ajout d'un fichier pour mettre les modifications

[kahina_paper1.git] / paper.tex

diff --git a/paper.tex b/paper.tex
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--- a/paper.tex
+++ b/paper.tex
@@ -82,13 +82,13 @@
 
 \begin{abstract}
 Polynomials are mathematical algebraic structures that play a great
 
 \begin{abstract}
 Polynomials are mathematical algebraic structures that play a great

-role in science and engineering. Finding roots of high degree
+role in science and engineering. Finding the roots of high degree

 polynomials is computationally demanding. In this paper, we present
 the results of a parallel implementation of the Ehrlich-Aberth
 algorithm for the root finding problem for high degree polynomials on
 GPU architectures. The main result of this
 work is to be able to solve high degree polynomials (up
 polynomials is computationally demanding. In this paper, we present
 the results of a parallel implementation of the Ehrlich-Aberth
 algorithm for the root finding problem for high degree polynomials on
 GPU architectures. The main result of this
 work is to be able to solve high degree polynomials (up

-to 1,000,000) very efficiently. We also compare the results with a
+to 1,000,000)  efficiently. We also compare the results with a

 sequential implementation and the Durand-Kerner method on full and
 sparse polynomials.
 \end{abstract}
 sequential implementation and the Durand-Kerner method on full and
 sparse polynomials.
 \end{abstract}


@@ -101,56 +101,57 @@ Polynomial root finding, Iterative methods, Ehrlich-Aberth, Durand-Kerner, GPU
 
 \linenumbers
 
 
 \linenumbers
 

-\section{The problem of finding roots of a polynomial}
-Polynomials are mathematical algebraic structures used in science and engineering to capture physical phenomenons and to express any outcome in the form of a function of some unknown variables. Formally speaking,  a polynomial $p(x)$ of degree \textit{n} having $n$ coefficients in the complex plane \textit{C} is :
+\section{The problem of finding the roots of a polynomial}
+Polynomials are mathematical algebraic structures used in science and engineering to capture physical phenomena and to express any outcome in the form of a function of some unknown variables. Formally speaking,  a polynomial $p(x)$ of degree \textit{n} having $n$ coefficients in the complex plane \textit{C} is :

 %%\begin{center}
 \begin{equation}
      {\Large p(x)=\sum_{i=0}^{n}{a_{i}x^{i}}}.
 \end{equation}
 %%\end{center}
 
 %%\begin{center}
 \begin{equation}
      {\Large p(x)=\sum_{i=0}^{n}{a_{i}x^{i}}}.
 \end{equation}
 %%\end{center}
 

-The root finding problem consists in finding the values of all the $n$ values of the variable $x$ for which \textit{p(x)} is nullified. Such values are called zeroes of $p$. If zeros are $\alpha_{i},\textit{i=1,...,n}$ the $p(x)$ can be written as :
+The root finding problem consists in finding the values of all the $n$ values of the variable $x$ for which \textit{p(x)} is nullified. Such values are called zeros of $p$. If zeros are $\alpha_{i},\textit{i=1,...,n}$ the $p(x)$ can be written as :

 \begin{equation}
      {\Large p(x)=a_{n}\prod_{i=1}^{n}(x-\alpha_{i}), a_{0} a_{n}\neq 0}.
 \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
 \begin{equation}
      {\Large p(x)=a_{n}\prod_{i=1}^{n}(x-\alpha_{i}), a_{0} a_{n}\neq 0}.
 \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}
 

-Often it is not be possible to solve such nonlinear equation
-root-finding problems analytically. When this occurs we turn to
+It is often impossible to solve such nonlinear equation
+root-finding problems analytically. When this occurs, we turn to

 numerical methods to approximate the solution. 
 Generally speaking, algorithms for solving problems can be divided into
 two main groups: direct methods and iterative methods.
 numerical methods to approximate the solution. 
 Generally speaking, algorithms for solving problems can be divided into
 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
-be solved by  direct methods. Since then, mathmathicians have
+
+Direct methods only exist 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 proved that polynomials of degree five or more could not
+be solved by  direct methods. Since then, mathematicians have

 focussed on numerical (iterative) methods such as the famous
 focussed on numerical (iterative) methods such as the famous

-Newton method, the Bernoulli method of the 18th, and the Graeffe method.
+Newton method, the Bernoulli method of the 18th century, and the Graeffe method.

 
 Later on, with the advent of electronic computers, other methods have
 been developed such as the Jenkins-Traub method, the Larkin method,
 
 Later on, with the advent of electronic computers, other methods have
 been developed such as the Jenkins-Traub method, the Larkin method,

-the Muller method, and several methods for simultaneous
+the Muller method, and several other methods for the simultaneous

 approximation of all the roots, starting with the Durand-Kerner (DK)
 method:
 %%\begin{center}
 \begin{equation}
 approximation of all the roots, starting with the Durand-Kerner (DK)
 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 $k$.
 
 


@@ -161,95 +162,106 @@ 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
+Aberth, Ehrlich and Farmer-Loizou~\cite{Loizou83} 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.
 
 
 the Ehrlich-Aberth method (EA) has a cubic order of convergence for simple roots whereas the Durand-Kerner has a quadratic order of convergence.
 
 

-Iterative methods raise several problem when implemented e.g.
-specific sizes of numbers must be used to deal with this
-difficulty. Moreover, the convergence time of iterative methods
+Moreover, the convergence times of iterative methods

 drastically increases like the degrees of high polynomials. It is expected that the
 drastically increases like the degrees of high polynomials. It is expected that the

-parallelization of these algorithms will improve the convergence
-time.
+parallelization of these algorithms will reduce the execution times.

 
 

-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
-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,
+Freeman~\cite{Freeman89} implemented and compared DK, EA and another method of the fourth order proposed
+by Farmer and Loizou~\cite{Loizou83}, on an 8-processor linear
+chain, for polynomials of degree 8. The third method often
+diverges, but the first two methods have speed-ups equal to 5.5. Later,

 Freeman and Bane~\cite{Freemanall90}  considered asynchronous
 algorithms, in which each processor continues to update its
 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
+approximations even though the latest values of other roots
+have not yet been received from the other processors.  In contrast,
+synchronous algorithms   wait the computation of all roots at a given
+iterations  before making a new one.
+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
-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.
+compute the roots of sparse polynomials of degree 10,000 in 430 seconds with only 8
+personal computers and 2 communications per iteration. Compared to sequential implementations
+where it takes up to 3,300 seconds to obtain the same results, the
+authors' work experiment show an interesting speedup.

 
 

-Very few works had been since this last work until the appearing of
+Few works have been conducted after those works until the appearance 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.  
-
-\section{The Sequential Aberth method}
+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 48,000. 
+
+
+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} concludes
+this paper and gives some hints for future research directions in this
+topic.
+
+\section{Ehrlich-Aberth method}

 \label{sec1}
 \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 
+A cubically convergent iteration method to find zeros of
+polynomials was proposed by O. Aberth~\cite{Aberth73}. The
+Ehrlich-Aberth method contains 4 main steps, presented in what
+follows.

 
 

-\begin{equation}
-w_{i}(z)=\prod_{j=1,j \neq i}^{n} (z-x_{j})
-\end{equation}
+%The Aberth method is a purely algebraic derivation. 
+%To illustrate the derivation, we let $w_{i}(z)$ be the product of linear factors 

 
 

-And let a rational function $R_{i}(z)$ be the correction term of the
-Weistrass method~\cite{Weierstrass03}
+%\begin{equation}
+%w_{i}(z)=\prod_{j=1,j \neq i}^{n} (z-x_{j})
+%\end{equation}

 
 

-\begin{equation}
-R_{i}(z)=\frac{p(z)}{w_{i}(z)} , i=1,2,...,n.
-\end{equation}
+%And let a rational function $R_{i}(z)$ be the correction term of the
+%Weistrass method~\cite{Weierstrass03}

 
 

-Differentiating the rational function $R_{i}(z)$ and applying the
-Newton method, we have:
+%\begin{equation}
+%R_{i}(z)=\frac{p(z)}{w_{i}(z)} , i=1,2,...,n.
+%\end{equation}

 
 

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

 
 

-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}


@@ -257,16 +269,16 @@ 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}
-
-Like for any iterative method, we need to choose $n$ initial guess points $z^{(0)}_{i}, i = 1, . . . , n.$
+\subsection{Vector $Z^{(0)}$ Initialization}
+\label{sec:vec_initialization}
+As for any iterative method, we need to choose $n$ initial guess points $z^{0}_{i}, i = 1, . . . , n.$

 The initial guess is very important since the number of steps needed by the iterative method to reach
 a given approximation strongly depends on it.
 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 Aberth iteration is started by selecting $n$
+In~\cite{Aberth73} the Ehrlich-Aberth iteration is started by selecting $n$

 equi-spaced points on a circle of center 0 and radius r, where r is
 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
 performed this choice by selecting complex numbers along different

-circles and relies on the result of~\cite{Ostrowski41}.
+circles which relies on the result of~\cite{Ostrowski41}.

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


@@ -281,39 +293,43 @@ 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 the 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}_{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=1,. . . .,n

 \end{equation}
 \end{equation}

-
+It can be noticed that this equation is equivalent to Eq.~\ref{Eq:EA},
+but we prefer the latter one because we can use it to improve the
+Ehrlich-Aberth method and find the roots of high degree polynomials. More
+details are given in 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 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}

-\forall i \in
-[1,n];\frac{z_{i}^{(k)}-z_{i}^{(k-1)}}{z_{i}^{(k)}}<\xi
+\forall i \in [1,n];\vert\frac{z_{i}^{k}-z_{i}^{k-1}}{z_{i}^{k}}\vert<\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}
 \label{sec2}

-The Ehrlich-Aberth method implementation suffers of 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
+With high degree polynomial, the Ehrlich-Aberth method implementation,
+as well as the Durand-Kerner implementation, 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 $|z|$ 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
 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
-~\cite{Karimall98} for the Durand-Kerner method, the authors
+for large degrees. This problem was discussed earlier in
+~\cite{Karimall98} for the Durand-Kerner method. The authors

 propose to use the logarithm and the exponential of a complex in order to compute the power at a high exponent.
 
 \begin{equation}
 propose to use the logarithm and the exponential of a complex in order to compute the power at a high exponent.
 
 \begin{equation}


@@ -325,60 +341,66 @@ propose to use the logarithm and the exponential of a complex in order to comput
 \begin{align}
 \label{defexpcomplex}
  \forall(x,y)\in R^{*2}; \exp(x+i.y) & = \exp(x).\exp(i.y)\\
 \begin{align}
 \label{defexpcomplex}
  \forall(x,y)\in R^{*2}; \exp(x+i.y) & = \exp(x).\exp(i.y)\\

-                                     & =\exp(x).\cos(y)+i.\exp(x).\sin(y)\label{defexpcomplex}
+                                     & =\exp(x).\cos(y)+i.\exp(x).\sin(y)\label{defexpcomplex1}

 \end{align}
 %%\end{equation}
 
 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
 \end{align}
 %%\end{equation}
 
 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}.
+manipulate lower absolute values and the roots for large polynomial degrees can be looked for successfully~\cite{Karimall98}.

 
 

-Applying this solution for the Aberth method we obtain the
-iteration function with logarithm:
+Applying this solution for the iteration function Eq.~\ref{Eq:Hi} of
+Ehrlich-Aberth method, we obtain the following 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}

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

 \end{equation}
 
 \end{equation}
 

-This solution is applied when the root except the circle unit, represented by the radius $R$ evaluated as:
+This solution is applied when the root except the circle unit, represented by the radius $R$ evaluated in C language as :

 \begin{equation}
 \begin{equation}

-\label{R}
-R = \exp( \log(DBL\_MAX) / (2*n) )
+\label{R.EL}
+R = exp(log(DBL\_MAX)/(2*n) );

 \end{equation}
 \end{equation}

- where $DBL\_MAX$ stands for the maximum representable double value.

 
 

-\section{The implementation of simultaneous methods in a parallel computer}
+
+%\begin{equation}
+
+%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}   
 \label{secStateofArt}   

-The main problem of simultaneous methods is that the necessary
+The main problem of simultaneous methods is that the 

 time needed for convergence is increased when we increase
 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

-iterations as by Miranker
+iterations as in Miranker

 ~\cite{Mirankar68,Mirankar71}, Schedler~\cite{Schedler72} and
 ~\cite{Mirankar68,Mirankar71}, Schedler~\cite{Schedler72} and

-Winogard~\cite{Winogard72}. The second approach aims at reducing the
+Winograd~\cite{Winogard72}. The second approach aims at reducing the

 computation time per iteration, as reported
 in~\cite{Benall68,Jana06,Janall99,Riceall06}. 
 
 There are many schemes for the simultaneous approximation of all roots of a given
 polynomial. Several works on different methods and issues of root
 computation time per iteration, as reported
 in~\cite{Benall68,Jana06,Janall99,Riceall06}. 
 
 There are many schemes for the simultaneous approximation of all roots of a given
 polynomial. Several works on different methods and issues of root

-finding have been reported in~\cite{Azad07, Gemignani07, Kalantari08, Skachek08, Zhancall08, Zhuall08}. However, Durand-Kerner and Ehrlich-Aberth methods are the most practical choices among
+finding have been reported in~\cite{Azad07, Gemignani07, Kalantari08,
+  Zhancall08, Zhuall08}. However, the Durand-Kerner and the Ehrlich-Aberth methods are the most practical choices among

 them~\cite{Bini04}. These two methods have been extensively
 them~\cite{Bini04}. These two methods have been extensively

-studied for parallelization due to their intrinsics, i.e. the
-computations involved in both methods has some inherent
+studied for parallelization due to their intrinsic parallelism, i.e. the
+computations involved in both methods have some inherent

 parallelism that can be suitably exploited by SIMD machines.
 parallelism that can be suitably exploited by SIMD machines.

-Moreover, they have fast rate of convergence (quadratic for the
+Moreover, they have fast a rate of convergence (quadratic for the

 Durand-Kerner and cubic for the Ehrlich-Aberth). Various parallel
 algorithms reported for these methods can be found
 in~\cite{Cosnard90, Freeman89,Freemanall90,Jana99,Janall99}.
 Durand-Kerner and cubic for the Ehrlich-Aberth). Various parallel
 algorithms reported for these methods can be found
 in~\cite{Cosnard90, Freeman89,Freemanall90,Jana99,Janall99}.


@@ -389,236 +411,283 @@ each processor to communicate its current approximation to all
 other processors at the end of each iteration (synchronous). Therefore they
 cause a high degree of memory conflict. Recently the author
 in~\cite{Mirankar71} proposed two versions of parallel algorithm
 other processors at the end of each iteration (synchronous). Therefore they
 cause a high degree of memory conflict. Recently the author
 in~\cite{Mirankar71} proposed two versions of parallel algorithm

-for the Durand-Kerner method, and Ehrlich-Aberth method on a model of
-Optoelectronic Transpose Interconnection System (OTIS).The
-algorithms are mapped on an OTIS-2D torus using N processors. This
-solution needs N processors to compute N roots, which is not
-practical for solving polynomials with large degrees.
-Until very recently, the literature doen not mention implementations able to compute the roots of
-large degree polynomials (higher then 1000) and within small or at least tractable times. Finding polynomial roots rapidly and accurately is the main objective of our work. 
+for the Durand-Kerner method, and the Ehrlich-Aberth method on a model of
+Optoelectronic Transpose Interconnection System (OTIS). The
+algorithms are mapped on an OTIS-2D torus using $N$ processors. This
+solution needs $N$ processors to compute $N$ roots, which is not
+practical for solving large degree polynomials.
+
+%Until very recently, the literature did not mention implementations
+%able to compute the roots of large degree polynomials (higher then
+%1000) and within small or at least tractable times.
+
+Finding polynomial roots rapidly and accurately is the main objective of our work. 

 With the advent of CUDA (Compute Unified Device
 Architecture), finding the roots of polynomials receives a new attention because of the new possibilities to solve higher degree polynomials in less time. 
 In~\cite{Kahinall14} we already proposed the first implementation
 of a root finding method on GPUs, that of the Durand-Kerner method. The main result showed
 that a parallel CUDA implementation is 10 times as fast as the
 sequential implementation on a single CPU for high degree
 With the advent of CUDA (Compute Unified Device
 Architecture), finding the roots of polynomials receives a new attention because of the new possibilities to solve higher degree polynomials in less time. 
 In~\cite{Kahinall14} we already proposed the first implementation
 of a root finding method on GPUs, that of the Durand-Kerner method. The 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 48000. In this paper we present a parallel implementation of Ehlisch-Aberth method on
-GPUs, which details are discussed in the sequel.
-
-
-\section {A CUDA parallel Ehrlich-Aberth method}
-In the following, we describe the parallel implementation of Ehrlich-Aberth method on GPU 
-for solving high degree polynomials. First, the hardware and software of the GPUs are presented. Then, a CUDA parallel Ehrlich-Aberth method are presented.
-
-\subsection{Background on the GPU architecture}
-A GPU is viewed as an accelerator for the data-parallel and
-intensive arithmetic computations. It draws its computing power
-from the parallel nature of its hardware and software
-architectures. A GPU is composed of hundreds of Streaming
-Processors (SPs) organized in several blocks called Streaming
-Multiprocessors (SMs). It also has a memory hierarchy. It has a
-private read-write local memory per SP, fast shared memory and
-read-only constant and texture caches per SM and a read-write
-global memory shared by all its SPs~\cite{NVIDIA10}.
-
-On a CPU equipped with a GPU, all the data-parallel and intensive
-functions of an application running on the CPU are off-loaded onto
-the GPU in order to accelerate their computations. A similar
-data-parallel function is executed on a GPU as a kernel by
-thousands or even millions of parallel threads, grouped together
-as a grid of thread blocks. Therefore, each SM of the GPU executes
-one or more thread blocks in SIMD fashion (Single  Instruction,
-Multiple Data) and in turn each SP of a GPU SM runs one or more
-threads within a block in SIMT fashion (Single Instruction,
-Multiple threads). Indeed at any given clock cycle, the threads
-execute the same instruction of a kernel, but each of them
-operates on different data.
- GPUs only work on data filled in their
-global memories and the final results of their kernel executions
-must be communicated to their CPUs. Hence, the data must be
-transferred in and out of the GPU. However, the speed of memory
-copy between the GPU and the CPU is slower than the memory
-bandwidths of the GPU memories and, thus, it dramatically affects
-the performances of GPU computations. Accordingly, it is necessary
-to limit as much as possible, data transfers between the GPU and its CPU during the
-computations.
-\subsection{Background on the CUDA Programming Model}
-
-The CUDA programming model is similar in style to a single program
-multiple-data (SPMD) software model. The GPU is viewed as a
-coprocessor that executes data-parallel kernel functions. CUDA
-provides three key abstractions, a hierarchy of thread groups,
-shared memories, and barrier synchronization. Threads have a three
-level hierarchy. A grid is a set of thread blocks that execute a
-kernel function. Each grid consists of blocks of threads. Each
-block is composed of hundreds of threads. Threads within one block
-can share data using shared memory and can be synchronized at a
-barrier. All threads within a block are executed concurrently on a
-multithreaded architecture.The programmer specifies the number of
-threads per block, and the number of blocks per grid. A thread in
-the CUDA programming language is much lighter weight than a thread
-in traditional operating systems. A thread in CUDA typically
-processes one data element at a time. The CUDA programming model
-has two shared read-write memory spaces, the shared memory space
-and the global memory space. The shared memory is local to a block
-and the global memory space is accessible by all blocks. CUDA also
-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.
-
-\subsection{ The implementation of Aberth method on GPU}
+polynomials of 48,000.
+%In this paper we present a parallel implementation of Ehrlich-Aberth
+%method on GPUs for sparse and full polynomials with high degree (up
+%to $1,000,000$).
+
+
+%% \section {A CUDA parallel Ehrlich-Aberth method}
+%% In the following, we describe the parallel implementation of Ehrlich-Aberth method on GPU 
+%% for solving high degree polynomials (up to $1,000,000$). First, the hardware and software of the GPUs are presented. Then, the CUDA parallel Ehrlich-Aberth method is presented.
+
+%% \subsection{Background on the GPU architecture}
+%% A GPU is viewed as an accelerator for the data-parallel and
+%% intensive arithmetic computations. It draws its computing power
+%% from the parallel nature of its hardware and software
+%% architectures. A GPU is composed of hundreds of Streaming
+%% Processors (SPs) organized in several blocks called Streaming
+%% Multiprocessors (SMs). It also has a memory hierarchy. It has a
+%% private read-write local memory per SP, fast shared memory and
+%% read-only constant and texture caches per SM and a read-write
+%% global memory shared by all its SPs~\cite{NVIDIA10}.
+
+%% On a CPU equipped with a GPU, all the data-parallel and intensive
+%% functions of an application running on the CPU are off-loaded onto
+%% the GPU in order to accelerate their computations. A similar
+%% data-parallel function is executed on a GPU as a kernel by
+%% thousands or even millions of parallel threads, grouped together
+%% as a grid of thread blocks. Therefore, each SM of the GPU executes
+%% one or more thread blocks in SIMD fashion (Single  Instruction,
+%% Multiple Data) and in turn each SP of a GPU SM runs one or more
+%% threads within a block in SIMT fashion (Single Instruction,
+%% Multiple threads). Indeed at any given clock cycle, the threads
+%% execute the same instruction of a kernel, but each of them
+%% operates on different data.
+%%  GPUs only work on data filled in their
+%% global memories and the final results of their kernel executions
+%% must be communicated to their CPUs. Hence, the data must be
+%% transferred in and out of the GPU. However, the speed of memory
+%% copy between the GPU and the CPU is slower than the memory
+%% bandwidths of the GPU memories and, thus, it dramatically affects
+%% the performances of GPU computations. Accordingly, it is necessary
+%% to limit as much as possible, data transfers between the GPU and its CPU during the
+%% computations.
+%% \subsection{Background on the CUDA Programming Model}
+
+%% The CUDA programming model is similar in style to a single program
+%% multiple-data (SPMD) software model. The GPU is viewed as a
+%% coprocessor that executes data-parallel kernel functions. CUDA
+%% provides three key abstractions, a hierarchy of thread groups,
+%% shared memories, and barrier synchronization. Threads have a three
+%% level hierarchy. A grid is a set of thread blocks that execute a
+%% kernel function. Each grid consists of blocks of threads. Each
+%% block is composed of hundreds of threads. Threads within one block
+%% can share data using shared memory and can be synchronized at a
+%% barrier. All threads within a block are executed concurrently on a
+%% multithreaded architecture.The programmer specifies the number of
+%% threads per block, and the number of blocks per grid. A thread in
+%% the CUDA programming language is much lighter weight than a thread
+%% in traditional operating systems. A thread in CUDA typically
+%% processes one data element at a time. The CUDA programming model
+%% has two shared read-write memory spaces, the shared memory space
+%% and the global memory space. The shared memory is local to a block
+%% and the global memory space is accessible by all blocks. CUDA also
+%% 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.
+
+\section{ Implementation of the Ehrlich-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}
-The main steps of Aberth method are shown in Algorithm.~\ref{alg1-seq} :
-  
-\begin{algorithm}[H]
-\label{alg1-seq}
-%\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)}
+%% \subsection{Sequential Ehrlich-Aberth algorithm}
+%% The main steps of Ehrlich-Aberth method are shown in Algorithm.~\ref{alg1-seq} :
+%% %\LinesNumbered  
+%% \begin{algorithm}[H]
+%% \label{alg1-seq}

 
 

-\KwOut {Z(The solution root's vector)}
+%% \caption{A sequential algorithm to find roots with the Ehrlich-Aberth method}

 
 

-\BlankLine
+%% \KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (error tolerance
+%%   threshold), $P$ (Polynomial to solve),$Pu$ (the derivative of P) $\Delta z_{max}$ (maximum value
+%%   of stop condition), k (number of iteration), n (Polynomial's degrees)}
+%% \KwOut {$Z$ (The solution root's vector), $ZPrec$ (the previous solution root's vector)}

 
 

-Initialization of the coefficients of the polynomial to solve\;
-Initialization of the solution vector $Z^{0}$\;
+%% \BlankLine

 
 

-\While {$\Delta z_{max}\succ \epsilon$}{
- Let $\Delta z_{max}=0$\;
-\For{$j \gets 0 $ \KwTo $n$}{
-$ZPrec\left[j\right]=Z\left[j\right]$\;
-$Z\left[j\right]=H\left(j,Z\right)$\;
-}
+%% Initialization of $P$\;
+%% Initialization of $Pu$\;
+%% Initialization of the solution vector $Z^{0}$\;
+%% $\Delta z_{max}=0$\;
+%%  k=0\;

 
 

-\For{$i \gets 0 $ \KwTo $n-1$}{
-$c=\frac{\left|Z\left[i\right]-ZPrec\left[i\right]\right|}{Z\left[i\right]}$\;
-\If{$c\succ\Delta z_{max}$ }{
-$\Delta z_{max}$=c\;}
-}
-}
-\end{algorithm}
+%% \While {$\Delta z_{max} > \varepsilon$}{
+%%  Let $\Delta z_{max}=0$\;
+%% \For{$j \gets 0 $ \KwTo $n$}{
+%% $ZPrec\left[j\right]=Z\left[j\right]$;// save Z at the iteration k.\

 
 

-~\\ 
-In this sequential algorithm, one CPU thread  executes all the steps. Let us look to the $3^{rd}$ step i.e. the execution of the iterative function, 2 sub-steps are needed. The first sub-step \textit{save}s the solution vector of the previous iteration, the second sub-step \textit{update}s or computes the new values of the roots vector.
-There exists two ways to execute the iterative function that we call a Jacobi one and a Gauss-Seidel one. With the Jacobi iteration, at iteration $k+1$ we need all the previous values $z^{(k)}_{i}$ to compute the new values $z^{(k+1)}_{i}$, that is :
+%% $Z\left[j\right]=H\left(j, Z, P, Pu\right)$;//update Z with the iterative function.\
+%% }
+%% k=k+1\;

 
 

-\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.
-\end{equation}
+%% \For{$i \gets 0 $ \KwTo $n-1$}{
+%% $c= testConverge(\Delta z_{max},ZPrec\left[j\right],Z\left[j\right])$\;
+%% \If{$c > \Delta z_{max}$ }{
+%% $\Delta z_{max}$=c\;}
+%% }

 
 

-With the Gauss-seidel iteration, we have:
-\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.
-\end{equation}
+%% }
+%% \end{algorithm}

 
 

-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 $z^{k+1}_{i}$.
+%% ~\\ 
+%% In this sequential algorithm, one CPU thread  executes all the steps. Let us look to the $3^{rd}$ step i.e. the execution of the iterative function, 2 sub-steps are needed. The first sub-step \textit{save}s the solution vector of the previous iteration, the second sub-step \textit{update}s or computes the new values of the roots vector.

 
 

-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.
+\subsection{Parallel implementation with CUDA }

 
 

-\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.
+In order to implement the Ehrlich-Aberth method in CUDA, it is
+possible to use the Jacobi scheme or the Gauss-Seidel one.  With the
+Jacobi iteration, at iteration $k+1$ we need all the previous values
+$z^{k}_{i}$ to compute the new values $z^{k+1}_{i}$, that is :

 
 

-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}
 \begin{equation}

-T_{iter}=n(T_{i}(n)+T_{j})+O(n).
+EAJ: 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=1,. . . .,n.

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

 
 \begin{equation}
 
 \begin{equation}

-\label{eq:T-global}
-T=\left[n\left(T_{i}(n)+T_{j}\right)+O(n)\right].K
+\label{eq:Aberth-H-GS}
+EAGS: 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^{i-1}_{j=1}\frac{1}{z^{k}_{i}-z^{k+1}_{j}}+\sum_{j=i+1}^{j=n}{\frac{1}{(z_{i}^{k}-z_{j}^{k})}})}, i=1,. . . .,n

 \end{equation}
 \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 }
-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.
+Using Eq.~\ref{eq:Aberth-H-GS} to update the vector solution
+\textit{Z}, we expect the Gauss-Seidel iteration to converge more
+quickly because, just as any Jacobi algorithm (for solving linear
+systems of equations), it uses the freshest computed roots $z^{k+1}_{i}$.
+
+%The $4^{th}$ step of the algorithm checks the convergence condition using Eq.~\ref{eq:Aberth-Conv-Cond}.
+%Both steps 3 and 4 use 1 thread to compute all the $n$ roots on CPU, which is very harmful for performance in case of the large degree polynomials.

 
 

-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. 
+%On the CPU,  both steps 3 and 4 contain the loop \verb=for= and a single thread executes all the instructions in the loop $n$ times. In this subsection, we explain how the GPU architecture can compute this loop and reduce the execution time.
+%In the GPU, the scheduler assigns the execution of this loop to a
+%group of threads organised as a grid of blocks with block containing a
+%number of threads. All threads within a block are executed
+%concurrently in parallel. The instructions run on the GPU are grouped
+%in special function called kernels. With CUDA, a programmer must
+%describe the kernel execution context:  the size of the Grid, the number of blocks and the number of threads per block.

 
 

-Algorithm~\ref{alg2-cuda} shows a sketch of the Aberth algorithm usind CUDA.
+%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 Ehrlich-Aberth method using CUDA.
+
+\begin{enumerate}

 \begin{algorithm}[H]
 \label{alg2-cuda}
 %\LinesNumbered
 \begin{algorithm}[H]
 \label{alg2-cuda}
 %\LinesNumbered

-\caption{CUDA Algorithm to find roots with the Aberth method}
+\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)}
+\KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (Error tolerance
+  threshold), P (Polynomial to solve), Pu (Derivative of P), $n$ (Polynomial degrees), $\Delta z_{max}$ (Maximum value of stop condition)}

 
 

-\KwOut {Z(The solution root's vector)}
+\KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}

 
 \BlankLine
 
 
 \BlankLine
 

-Initialization of the coeffcients of the polynomial to solve\;
-Initialization of the solution vector $Z^{0}$\;
-Allocate and copy initial data to the GPU global memory\;
+\item Initialization of the of P\;
+\item Initialization of the of Pu\;
+\item Initialization of the solution vector $Z^{0}$\;
+\item Allocate and copy initial data to the GPU global memory\;
+\item k=0\;
+\While {$\Delta z_{max} > \epsilon$}{
+\item Let $\Delta z_{max}=0$\;
+\item $ kernel\_save(ZPrec,Z)$\;
+\item  k=k+1\;
+\item $ kernel\_update(Z,P,Pu)$\;
+\item $kernel\_testConverge(\Delta z_{max},Z,ZPrec)$\;

 
 

-\While {$\Delta z_{max}\succ \epsilon$}{
- Let $\Delta z_{max}=0$\;
-$ kernel\_save(d\_z^{k-1})$\;
-$ kernel\_update(d\_z^{k})$\;
-$kernel\_testConverge(\Delta z_{max},d_z^{k},d_z^{k-1})$\;

 }
 }

+\item Copy results from GPU memory to CPU memory\;

 \end{algorithm}
 \end{algorithm}

+\end{enumerate}

 ~\\ 
 
 ~\\ 
 

-After the initialisation 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=(do ... 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 $z^{k}$, according to Algorithm~\ref{alg3-update}. We notice that the update kernel is called in two forms, separated with the value of \emph{R} which determines the radius beyond which we apply the logarithm computation of the power of a complex. 
+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 7 of
+Algorithm~\ref{alg2-cuda} consists in saving the vector of
+polynomial roots found at the previous time-step in GPU memory, in
+order to check the convergence of the roots after each iteration (line
+10, Algorithm~\ref{alg2-cuda}).
+
+The second kernel executes the iterative function and updates
+$Z$, 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
 
 \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{$(\left|Z\right|<= R)$}{
+$kernel\_update(Z,P,Pu)$\;}

 {
 {

-$kernel\_update\_Log(d\_z^{k})$\;
+$kernel\_update\_ExpoLog(Z,P,Pu)$\;

 }
 \end{algorithm}
 
 }
 \end{algorithm}
 

-The first form executes formula \ref{eq:SimplePolynome} 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 formulas (Eq.~\ref{deflncomplex},Eq.~\ref{defexpcomplex}). The radius $R$ is evaluated as :
+If the modulus
+of the current complex is less than a given value called the
+radius i.e. ($ |z^{k}_{i}|<= R$), then the classical form of the EA
+function Eq.~\ref{Eq:Hi} is executed else the EA.EL
+function Eq.~\ref{Log_H2} is executed.
+(with Eq.~\ref{deflncomplex}, Eq.~\ref{defexpcomplex}). The radius $R$ is evaluated as in Eq.~\ref{R.EL}.
+
+The last kernel checks the convergence of the roots after each update
+of $Z^{k}$, according to formula Eq.~\ref{eq:Aberth-Conv-Cond}. We used the functions of the CUBLAS Library (CUDA Basic Linear Algebra Subroutines) to implement this kernel. 
+
+The kernel terminates its computations when all the roots have
+converged. It should be noticed that, as blocks of threads are
+scheduled automatically by the GPU, we have absolutely no control on
+the order of the blocks. Consequently, our algorithm is executed more
+or less with the asynchronous iteration model, where blocks of roots
+are updated in a non deterministic way. As the Durand-Kerner method
+has been proved to converge with asynchronous iterations, we think it
+is similar with the Ehrlich-Aberth method, but we did not try to prove
+this in that paper. Another consequence of that, is that several
+executions of our algorithm with the same polynomial do not
+necessarily give the same result (but roots have the same accuracy)
+and the same number of iterations (even if the variation is not very
+significant).
+
+

 
 

-$$R = \exp( \log(DBL\_MAX) / (2*n) )$$ where $DBL\_MAX$ stands for the maximum representable double value.

 
 

-The last kernel verifies the convergence of the roots after each update of $Z^{(k)}$, according to formula. We used the functions of the CUBLAS Library (CUDA Basic Linear Algebra Subroutines) to implement this kernel. 

 
 

-The kernels terminate it computations when all the roots converge. Finally, the solution of the root finding problem is copied back from GPU global memory to CPU memory. We use the communication functions of CUDA for the memory allocation in the GPU \verb=(cudaMalloc())= and for data transfers from the CPU memory to the GPU memory \verb=(cudaMemcpyHostToDevice)=
-or from GPU memory to CPU memory \verb=(cudaMemcpyDeviceToHost))=. 

 %%HIER END MY REVISIONS (SIDER)
 \section{Experimental study}
 %%HIER END MY REVISIONS (SIDER)
 \section{Experimental study}

-
-\subsection{Definition of the used polynomials }
-We study two categories of polynomials : the sparse polynomials and the full polynomials.
-\paragraph{A sparse polynomial}: is a polynomial for which only some coefficients are not null. We use in the following polonymial for which the roots are distributed on 2 distinct circles :
+\label{sec6}
+%\subsection{Definition of the used polynomials }
+We study two categories of polynomials: sparse polynomials and the full polynomials.\\
+{\it A sparse polynomial} is a polynomial for which only some
+coefficients are not null. In this paper, we consider sparse polynomials for which the roots are distributed on 2 distinct circles:

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

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

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


@@ -626,100 +695,182 @@ We study two categories of polynomials : the sparse polynomials and the full pol
 \begin{equation}
      {\Large \forall a_{i} \in C, i\in N;  p(x)=\sum^{n}_{i=0} a_{i}.x^{i}} 
 \end{equation}
 \begin{equation}
      {\Large \forall a_{i} \in C, i\in N;  p(x)=\sum^{n}_{i=0} a_{i}.x^{i}} 
 \end{equation}

-With this form, we can have until \textit{n} non zero terms whereas the sparse ones have just two non zero terms.
-
-\subsection{The study condition} 
-The our experiences results concern two parameters which are
-the polynomial degree and the execution time of our program
-to converge on the solution. The polynomial degree allows us
-to validate that our algorithm is powerful with high degree
-polynomials. The execution time remains the
-element-key which justifies our work of parallelization.
-       For our tests we used a CPU Intel(R) Xeon(R) CPU
-E5620@2.40GHz and a GPU K40 (with 6 Go of ram).
-
-
-\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. 
-\subsubsection{The execution time in seconds of Ehrlich-Aberth algorithm on CPU core vs. on a Tesla GPU}
-
-
-\begin{figure}[H]
+%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} 
+%Two parameters are studied are
+%the polynomial degree and the execution time of our program
+%to converge on the solution. The polynomial degree allows us
+%to validate that our algorithm is powerful with high degree
+%polynomials. The execution time remains the
+%element-key which justifies our work of parallelization.
+For our tests, a CPU Intel(R) Xeon(R) CPU
+E5620@2.40GHz and a GPU K40 (with 6 Go of ram) are used. 
+
+
+%\subsection{Comparative study}
+%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 of 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
+to $10^{-7}$.
+%Since we were more interested in the comparison of the
+%performance behaviors of Ehrlich-Aberth and Durand-Kerner methods on
+%CPUs versus on GPUs.
+The initialization values of the vector solution
+of the methods are given in Section~\ref{sec:vec_initialization}.
+
+\subsection{Comparison of execution times of the Ehrlich-Aberth method
+  on a CPU with OpenMP (1 core and 4 cores) vs. on a Tesla GPU}
+
+\begin{figure}[htbp]

 \centering
 \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{Comparison of execution times  of the Ehrlich-Aberth method
+  on a CPU with OpenMP (1 core, 4 cores) and on a Tesla GPU}

 \label{fig:01}
 \end{figure}
 \label{fig:01}
 \end{figure}

+%%Figure 1 %%show a comparison of execution time between the parallel
+%%and sequential version of the Ehrlich-Aberth algorithm with sparse
+%%polynomial exceed 100000,
+
+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 K40 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. On the other hand, the GPU implementation for the same
+polynomials do not take more 100s. With the GPU
+we can solve high degree polynomials very quickly up to degree 1,000,000. We can also notice that the GPU implementation are
+ almost 40 times faster then the implementation on the CPU (4 cores).
+

 
 

-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 methods 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 sequential implementation exceed 16,000 s for 450,000 degrees polynomials, in counterpart  the GPU implementation for the same polynomials need only 350 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. 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.

  
 
  
 

+%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)}
-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.
+\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). 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 noticed 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
+10 different sparse and full polynomials of degree 50,000 and of degree 500,000.

 
 

-\begin{figure}[H]
+\begin{figure}[htbp]

 \centering
   \includegraphics[width=0.8\textwidth]{figures/influence_nb_threads}
 \caption{Influence of the number of threads on the execution times of different polynomials (sparse and full)}
 \centering
   \includegraphics[width=0.8\textwidth]{figures/influence_nb_threads}
 \caption{Influence of the number of threads on the execution times of different polynomials (sparse and full)}

-\label{fig:01}
+\label{fig:02}

 \end{figure}
 
 \end{figure}
 

-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.
+Figure~\ref{fig:02} shows that, the best execution time for both
+sparse and full polynomial are given when the threads number varies
+between 64 and 256 threads per block. We notice that with small
+polynomials the best number of threads per block is 64, whereas for large polynomials the best number of threads per block is
+256. However, in the following experiments we specify that the number
+of threads per block is 256.
+

 
 

-\subsubsection{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.   
-\begin{figure}[H]
+In this experiment we report the performance of the exp-log solution described in Section~\ref{sec2} to compute high degree polynomials.   
+\begin{figure}[htbp]

 \centering
   \includegraphics[width=0.8\textwidth]{figures/sparse_full_explog}
 \centering
   \includegraphics[width=0.8\textwidth]{figures/sparse_full_explog}

-\caption{The impact of exp-log solution to compute very high degrees of  polynomial.}
-\label{fig:01}
+\caption{The impact of exp-log solution to compute  high degree  polynomials}
+\label{fig:03}

 \end{figure}
 
 \end{figure}
 

-The figure 3, show a comparison between the execution time of the Ehrlich-Aberth algorithm applying log-exp solution and the execution time of the Ehrlich-Aberth algorithm without applying log-exp 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
+degree inferior to 4,000 and sparse polynomials inferior to 150,000. We
+also clearly show that the classical version (without exp-log) of
+Ehrlich-Aberth algorithm does not converge after these degrees with
+sparse and full polynomials. On the contrary, the new version of
+the Ehrlich-Aberth algorithm with the exp-log solution can solve
+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 . 

 
 

-%\begin{figure}[H]
-\%centering
-  %\includegraphics[width=0.8\textwidth]{figures/log_exp_Sparse}
-%\caption{The impact of exp-log solution to compute very high degrees of  polynomial.}
-%\label{fig:01}
-%\end{figure}

 
 

-%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}
-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]
+\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 both sparse and full polynomials.  
+
+\begin{figure}[htbp]

 \centering
   \includegraphics[width=0.8\textwidth]{figures/EA_DK}
 \centering
   \includegraphics[width=0.8\textwidth]{figures/EA_DK}

-\caption{The execution time of Ehrlich-Aberth versus Durand-Kerner algorithm on GPU}
-\label{fig:01}
+\caption{Execution times of the  Durand-Kerner and the Ehrlich-Aberth methods on GPU}
+\label{fig:04}

 \end{figure}
 
 \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, being on average 25 times faster. Then, when degrees of
+polynomials exceed 500,000 the execution times with DK are very long.
+
+%with double precision not exceed $10^{-5}$.

 
 

-\begin{figure}[H]
+\begin{figure}[htbp]

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

-\caption{The iteration number of Ehrlich-Aberth versus Durand-Kerner algorithm}
-\label{fig:01}
+\caption{The number of iterations to converge for the Ehrlich-Aberth
+  and the Durand-Kerner methods}
+\label{fig:05}

 \end{figure}
 
 \end{figure}
 

-\bibliography{mybibfile}
+Figure~\ref{fig:05} shows the evaluation of the number of iterations according
+to the degree of polynomials for both EA and DK algorithms. We can see
+that the number of iterations of DK is of order 100 while EA is of order
+10. Indeed the computation of the derivative of P in the iterative function (Eq.~\ref{Eq:Hi}) executed by EA
+allows the algorithm to converge faster. On the contrary, the
+DK operator (Eq.~\ref{DK}) needs low operations, consequently low
+execution times per iteration, but it needs more iterations to converge.

 
 
 
 

-\section{Conclusion and perspective}
+
+
+ \section{Conclusion and perspectives}
+\label{sec7}
+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 more efficient in
+GPU than 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  using 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 a multi-GPU machine or
+with a cluster of GPUs. It may also be interesting to study the
+implementation of other root finding polynomial methods on GPU.
+
+
+
+\bibliography{mybibfile}

 
 \end{document}
 
 \end{document}