we prefer used the last one seen the advantage of its use to improve the Aberth method. More detail in the section ~\ref{sec2}.    
+we notice that the function iterative in Eq.~\ref{Eq:Hi} it the same those presented in Eq.~\ref{Eq:EA}, but we prefer used the last one seen the advantage of its use to improve the Ehrlich-Aberth method and resolve very high degrees polynomials. More detail in the section ~\ref{sec2}.    

 \subsection{Convergence Condition}
 \subsection{Convergence Condition}

-The convergence condition determines the termination of the algorithm. It consists in stopping from running
-the iterative function $H_{i}(z)$ when the roots are sufficiently stable. We consider that the method
-converges sufficiently when :
+The convergence condition determines the termination of the algorithm. It consists in stopping from running the iterative function  when the roots are sufficiently stable. We consider that the method converges sufficiently when:

 
 \begin{equation}
 \label{eq:Aberth-Conv-Cond}
 
 \begin{equation}
 \label{eq:Aberth-Conv-Cond}


@@ -302,7 +303,7 @@ converges sufficiently when :
 \end{equation}
 
 
 \end{equation}
 
 

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

 \label{sec2}
 The Ehrlich-Aberth method implementation suffers of overflow problems. This
 situation occurs, for instance, in the case where a polynomial
 \label{sec2}
 The Ehrlich-Aberth method implementation suffers of overflow problems. This
 situation occurs, for instance, in the case where a polynomial


@@ -336,7 +337,7 @@ iteration function with 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}

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


@@ -360,7 +361,7 @@ R = exp(log(DBL_MAX)/(2*n) );
 %\end{equation}
  where \verb=DBL_MAX= stands for the maximum representable \verb=double= value.
 
 %\end{equation}
  where \verb=DBL_MAX= stands for the maximum representable \verb=double= value.
 

-\section{The implementation of simultaneous methods in a parallel computer}
+\section{Implementation of simultaneous methods in a parallel computer}

 \label{secStateofArt}   
 The main problem of simultaneous methods is that the necessary
 time needed for convergence is increased when we increase
 \label{secStateofArt}   
 The main problem of simultaneous methods is that the necessary
 time needed for convergence is increased when we increase


@@ -476,7 +477,7 @@ read-only caches.
 
 
 
 
 
 

-\subsubsection{A sequential Aberth algorithm}
+\subsection{A sequential Aberth algorithm}

 The main steps of Aberth method are shown in Algorithm.~\ref{alg1-seq} :
   
 \begin{algorithm}[H]
 The main steps of Aberth method are shown in Algorithm.~\ref{alg1-seq} :
   
 \begin{algorithm}[H]


@@ -484,9 +485,7 @@ The main steps of Aberth method are shown in Algorithm.~\ref{alg1-seq} :
 %\LinesNumbered
 \caption{A sequential algorithm to find roots with the Aberth method}
 
 %\LinesNumbered
 \caption{A sequential algorithm to find roots with the Aberth method}
 

-\KwIn{$Z^{0}$(Initial root's vector),$\varepsilon$ (error
-tolerance threshold),P(Polynomial to solve)}
-
+\KwIn{$Z^{0}$(Initial root's vector),$\varepsilon$ (error tolerance threshold),P(Polynomial to solve)}

 \KwOut {Z(The solution root's vector)}
 
 \BlankLine
 \KwOut {Z(The solution root's vector)}
 
 \BlankLine


@@ -503,7 +502,7 @@ $Z\left[j\right]=H\left(j,Z\right)$\;
 
 \For{$i \gets 0 $ \KwTo $n-1$}{
 $c=\frac{\left|Z\left[i\right]-ZPrec\left[i\right]\right|}{Z\left[i\right]}$\;
 
 \For{$i \gets 0 $ \KwTo $n-1$}{
 $c=\frac{\left|Z\left[i\right]-ZPrec\left[i\right]\right|}{Z\left[i\right]}$\;

-\If{$c\succ\Delta z_{max}$ }{
+\If{$c > \Delta z_{max}$ }{

 $\Delta z_{max}$=c\;}
 }
 }
 $\Delta z_{max}$=c\;}
 }
 }


@@ -514,51 +513,26 @@ In this sequential algorithm, one CPU thread  executes all the steps. Let us loo
 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 :
 
 \begin{equation}
 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 :
 
 \begin{equation}

-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.
+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.

 \end{equation}
 
 With the Gauss-Seidel iteration, we have:
 \begin{equation}
 \label{eq:Aberth-H-GS}
 \end{equation}
 
 With the Gauss-Seidel iteration, we have:
 \begin{equation}
 \label{eq:Aberth-H-GS}

-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.
+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 %%
 \end{equation}
 %%Here a finiched my revision %%

-Using Equation.~\ref{eq:Aberth-H-GS} for the update sub-step of $H(i,z^{k+1})$, we expect the Gauss-Seidel iteration to converge more quickly because, just as its ancestor (for solving linear systems of equations), it uses the most fresh computed roots $z^{k+1}_{i}$.
+Using Equation.~\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 its ancestor (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}.
 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 Equation.~\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.
 

-\paragraph{The execution time}
-Let $T_{i}(n)$ be the time to compute one new root value at step 3, $T_{i}$ depends on the polynomial's degree $n$. When $n$ increase $T_{i}(n)$ increases too. We need $n.T_{i}(n)$ to compute all the new values in one iteration at step 3.
-
-Let $T_{j}$ be the time needed to check the convergence of one root value at the step 4, so we need $n.T_{j}$ to compute global convergence condition in each iteration at step 4.
-
-Thus, the execution time for both steps 3 and 4 is:
-\begin{equation}
-T_{iter}=n(T_{i}(n)+T_{j})+O(n).
-\end{equation}
-Let $K$ be the number of iterations necessary to compute all the roots, so the total execution time $T$ can be given as:
-
-\begin{equation}
-\label{eq:T-global}
-T=\left[n\left(T_{i}(n)+T_{j}\right)+O(n)\right].K
-\end{equation}
-The execution time increases with the increasing of the polynomial degree, which justifies to parallelize these steps  in order to reduce the global execution time. In the following,  we explain how we did parallelize these steps on a GPU architecture using the CUDA platform.

 
 

-\subsubsection{A Parallel implementation with CUDA }
+\subsection{A 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 schduler 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. It's up to the programmer, to describe the execution context, that is the size of the Grid, the number of blocks and the number of threads per block upon the call of a given kernel, according to a special syntax defined by 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 schduler 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. It's up to the programmer, to describe the execution context, that is the size of the Grid, the number of blocks and the number of threads per block upon the call of a given kernel, according to a special syntax defined by CUDA.
 

-Let N be the number of threads executed in parallel,  Equation.~\ref{eq:T-global} becomes then : 
-
-\begin{equation}
-T=\left[\frac{n}{N}\left(T_{i}(n)+T_{j}\right)+O(n)\right].K.
-\end{equation}
-
-In theory, total execution time $T$ on GPU is speed up $N$ times as  $T$ on CPU. We will see at what extent this is true in the experimental study hereafter. 
-~\\
-~\\
-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 ``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. 

 
 Algorithm~\ref{alg2-cuda} shows a sketch of the Aberth algorithm usind CUDA.
 
 
 Algorithm~\ref{alg2-cuda} shows a sketch of the Aberth algorithm usind CUDA.
 


@@ -594,7 +568,7 @@ The second kernel executes the iterative function $H$ and updates $z^{k}$, accor
 \begin{algorithm}[H]
 \label{alg3-update}
 %\LinesNumbered
 \begin{algorithm}[H]
 \label{alg3-update}
 %\LinesNumbered

-\caption{A global Algorithm for the iterative function}
+\caption{Kernel update}

 
 \eIf{$(\left|Z^{(k)}\right|<= R)$}{
 $kernel\_update(d\_z^{k})$\;}
 
 \eIf{$(\left|Z^{(k)}\right|<= R)$}{
 $kernel\_update(d\_z^{k})$\;}


@@ -614,15 +588,15 @@ or from GPU memory to CPU memory \verb=(cudaMemcpyDeviceToHost))=.
 %%HIER END MY REVISIONS (SIDER)
 \section{Experimental study}
 \label{sec6}
 %%HIER END MY REVISIONS (SIDER)
 \section{Experimental study}
 \label{sec6}

-\subsection{Definition of the used polynomials }
+%\subsection{Definition of the used polynomials }

 We study two categories of polynomials : the sparse polynomials and the full 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 polonymial for which the roots are distributed on 2 distinct circles :
+\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 :

 \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}
 
 
 \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:
+\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:

 %%\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}


@@ -632,7 +606,7 @@ We study two categories of polynomials : the sparse polynomials and the full pol
 \end{equation}
 With this form, we can have until \textit{n} non zero terms whereas the sparse ones have just two non zero terms.
 
 \end{equation}
 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
 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


@@ -643,13 +617,13 @@ element-key which justifies our work of parallelization.
 E5620@2.40GHz and a GPU K40 (with 6 Go of ram).
 
 
 E5620@2.40GHz and a GPU K40 (with 6 Go of ram).
 
 

-\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.
 
 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.
 
 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. 
 In this section, we discuss the performance Ehrlich-Aberth method  of root finding polynomials implemented 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 time, 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. 

-\subsubsection{The execution time in seconds of Ehrlich-Aberth algorithm on CPU OpenMP (1 core, 4 cores) vs. on a Tesla GPU}
+\subsection{The execution time in seconds of Ehrlich-Aberth algorithm on CPU OpenMP (1 core, 4 cores) vs. on a Tesla GPU}

 
 
 %\begin{figure}[H]
 
 
 %\begin{figure}[H]


@@ -670,7 +644,7 @@ We report respectively the execution time of the Ehrlich-Aberth method implement
  
  %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.
 
  
  %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.
 

-\subsubsection{Influence of the number of threads on the execution times of different polynomials (sparse and full)}
+\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.
 
 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.
 


@@ -683,7 +657,7 @@ 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.
 

-\subsubsection{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.   
 \begin{figure}[H]
 
 In this experiment we report the performance of log.exp solution describe in ~\ref{sec2} to compute very high degrees polynomials.   
 \begin{figure}[H]


@@ -708,7 +682,7 @@ in fact, when the modulus of the roots are up than \textit{R} given in ~\ref{R},
 %we report the performances of the exp.log for the Ehrlich-Aberth algorithm for solving very high degree of polynomial. 
 
  
 %we report the performances of the exp.log for the Ehrlich-Aberth algorithm for solving very high degree of polynomial. 
 
  

-\subsubsection{A comparative study between Ehrlich-Aberth algorithm and Durand-kerner algorithm}
+\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]
 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]


@@ -735,6 +709,7 @@ This figure show the execution time of the both algorithm EA and DK with sparse
 
 
 \section{Conclusion and perspective}
 
 
 \section{Conclusion and perspective}

+

 \label{sec7}
 \bibliography{mybibfile}
 
 \label{sec7}
 \bibliography{mybibfile}