corrections

[kahina_paper1.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index d9d72c31c05b570457dfed406a93e3964c5efa97..d9c332420842844c03ce34d20615eaa0123df349 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -115,14 +115,14 @@ The root finding problem consists in finding the values of all the $n$ values of
 \end{equation}
 
 The problem of finding a root is equivalent to that of solving a fixed-point problem. To see this, consider the fixed-point problem of finding the $n$-dimensional
 \end{equation}
 
 The problem of finding a root is equivalent to that of solving a fixed-point problem. To see this, consider the fixed-point problem of finding the $n$-dimensional

-vector $x$ such that
+vector $x$ such that :

 \begin{center}
 $x=g(x)$
 \end{center}
 where $g : C^{n}\longrightarrow C^{n}$. Usually, we can easily
 rewrite this fixed-point problem as a root-finding problem by
 setting $f(x) = x-g(x)$ and likewise we can recast the
 \begin{center}
 $x=g(x)$
 \end{center}
 where $g : C^{n}\longrightarrow C^{n}$. Usually, we can easily
 rewrite this fixed-point problem as a root-finding problem by
 setting $f(x) = x-g(x)$ and likewise we can recast the

-root-finding problem into a fixed-point problem by setting
+root-finding problem into a fixed-point problem by setting :

 \begin{center}
 $g(x)= f(x)-x$.
 \end{center}
 \begin{center}
 $g(x)= f(x)-x$.
 \end{center}


@@ -136,7 +136,7 @@ two main groups: direct methods and iterative methods.
 Direct methods exist only for $n \leq 4$, solved in closed form by G. Cardano
 in the mid-16th century. However, N. H. Abel in the early 19th
 century showed that polynomials of degree five or more could not
 Direct methods exist only for $n \leq 4$, solved in closed form by G. Cardano
 in the mid-16th century. However, N. H. Abel in the early 19th
 century showed that polynomials of degree five or more could not

-be solved by  direct methods. Since then, mathmathicians have
+be solved by  direct methods. Since then, mathematicians have

 focussed on numerical (iterative) methods such as the famous
 Newton method, the Bernoulli method of the 18th, and the Graeffe method.
 
 focussed on numerical (iterative) methods such as the famous
 Newton method, the Bernoulli method of the 18th, and the Graeffe method.
 


@@ -147,10 +147,11 @@ 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)}
+\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}
 \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 $k$.
 
 


@@ -161,14 +162,15 @@ 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}
 %%\begin{center}
 \begin{equation}

- Z_i^{k+1}=Z_i^k-\frac{1}{{\frac {P'(Z_i^k)} {P(Z_i^k)}}-{\sum_{i\neq j}(Z_i^k-Z_j^k)}}.
+\label{Eq:EA}
+ 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}
 \end{equation}
 %%\end{center}

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

 
 Aberth, Ehrlich and Farmer-Loizou~\cite{Loizon83} have proved that
 the Ehrlich-Aberth method (EA) has a cubic order of convergence for simple roots whereas the Durand-Kerner has a quadratic order of convergence.
 
 Aberth, Ehrlich and Farmer-Loizou~\cite{Loizon83} have proved that
 the Ehrlich-Aberth method (EA) has a cubic order of convergence for simple roots whereas the Durand-Kerner has a quadratic order of convergence.


@@ -181,75 +183,82 @@ drastically increases like the degrees of high polynomials. It is expected that
 parallelization of these algorithms will improve the convergence
 time.
 
 parallelization of these algorithms will improve the convergence
 time.
 

-Many authors have dealt with the parallelisation of
+Many authors have dealt with the parallelization of

 simultaneous methods, i.e. that find all the zeros simultaneously. 
 simultaneous methods, i.e. that find all the zeros simultaneously. 

-Freeman~\cite{Freeman89} implemeted and compared DK, EA and another method of the fourth order proposed
-by Farmer and Loizou~\cite{Loizon83}, on a 8- processor linear
+Freeman~\cite{Freeman89} implemented and compared DK, EA and another method of the fourth order proposed
+by Farmer and Loizou~\cite{Loizon83}, on a 8-processor linear

 chain, for polynomials of degree up to 8. The third method often
 chain, for polynomials of degree up to 8. The third method often

-diverges, but the first two methods have speed-up 5.5
-(speed-up=(Time on one processor)/(Time on p processors)). Later,
+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 $z_i((k))$
 have not been received from the other processors, in contrast with synchronous algorithms where it would wait those values before making a new iteration.
 Freeman and Bane~\cite{Freemanall90}  considered asynchronous
 algorithms, in which each processor continues to update its
 approximations even though the latest values of other $z_i((k))$
 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 et al. ~\cite{Raphaelall01} proposed two methods of parallelisation for
+Couturier and al.~\cite{Raphaelall01} proposed two methods of parallelization for

 a shared memory architecture and for distributed memory one. They were able to
 a shared memory architecture and for distributed memory one. They were able to

-compute the roots of polynomials of degree 10000 in 430 seconds with only 8
+compute the roots of sparse polynomials of degree 10000 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, indeed.
+where it takes up to 3300 seconds to obtain the same results, the authors show an interesting speedup.

 
 

-Very few works had been since this last work until the appearing of
+Very few works had been performed since this last work until the appearing of

 the Compute Unified Device Architecture (CUDA)~\cite{CUDA10}, a
 parallel computing platform and a programming model invented by
 the Compute Unified Device Architecture (CUDA)~\cite{CUDA10}, a
 parallel computing platform and a programming model invented by

-NVIDIA. The computing power of GPUs (Graphics Processing Unit) has exceeded that of
-of CPUs. However, CUDA adopts a totally new computing architecture to use the
+NVIDIA. The computing power of GPUs (Graphics Processing Unit) has exceeded that of CPUs. However, CUDA adopts a totally new computing architecture to use the

 hardware resources provided by GPU in order to offer a stronger
 computing ability to the massive data computing.
 
 
 hardware resources provided by GPU in order to offer a stronger
 computing ability to the massive data computing.
 
 

-Ghidouche et al. ~\cite{Kahinall14} proposed an implementation of the
+Ghidouche and al~\cite{Kahinall14} proposed an implementation of the

 Durand-Kerner method on GPU. Their main
 Durand-Kerner method on GPU. Their main

-result showed that a parallel CUDA implementation is 10 times as fast as
-the sequential implementation on a single CPU for high degree
-polynomials of about 48000. To our knowledge, it is the first time such high degree polynomials are numerically solved.
-
-
-In this paper, we focus on the implementation of the Aberth method for
-high degree polynomials on GPU. The paper is organised as fellows. Initially, we recall the Aberth method in Section.\ref{sec1}. Improvements for the 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.4 we propose a parallel implementation of the Aberth method on GPU and discuss it. Section 5 presents and investigates our implementation and experimental study results. Finally, Section 6 concludes this paper and gives some hints for future research directions in this topic.  
+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. 
+
+
+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
+follows. Initially, we recall the Ehrlich-Aberth method in Section
+\ref{sec1}. Improvements for the Ehrlich-Aberth method are proposed in
+Section \ref{sec2}. Related work to the implementation of simultaneous
+methods using a parallel approach is presented in Section
+\ref{secStateofArt}.  In Section \ref{sec5} we propose a parallel
+implementation of the Ehrlich-Aberth method on GPU and discuss
+it. Section \ref{sec6} presents and investigates our implementation
+and experimental study results. Finally, Section\ref{sec7} 6 concludes
+this paper and gives some hints for future research directions in this
+topic.

 
 \section{The Sequential Aberth method}
 \label{sec1}
 A cubically convergent iteration method for finding zeros of
 
 \section{The Sequential Aberth method}
 \label{sec1}
 A cubically convergent iteration method for finding zeros of

-polynomials was proposed by O.Aberth~\cite{Aberth73}. The Aberth
-method is a purely algebraic derivation. To illustrate the
-derivation, we let $w_{i}(z)$ be the product of linear factors 
+polynomials was proposed by O. Aberth~\cite{Aberth73}. In the fellowing we present the main stages of the running of the Aberth method.
+%The Aberth method is a purely algebraic derivation. 
+%To illustrate the derivation, we let $w_{i}(z)$ be the product of linear factors 

 
 

-\begin{equation}
-w_{i}(z)=\prod_{j=1,j \neq i}^{n} (z-x_{j})
-\end{equation}
+%\begin{equation}
+%w_{i}(z)=\prod_{j=1,j \neq i}^{n} (z-x_{j})
+%\end{equation}

 
 

-And let a rational function $R_{i}(z)$ be the correction term of the
-Weistrass method~\cite{Weierstrass03}
+%And let a rational function $R_{i}(z)$ be the correction term of the
+%Weistrass method~\cite{Weierstrass03}

 
 

-\begin{equation}
-R_{i}(z)=\frac{p(z)}{w_{i}(z)} , i=1,2,...,n.
-\end{equation}
+%\begin{equation}
+%R_{i}(z)=\frac{p(z)}{w_{i}(z)} , i=1,2,...,n.
+%\end{equation}

 
 

-Differentiating the rational function $R_{i}(z)$ and applying the
-Newton method, we have:
+%Differentiating the rational function $R_{i}(z)$ and applying the
+%Newton method, we have:

 
 

-\begin{equation}
-\frac{R_{i}(z)}{R_{i}^{'}(z)}= \frac{p(z)}{p^{'}(z)-p(z)\frac{w_{i}(z)}{w_{i}^{'}(z)}}= \frac{p(z)}{p^{'}(z)-p(z) \sum _{j=1,j \neq i}^{n}\frac{1}{z-x_{i}}}, i=1,2,...,n
-\end{equation}
-
-Substituting $x_{j}$ for z we obtain the Aberth iteration method.
+%\begin{equation}
+%\frac{R_{i}(z)}{R_{i}^{'}(z)}= \frac{p(z)}{p^{'}(z)-p(z)\frac{w_{i}(z)}{w_{i}^{'}(z)}}= \frac{p(z)}{p^{'}(z)-p(z) \sum _{j=1,j \neq i}^{n}\frac{1}{z-x_{j}}}, i=1,2,...,n
+%\end{equation}
+%where R_{i}^{'}(z)is the rational function derivative of F evaluated in the point z 
+%Substituting $x_{j}$ for $z_{j}$ we obtain the Aberth iteration method.% 

 
 

-In the fellowing we present the main stages of the running of the Aberth method.

 
 \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}


@@ -264,7 +273,7 @@ The initial guess is very important since the number of steps needed by the iter
 a given approximation strongly depends on it.
 In~\cite{Aberth73} the Aberth iteration is started by selecting $n$
 equi-spaced points on a circle of center 0 and radius r, where r is
 a given approximation strongly depends on it.
 In~\cite{Aberth73} the Aberth iteration is started by selecting $n$
 equi-spaced points on a circle of center 0 and radius r, where r is

-an upper bound to the moduli of the zeros. Later, Bini et al.~\cite{Bini96}
+an upper bound to the moduli of the zeros. Later, Bini and al.~\cite{Bini96}

 performed this choice by selecting complex numbers along different
 circles and relies on the result of~\cite{Ostrowski41}.
 
 performed this choice by selecting complex numbers along different
 circles and relies on the result of~\cite{Ostrowski41}.
 


@@ -281,29 +290,30 @@ 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 $H_{i}$}
-The operator used by the Aberth method is corresponding to the
-following equation 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}
 
 \begin{equation}

-H_{i}(z)=z_{i}-\frac{1}{\frac{p^{'}(z_{i})}{p(z_{i})}-\sum_{j\neq
-i}{\frac{1}{z_{i}-z_{j}}}}
+\label{Eq:Hi}
+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}
 \end{equation}

-
+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}
 \forall i \in
 
 \begin{equation}
 \label{eq:Aberth-Conv-Cond}
 \forall i \in

-[1,n];\frac{z_{i}^{(k)}-z_{i}^{(k-1)}}{z_{i}^{(k)}}<\xi
+[1,n];\frac{z_{i}^{k}-z_{i}^{k-1}}{z_{i}^{k}}<\xi

 \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


@@ -332,36 +342,40 @@ propose to use the logarithm and the exponential of a complex in order to comput
 Using the logarithm (eq.~\ref{deflncomplex}) and the exponential (eq.~\ref{defexpcomplex}) operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations
 manipulate lower absolute values and the roots for large polynomial's degrees can be looked for successfully~\cite{Karimall98}.
 
 Using the logarithm (eq.~\ref{deflncomplex}) and the exponential (eq.~\ref{defexpcomplex}) operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations
 manipulate lower absolute values and the roots for large polynomial's degrees can be looked for successfully~\cite{Karimall98}.
 

-Applying this solution for the Aberth method we obtain the
+Applying this solution for the Ehrlich-Aberth method we obtain the

 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}
 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}

-H_{i}(z)=z_{i}^{k}-\exp \left(\ln \left(
-p(z_{k})\right)-\ln\left(p(z_{k}^{'})\right)- \ln
-\left(1-Q(z_{k})\right)\right),
+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}
 
 where:
 
 \begin{equation}
 \label{Log_H1}
 \end{equation}
 
 where:
 
 \begin{equation}
 \label{Log_H1}

-Q(z_{k})=\exp\left( \ln (p(z_{k}))-\ln(p(z_{k}^{'}))+\ln \left(
-\sum_{k\neq j}^{n}\frac{1}{z_{k}-z_{j}}\right)\right).
+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).

 \end{equation}
 
 \end{equation}
 

-This solution is applied when the root except the circle unit, represented by the radius $R$ evaluated as:
-\begin{equation}
-\label{R}
-R = \exp( \log(DBL\_MAX) / (2*n) )
-\end{equation}
- where $DBL\_MAX$ stands for the maximum representable double value.
+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{equation}

 
 

-\section{The implementation of simultaneous methods in a parallel computer}
+%R = \exp( \log(DBL\_MAX) / (2*n) )
+%\end{equation}
+ where \verb=DBL_MAX= stands for the maximum representable \verb=double= value.
+
+\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

-the degree of the polynomial. The parallelisation of these
+the degree of the polynomial. The parallelization of these

 algorithms is expected to improve the convergence time.
 Authors usually adopt one of the two following approaches to parallelize root
 finding algorithms. The first approach aims at reducing the total number of
 algorithms is expected to improve the convergence time.
 Authors usually adopt one of the two following approaches to parallelize root
 finding algorithms. The first approach aims at reducing the total number of


@@ -466,13 +480,14 @@ provides two read-only memory spaces, the constant space and the
 texture space, which reside in external DRAM, and are accessed via
 read-only caches.
 
 texture space, which reside in external DRAM, and are accessed via
 read-only caches.
 

-\subsection{ The implementation of Aberth method on GPU}
+\section{ The implementation of Aberth method on GPU}
+\label{sec5}

 %%\subsection{A CUDA implementation of the Aberth's method }
 %%\subsection{A GPU implementation of the Aberth's method }
 
 
 
 %%\subsection{A CUDA implementation of the Aberth's method }
 %%\subsection{A GPU implementation of the Aberth's method }
 
 
 

-\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]


@@ -480,9 +495,7 @@ 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


@@ -499,7 +512,7 @@ $Z\left[j\right]=H\left(j,Z\right)$\;
 
 \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\;}
 }
 }


@@ -510,51 +523,26 @@ 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}

-H(i,z^{k+1})=\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}
 
 \end{equation}
 

-With the Gauss-seidel iteration, we have:
+With the Gauss-Seidel iteration, we have:

 \begin{equation}
 \label{eq:Aberth-H-GS}
 \begin{equation}
 \label{eq:Aberth-H-GS}

-H(i,z^{k+1})=\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}
 \end{equation}

-
-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 $z^{k+1}_{i}$.
+%%Here a finiched my revision %%
+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 parallelise these steps  in order to reduce the global execution time. In the following,  we explain how we did parrallelize 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.
 


@@ -590,7 +578,7 @@ 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{$(\left|Z^{(k)}\right|<= R)$}{
 $kernel\_update(d\_z^{k})$\;}


@@ -609,16 +597,16 @@ The kernels terminate it computations when all the roots converge. Finally, the
 or from GPU memory to CPU memory \verb=(cudaMemcpyDeviceToHost))=. 
 %%HIER END MY REVISIONS (SIDER)
 \section{Experimental study}
 or from GPU memory to CPU memory \verb=(cudaMemcpyDeviceToHost))=. 
 %%HIER END MY REVISIONS (SIDER)
 \section{Experimental study}

-
-\subsection{Definition of the used polynomials }
+\label{sec6}
+%\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}


@@ -628,7 +616,7 @@ 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


@@ -639,31 +627,36 @@ 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 core 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]
+%\centering
+ % \includegraphics[width=0.8\textwidth]{figures/Compar_EA_algorithm_CPU_GPU}
+%\caption{The execution time in seconds of Ehrlich-Aberth algorithm on CPU core vs. on a Tesla GPU}
+%\label{fig:01}
+%\end{figure}
+

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

-  \includegraphics[width=0.8\textwidth]{figures/Compar_EA_algorithm_CPU_GPU}
-\caption{The execution time in seconds of Ehrlich-Aberth algorithm on CPU core vs. on a Tesla GPU}
+  \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}

 \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, 
 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 the execution times of the Ehrlich-Aberth method implemented on one core of the Quad-Core Xeon E5620 CPU and those of the same methods 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. 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 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 1,000,000 polynomials degrees GPU implementation not reach 2,300 s degrees. While CPU implementation need more than 10 hours.
-with an execution time under to 2500 s CPU implementation can resolve polynomials degrees of only 200,000 s, 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. Furthermore, we verify that the number of iterations is the same. This reduction of time allows us to compute roots of polynomial of more important degree at the same time than with a CPU.
+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.

  
  

+ %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. 
 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 K40c 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 polynomials of size 50000 and 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 50000 and 10 different polynomials of size 500000 degrees.

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


@@ -674,7 +667,7 @@ 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]


@@ -699,7 +692,7 @@ 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]


@@ -718,20 +711,16 @@ This figure show the execution time of the both algorithm EA and DK with sparse
 \label{fig:01}
 \end{figure}
 
 \label{fig:01}
 \end{figure}
 

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

 
 

-\begin{figure}[H]
-\centering
-  \includegraphics[width=0.8\textwidth]{figures/openMP-GPU}
-\caption{The execution time in seconds of Ehrlich-Aberth algorithm on OpenMP(1 core, 4 cores) vs. on a Tesla GPU}
-\label{fig:01}
-\end{figure}

 
 
 
 
 \section{Conclusion and perspective}
 
 
 
 
 
 \section{Conclusion and perspective}
 

+\label{sec7}

 \bibliography{mybibfile}
 
 \end{document}
 \bibliography{mybibfile}
 
 \end{document}