X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/kahina_paper1.git/blobdiff_plain/761635b66f95dbbc02078e5129bed61f2b41a289..1245c0d9665437ae2a44d0393333555b940e4587:/paper.tex?ds=sidebyside

diff --git a/paper.tex b/paper.tex
index e6a1e37..1ba7a17 100644
--- a/paper.tex
+++ b/paper.tex
@@ -1,12 +1,13 @@
 \documentclass[review]{elsarticle}
 
 \usepackage{lineno,hyperref}
-\usepackage[utf8]{inputenc}
+%%\usepackage[utf8]{inputenc}
 %%\usepackage[T1]{fontenc}
 %%\usepackage[french]{babel}
-
+\usepackage{float} 
 \usepackage{amsmath,amsfonts,amssymb}
 \usepackage[ruled,vlined]{algorithm2e}
+%\usepackage[french,boxed,linesnumbered]{algorithm2e}
 \usepackage{array,multirow,makecell}
 \setcellgapes{1pt}
 \makegapedcells
@@ -53,12 +54,12 @@
 
 \begin{frontmatter}
 
-\title{A parallel  root finding polynomial on GPU}
+\title{Rapid solution of very high degree polynomials root finding using GPU}
 
 %% Group authors per affiliation:
-%\author{Elsevier\fnref{myfootnote}}
-%\address{Radarweg 29, Amsterdam}
-%\fntext[myfootnote]{Since 1880.}
+\author{Elsevier\fnref{myfootnote}}
+\address{Radarweg 29, Amsterdam}
+\fntext[myfootnote]{Since 1880.}
 
 %% or include affiliations in footnotes:
 \author[mymainaddress]{Ghidouche Kahina\corref{mycorrespondingauthor}}
@@ -66,7 +67,7 @@
 \cortext[mycorrespondingauthor]{Corresponding author}
 \ead{kahina.ghidouche@gmail.com}
 
-\author[mysecondaryaddress]{Couturier RaphaÃ«l\corref{mycorrespondingauthor}}
+\author[mysecondaryaddress]{Couturier Raphael\corref{mycorrespondingauthor}}
 %%\cortext[mycorrespondingauthor]{Corresponding author}
 \ead{raphael.couturier@univ-fcomte.fr}
 
@@ -74,150 +75,150 @@
 %%\cortext[mycorrespondingauthor]{Corresponding author}
 \ead{ar.sider@univ-bejaia.dz}
 
-\address[mymainaddress]{Department of informatics, University of
-  BÃ©jaia, Algeria}
-\address[mysecondaryaddress]{FEMTO-ST Institute, University of
-  Bourgogne Franche-Comte }
+\address[mymainaddress]{Department of informatics,University of Bejaia,Algeria}
+\address[mysecondaryaddress]{FEMTO-ST Institute, University of Franche-ComptÃ© }
 
 \begin{abstract}
-in this article we present a parallel implementation
-of the Aberth algorithm for the problem root finding for
-high degree polynomials on GPU architecture (Graphics
-Processing Unit).
+Polynomials are mathematical algebraic structures that play a great role in science and engineering. But the process of solving them  for high and large degrees is computationally demanding and still not solved. In this paper, we present the results of a parallel implementation of the Ehrlich-Aberth algorithm for the problem root finding for
+high degree polynomials on GPU architectures (Graphics Processing Unit). The main result of this work is to be able to solve high and very large degree polynomials (up to 100000) very efficiently. We also compare the results with a sequential implementation and the Durand-Kerner method on full and sparse polynomials.
 \end{abstract}
 
 \begin{keyword}
-root finding of polynomials, high degree, iterative methods, Durant-Kerner, GPU, CUDA, CPU , Parallelization
+root finding of polynomials, high degree, iterative methods, Ehrlich-Aberth, Durant-Kerner, GPU, CUDA, CPU , Parallelization
 \end{keyword}
 
 \end{frontmatter}
 
 \linenumbers
 
-\section{Root finding problem}
-We consider a polynomial of degree \textit{n} having coefficients
-in the complex \textit{C} and zeros $\alpha_{i},\textit{i=1,...,n}$. 
+\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 :
 %%\begin{center}
 \begin{equation}
-     {\Large p(x)=\sum{a_{i}x^{i}}=a_{n}\prod(x-\alpha_{i}),a_{0} a_{n}\neq 0}
+     {\Large p(x)=\sum_{i=0}^{n}{a_{i}x^{i}}}.
 \end{equation}
 %%\end{center}
 
- The root finding problem consist to find
-all n root of \textit{p(x)}. the problem of finding a root is
-equivalent to the problem of finding a fixed-point. To see this
-consider the fixed-point problem of finding the n-dimensional
-vector x such that
+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 :
+\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
 \begin{center}
-$x=g(x).  $
+$x=g(x)$
 \end{center}
-Where $g : C^{n}\longrightarrow C^{n}$. Note that we can easily
+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
+setting $f(x) = x-g(x)$ and likewise we can recast the
 root-finding problem into a fixed-point problem by setting
 \begin{center}
-$g(x)= f(x)-x$
+$g(x)= f(x)-x$.
 \end{center}
-Often it will not be possible to solve such nonlinear equation
+
+Often it is not be possible 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 numerically can be divided into
+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
+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  directs methods. Since then researchers have
-concentrated on numerical (iterative) methods such as the famous
+be solved by  directs methods. Since then, mathmathicians have
+focussed on numerical (iterative) methods such as the famous
 Newton's method, Bernoulli's method of the 18th, and Graeffe's.
-With the advent of electronic computers, different methods has
-been developed such as the Jenkins-Traub method, Larkin s method,
+
+Later on, with the advent of electronic computers, other methods has
+been developed such as the Jenkins-Traub method, Larkin's method,
 Muller's method, and several methods for simultaneous
-approximation of all the roots, starting with the Durand-Kerner
-method:
+approximation of all the roots, starting with the Durand-Kerner (DK)
+method :
 %%\begin{center}
 \begin{equation}
  Z_{i}=Z_{i}-\frac{P(Z_{i})}{\prod_{i\neq j}(z_{i}-z_{j})}
 \end{equation}
 %%\end{center}
 
-This formula is mentioned for the first time from
+This formula is mentioned for the first time by
 Weiestrass~\cite{Weierstrass03} as part of the fundamental theorem
-of Algebra and is rediscovered from Ilieff~\cite{Ilie50},
+of Algebra and is rediscovered by Ilieff~\cite{Ilie50},
 Docev~\cite{Docev62}, Durand~\cite{Durand60},
-Kerner~\cite{Kerner66}. Another method discovered from
+Kerner~\cite{Kerner66}. Another method discovered by
 Borsch-Supan~\cite{ Borch-Supan63} and also described and brought
-in the following form from Ehrlich~\cite{Ehrlich67} and
-Aberth~\cite{Aberth73}.
+in the following form by Ehrlich~\cite{Ehrlich67} and
+Aberth~\cite{Aberth73} uses a different iteration formula given as fellows :
 %%\begin{center}
 \begin{equation}
- Z_{i}=Z_{i}-\frac{1}{{\frac {P'(Z_{i})} {P(Z_{i})}}-{\sum_{i\neq j}(z_{i}-z_{j})}}
+ Z_{i}=Z_{i}-\frac{1}{{\frac {P'(Z_{i})} {P(Z_{i})}}-{\sum_{i\neq j}(z_{i}-z_{j})}}.
 \end{equation}
 %%\end{center}
 
 Aberth, Ehrlich and Farmer-Loizou~\cite{Loizon83} have proved that
-the above method has cubic order of convergence for simple roots.
+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
-drastically increase like the degrees of high polynomials. The
+difficulty. Moreover, the convergence time of iterative methods
+drastically increases like the degrees of high polynomials. It is expected that the
 parallelization of these algorithms will improve the convergence
 time.
 
-Many authors have treated the problem of parallelization of
-simultaneous methods. Freeman~\cite{Freeman89} has tested the DK
-method, EA method and another method of the fourth order proposed
-from Farmer and Loizou~\cite{Loizon83},on a 8- processor linear
-chain, for polynomial of degree up to 8. The third method often
+Many authors have dealt with the parallelisation of
+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 and Bane~\cite{Freemanall90}  consider asynchronous
+(speed-up=(Time on one processor)/(Time on p processors)). Later,
+Freeman and Bane~\cite{Freemanall90}  considered asynchronous
 algorithms, in which each processor continues to update its
-approximations even although the latest values of other $z_i((k))$
-have not received from the other processors, in difference with
-the synchronous version where it would wait.
-in~\cite{Raphaelall01}proposed two methods of parallelization for
-architecture with shared memory and distributed memory,it able to
-compute the root of polynomial degree  10000 on 430 s with only 8
-pc and 2 communications per iteration. Compare to the sequential
-it take 3300 s to obtain the same results.
-
-After this few works discuses this problem until the apparition of
-the Compute Unified Device Architecture (CUDA)~\cite{CUDA10},a
+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
+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.
+
+Very few works had been 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
-NVIDIA. The computing ability of GPU has exceeded the counterpart
-of CPU. It is a waste of resource to be just a graphics card for
-GPU. CUDA adopts a totally new computing architecture to use the
+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
 hardware resources provided by GPU in order to offer a stronger
 computing ability to the massive data computing.
 
 
-Indeed,~\cite{Kahinall14}proposed the implementation of the
-Durand-Kerner method on GPU (Graphics Processing Unit). The main
-result prove that a parallel implementation is 10 times as fast as
+Ghidouche et al. ~\cite{Kahinall14} proposed an implementation of the
+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 that is greater than about 48000.
-\paragraph{}
-The mean part of our work is to implement the Aberth method for the problem root finding for
-high degree polynomials on GPU architecture (Graphics Processing Unit). Initially we present the Aberth method in section 1. Amelioration of Aberth method was proposed in section 2. A related works for the implementation of simultaneous methods in a parallel computer was discuss in section 3. Section 4 we propose a parallel implementation of Aberth method on GPU. Section 5, we present our result and discuss it. Finally, in Section 6, we present our conclusions and future research directions.  
+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{Aberth method}
+\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 factor 
+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}
 
-And rational function $R_{i}(z)$ be the correction term of
-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
+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
@@ -227,32 +228,33 @@ Newton method, we have:
 \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
+Substituting $x_{j}$ for z we obtain the Aberth iteration method.
 
-Let present the means stages of Aberth method.
+In the fellowing we present the main stages of the running of the Aberth method.
 
 \subsection{Polynomials Initialization}
- The initialization of polynomial P(z) with complex coefficients
- are given by:
+The initialization of a polynomial p(z) is done by setting each of the $n$ complex coefficients $a_{i}$
+:
 
 \begin{equation}
+\label{eq:SimplePolynome}
   p(z)=\sum{a_{i}z^{n-i}} , a_{n} \neq 0,a_{0}=1, a_{i}\subset C
 \end{equation}
 
 
-\subsection{Vector $Z^{(0)}$ Initialization}
+\subsection{Vector $z^{(0)}$ Initialization}
 
-The choice of the initial points $z^{(0)}_{i}, i = 1, . . . , n.$
-from which starting the iteration  (2) or (3), is rather delicate
-since the number of steps needed by the iterative method to reach
+Like for any iterative method, we need to choose $n$ initial guess points $z^{(0)}_{i}, i = 1, . . . , n.$
+The initial guess is very important since the number of steps needed by the iterative method to reach
 a given approximation strongly depends on it.
-In~\cite{Aberth73}the Aberth iteration is started by selecting n
-equispaced points on a circle of center 0 and radius r, where r is
-an upper bound to the moduli of the zeros. After,~\cite{Bini96}
-performs this choice by selecting complex numbers along different
+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}
+performed this choice by selecting complex numbers along different
 circles and relies on the result of~\cite{Ostrowski41}.
 
 \begin{equation}
+\label{eq:radiusR}
 %%\begin{align}
 \sigma_{0}=\frac{u+v}{2};u=\frac{\sum_{i=1}^{n}u_{i}}{n.max_{i=1}^{n}u_{i}};
 v=\frac{\sum_{i=0}^{n-1}v_{i}}{n.min_{i=0}^{n-1}v_{i}};\\
@@ -265,127 +267,133 @@ v_{i}=\frac{|\frac{a_{n}}{a_{i}}|^{\frac{1}{n-i}}}{2}.
 \end{equation}
 
 \subsection{Iterative Function $H_{i}$}
-The operator used with Aberth method is corresponding to the
+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.
 
 \begin{equation}
-H_{i}(z)=z_{i}-\frac{1}{\frac{P^{'}(z_{i})}{P(z_{i})}-\sum_{j\neq
+H_{i}(z)=z_{i}-\frac{1}{\frac{p^{'}(z_{i})}{p(z_{i})}-\sum_{j\neq
 i}{\frac{1}{z_{i}-z_{j}}}}
 \end{equation}
 
-\subsection{Convergence condition}
-Determines the success of the termination. It consists in stopping
-the iterative function $H_{i}(z)$ when the root are stable, the method
-converge sufficiently:
+\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 :
 
 \begin{equation}
+\label{eq:Aberth-Conv-Cond}
 \forall i \in
 [1,n];\frac{z_{i}^{(k)}-z_{i}^{(k-1)}}{z_{i}^{(k)}}<\xi
 \end{equation}
 
 
-\section{Amelioration of Aberth method }
-The Aberth method implementation suffer of overflow problems. This
+\section{Improving the Ehrlich-Aberth Method}
+\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 large degree is computed at a
-point $\xi$ where $|\xi| > 1$. Indeed the limited number in the
-mantissa of floating takings the computation of P(z) wrong when z
-is large. for example $(10^{50}) +1+ (- 10^{50})$ will give result
-0 instead of 1 in reality. Consequently we can not compute the roots
-for large polynomial's degree. This problem was discuss in
+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
+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
-propose to use the logarithm and the exponential of a complex:
+propose to use the logarithm and the exponential of a complex in order to compute the power at a high exponent.
 
 \begin{equation}
+\label{deflncomplex}
  \forall(x,y)\in R^{*2}; \ln (x+i.y)=\ln(x^{2}+y^{2})
 2+i.\arcsin(y\sqrt{x^{2}+y^{2}})_{\left] -\pi, \pi\right[ }
 \end{equation}
 %%\begin{equation}
 \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)
+                                     & =\exp(x).\cos(y)+i.\exp(x).\sin(y)\label{defexpcomplex}
 \end{align}
 %%\end{equation}
 
-The application of logarithm can replace any multiplications and
-divisions with additions and subtractions. Consequently, it
-manipulates lower absolute values and can be compute the roots for
-large polynomial's degree exceed~\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
 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)
+\left(1-Q(z_{k})\right)\right),
 \end{equation}
-Where:
+
+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)
+\sum_{k\neq j}^{n}\frac{1}{z_{k}-z_{j}}\right)\right).
 \end{equation}
 
-This solution is applying when it is necessary
+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.
 
 \section{The implementation of simultaneous methods in a parallel computer}
-    The main problem of the simultaneous methods is that the necessary
-time needed for the convergence is increased with the increasing
-of the degree of the polynomial. The parallelization of these
-algorithms will improve the convergence time. Researchers usually
-adopt one of the two following approaches to parallelize root
-finding algorithms. One approach is to reduce the total number of
-iterations as implemented by Miranker
+\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
+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
 ~\cite{Mirankar68,Mirankar71}, Schedler~\cite{Schedler72} and
-Winogard~\cite{Winogard72}. Another approach is to reduce the
+Winogard~\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 simultaneous approximations of all roots of a given
+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 methods are the most practical choices among
+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
 them~\cite{Bini04}. These two methods have been extensively
-studied for parallelization due to their following advantages. The
-computation involved in these methods has some inherent
+studied for parallelization due to their intrinsics, i.e. the
+computations involved in both methods has some inherent
 parallelism that can be suitably exploited by SIMD machines.
 Moreover, they have fast rate of convergence (quadratic for the
-Durand-Kerner method and cubic for the Ehrlich). Various parallel
+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}.
+in~\cite{Cosnard90, Freeman89,Freemanall90,Jana99,Janall99}.
 Freeman and Bane~\cite{Freemanall90} presented two parallel
 algorithms on a local memory MIMD computer with the compute-to
 communication time ratio O(n). However, their algorithms require
 each processor to communicate its current approximation to all
-other processors at the end of each iteration. Therefore they
+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 Aberth method on model of
+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 need N processors to compute N roots, that it is not
-practical (is not suitable to compute large polynomial's degrees).
-Until then, the related works are not able to compute the root of
-the large polynomial's degrees (higher then 1000) and with small
-time.
-
-    Finding polynomial roots rapidly and accurately it is our
-objective, with the apparition of the CUDA(Compute Unified Device
-Architecture), finding the roots of polynomials becomes rewarding
-and very interesting, 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. In~\cite{Kahinall14} we proposed the first implantation
-of the root finding polynomials method on GPU (Graphics Processing
-Unit),which is the Durand-Kerner method. The main result prove
-that a parallel implementation is 10 times as fast as the
+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. 
+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 that is greater than about 48000. Indeed, in this
-paper we present a parallel implementation of Aberth method on
-GPU, more details are discussed in the following of this paper.
+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 parallel implementation of Aberth method}
+\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
@@ -396,9 +404,9 @@ 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}
+global memory shared by all its SPs~\cite{NVIDIA10}.
 
-    On a CPU equipped with a GPU, all the data-parallel and intensive
+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
@@ -417,12 +425,12 @@ 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 data transfers between the GPU and its CPU during the
+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) softwaremodel. The GPU is treated as a
+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
@@ -450,12 +458,12 @@ read-only caches.
 
 
 \subsubsection{A sequential Aberth algorithm}
-The means steps of Aberth method can expressed as an algorithm
-like:
+The main steps of Aberth method are shown in Algorithm.~\ref{alg1-seq} :
   
 \begin{algorithm}[H]
-\LinesNumbered
-\caption{Algorithm to find root polynomial with Aberth method}
+\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)}
@@ -464,7 +472,7 @@ tolerance threshold),P(Polynomial to solve)}
 
 \BlankLine
 
-Initialization of the parameter of the polynomial to solve\;
+Initialization of the coefficients of the polynomial to solve\;
 Initialization of the solution vector $Z^{0}$\;
 
 \While {$\Delta z_{max}\succ \epsilon$}{
@@ -483,57 +491,62 @@ $\Delta z_{max}$=c\;}
 \end{algorithm}
 
 ~\\ 
-In this sequential algorithm one thread CPU execute all steps. Let see the step 3 the execution of the iterative function, 2 instructions are needed, the first instruction \textit{save} the solution vector for the previous iteration, the second instruction \textit{update} or compute a new values of the roots.
-We have two manner to execute the iterative function, taking a Jacobi iteration who need all the previous value $z^{(k)}_{i}$ to compute the new value $z^{(k+1)}_{i}$we have:
+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 :
 
 \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}
 
-Or with the Gauss-seidel iteration, we have:
+With the Gauss-seidel iteration, we have:
 \begin{equation}
-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.
+\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}
 
-In formula(16), the Gauss-seidel iteration converge more quickly because they used the most fresh computed root $z^{k+1}_{i}$ , at this reason we used Gauss-seidel iteration.
+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 $z^{k+1}_{i}$.
 
-The steps 4 of the Aberth method compute the convergence of the roots, using(9) formula.
-Both steps 3 and 4 use 1 thread to compute N roots on CPU, which is harmful for the large polynomial's roots finding.
+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 $T_{i}(N)$: the time to compute one new root's value of the step 3,$T_{i}$ depend on the polynomial's degrees N, when N increase $T_{i}$ increase to. We need $N.T_{i}(N)$ to compute all the new root's value in one iteration on the step 3.
+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 $T_{j}$: the time to compute one root's convergence value of the step 4, we need $N.T_{j}$ to compute all the root's convergence value in one iteration on the step 4.
+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.
 
-The execution time for both steps 3 and 4 can see like:
+Thus, the execution time for both steps 3 and 4 is:
 \begin{equation}
-T_{exe}=N(T_{i}(N)+T_{j})+O(n).
+T_{iter}=n(T_{i}(n)+T_{j})+O(n).
 \end{equation}
-Let Nbr\_iter the number of iteration necessary to compute all the roots, so the total execution time $Total\_time_{exe}$ can give like:
+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}
-Total\_time_{exe}=\left[N\left(T_{i}(N)+T_{j}\right)+O(n)\right].Nbr\_iter
+\label{eq:T-global}
+T=\left[n\left(T_{i}(n)+T_{j}\right)+O(n)\right].K
 \end{equation}
-The execution time increase with the increasing of the polynomial's root, which take necessary to parallelize this step to reduce the execution time. In the following paper you explain how we parrallelize  this step using GPU architecture with CUDA platform.
+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{Parallelize the steps on GPU }
-On the CPU Aberth algorithm both steps 3 and 4 contain the loop \verb=for=, it use one thread to execute all the instruction in the loop N times. Here we explain how the GPU architecture can compute this loop and reduce the execution time.
-The GPU architecture assign the execution of this loop to a groups of parallel threads organized as a grid of blocks each block contain a number of threads. All threads within a block are executed concurrently in parallel. The instruction are executed as a kernel.
+\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.
 
-Let nbr\_thread be the number of threads executed in parallel, so you can easily transform the (18)formula like this: 
+Let N be the number of threads executed in parallel,  Equation.~\ref{eq:T-global} becomes then : 
 
 \begin{equation}
-Total\_time_{exe}=\left[\frac{N}{nbr\_thread}\left(T_{i}(N)+T_{j}\right)+O(n)\right].Nbr\_iter.
+T=\left[\frac{n}{N}\left(T_{i}(n)+T_{j}\right)+O(n)\right].K.
 \end{equation}
 
-In theory, the $Total\_time_{exe}$ on GPU is speed up nbr\_thread times as a $Total\_time_{exe}$ on CPU. We show more details in the experiment part. 
+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 platform, All the instruction of the loop \verb=for= are executed by the GPU as a kernel form. A kernel is a procedure written in CUDA and defined by a heading \verb=__global__=, which means that it is to be executed by the GPU. The following algorithm see the Aberth algorithm on 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 ``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.
 
 \begin{algorithm}[H]
-\LinesNumbered
-\caption{Algorithm to find root polynomial with Aberth method}
+\label{alg2-cuda}
+%\LinesNumbered
+\caption{CUDA Algorithm to find roots with the Aberth method}
 
 \KwIn{$Z^{0}$(Initial root's vector),$\varepsilon$ (error
 tolerance threshold),P(Polynomial to solve)}
@@ -542,25 +555,26 @@ tolerance threshold),P(Polynomial to solve)}
 
 \BlankLine
 
-Initialization of the parameter of the polynomial to solve\;
+Initialization of the coeffcients of the polynomial to solve\;
 Initialization of the solution vector $Z^{0}$\;
-Allocate and fill the data in the global memory GPU\;
+Allocate and copy initial data to the GPU global memory\;
 
 \While {$\Delta z_{max}\succ \epsilon$}{
  Let $\Delta z_{max}=0$\;
-$ kernel\_save(d\_Z^{k-1})$\;
+$ kernel\_save(d\_z^{k-1})$\;
 $ kernel\_update(d\_z^{k})$\;
-$kernel\_testConverge (d_?z_{max},d_Z^{k},d_Z^{k-1})$\;
+$kernel\_testConverge(\Delta z_{max},d_z^{k},d_z^{k-1})$\;
 }
 \end{algorithm}
 ~\\ 
 
-After the initialization step, all data of the root finding problem to be solved must be copied from the CPU memory to the GPU global memory, because the GPUs only work on the data filled 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 \textit{save} in line( 6, Algorithm 2) consist to save the vector of polynomial's root found at the previous time step on GPU memory, in order to test the convergence of the root at each iteration in line (8, Algorithm 2).
+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 and update Z(k),as formula (), we notice that the kernel update are called in two forms,  separated with  the value of \emph{R} which determines the radius beyond which we apply the logarithm formula like this: 
+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. 
 
 \begin{algorithm}[H]
-\LinesNumbered
+\label{alg3-update}
+%\LinesNumbered
 \caption{A global Algorithm for the iterative function}
 
 \eIf{$(\left|Z^{(k)}\right|<= R)$}{
@@ -570,42 +584,36 @@ $kernel\_update\_Log(d\_z^{k})$\;
 }
 \end{algorithm}
 
-The first form execute the formula(8) if all the module's $( |Z(k)|<= R)$, else the kernel execute the formulas(13,14).the radius R was computed like:
+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 :
 
-$$R = \exp( \log(DBL\_MAX) / (2*(double)P.degrePolynome) )$$
+$$R = \exp( \log(DBL\_MAX) / (2*n) )$$ where $DBL\_MAX$ stands for the maximum representable double value.
 
-The last kernel verify the convergence of the root after each update of $Z^{(k)}$, as formula(), we used the function of the CUBLAS Library (CUDA Basic Linear Algebra Subroutines) to implement this kernel. 
+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 terminates its computations when all the root are converged. Finally, the solution of the  root finding problem is copied back from the GPU global memory to the CPU memory. We use the communication functions of CUDA for the memory allocations in the GPU \verb=(cudaMalloc())= and the data transfers from the CPU memory to the GPU memory \verb=(cudaMemcpyHostToDevice)=
-or from the GPU memory to the CPU memory \verb=(cudaMemcpyDeviceToHost))=. 
-\subsection{Experimental study}
+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}
 
-\subsubsection{Definition of the polynomial used}
-We use a polynomial of the following form for which the
-roots are distributed on 2 distinct circles:
+\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 :
 \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})
+	\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}
 
-This form makes it possible to associate roots having two
-different modules and thus to work on a polynomial constitute
-of four non zero terms.
-\\
- An other form of the polynomial 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}
-     {\Large \forall a_{i} \in C; p(x)=\sum^{n-1}_{i=1} a_{i}.x^{i}} 
+     {\Large \forall a_{i} \in C, i\in N;  p(x)=\sum^{n}_{i=0} a_{i}.x^{i}} 
 \end{equation}
-with this formula, we can have until \textit{n} non zero terms.
-
-\subsubsection{The study condition} 
-In order to have representative average values, for each
-point of our curves we measured the roots finding of 10
-different polynomials.
+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
@@ -613,65 +621,85 @@ 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 Tesla C2070 (with 6 Go of ram)
-
-\subsubsection{Comparative study}
-We initially carried out the convergence of Aberth algorithm with various sizes of polynomial, in second we evaluate the influence of the size of the threads per block....
-
-\paragraph{Aberth algorithm on CPU and GPU}
-
-\begin{table}[!ht]
-	\centering
-		\begin{tabular} {|R{2cm}|L{2.5cm}|L{2.5cm}|L{1.5cm}|L{1.5cm}|}
-			\hline Polynomial's degrees & $T_{exe}$ on CPU & $T_{exe}$ on GPU & CPU iteration & GPU iteration\\
-				\hline 5000 & 1.90 & 0.40 & 18 & 17\\
-				\hline 10000 & 172.723 & 0.59 & 21 & 24\\
-				\hline 20000 & 172.723 & 1.52 & 21 & 25\\
-				\hline 30000 & 172.723 & 2.77 & 21 & 33\\
-				\hline 50000 & 172.723 & 3.92 & 21 & 18\\
-				\hline 500000 & $>$1h & 497.109 &  & 24\\
-				\hline 1000000 & $>$1h & 1,524.51& & 24\\
-				\hline 
-		\end{tabular}
-	\caption{the convergence of Aberth algorithm}
-	\label{tab:theConvergenceOfAberthAlgorithm}
-\end{table}
+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]
+\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}
+
+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.
  
-\paragraph{The impact of the thread's number into the convergence of Aberth  algorithm}
-
-\begin{table}[!h]
-	\centering
-		\begin{tabular} {|R{2.5cm}|L{2.5cm}|L{2.5cm}|}
-			\hline Thread's numbers & Execution time &Number of iteration\\
-				\hline 1024 & 523 & 27\\
-				\hline 512 & 449.426 & 24\\
-				\hline 256 & 440.805 & 24\\
-				\hline 128 & 456.175 & 22\\
-				\hline 64 & 472.862 & 23\\
-				\hline 32 & 830.152 & 24\\
-				\hline 8 & 2632.78 & 23 \\
-				\hline
-				\end{tabular}
-				\caption{The impact of the thread's number into the convergence of Aberth  algorithm}
-	\label{tab:Theimpactofthethread'snumberintotheconvergenceofAberthalgorithm}
-		
-\end{table}
-
-\paragraph{A comparative study between Aberth and Durand-kerner algorithm}
-\begin{table}[htbp]
-	\centering
-		\begin{tabular} {|R{2cm}|L{2.5cm}|L{2.5cm}|L{1.5cm}|L{1.5cm}|}
-			\hline Polynomial's degrees & Aberth $T_{exe}$ & D-Kerner $T_{exe}$ & Aberth iteration & D-Kerner iteration\\
-			\hline 5000 &  0.40 & 3.42 & 17 & 138 \\
-			\hline 50000 & 3.92 & 385.266 & 17 & 823\\
-			\hline 500000 & 497.109 & 4677.36 & 24 & 214\\
-			\hline					
-					\end{tabular}
-	\caption{Aberth algorithm compare to Durand-Kerner algorithm}
-	\label{tab:AberthAlgorithCompareToDurandKernerAlgorithm}
-\end{table}
 
 
+\subsubsection{Influence of the number of threads on the execution times of different polynomials (sparse and full)}
+It is also interesting to see the influence of the number of threads per block on the execution time. 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.
+
+\begin{figure}[H]
+\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}
+\end{figure}
+
+The figure 2 show that, the best execution time for both sparse and full polynomial are given while the threads number varies between 64 and 256 threads per bloc. We notice that with small polynomials the number of threads per block is 64, Whereas, the large polynomials the 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}
+
+In this experiment we report the performance of log.exp solution describe in ~\ref{sec2} to compute very high degrees polynomials.   
+\begin{figure}[H]
+\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}
+\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 polynomials degrees. We can see that the execution time for the both algorithms are the same while the polynomials degrees are less than 4500. After,we show clearly that the classical version of Ehrlich-Aberth algorithm (without applying log.exp) stop to converge and can not solving polynomial exceed 4500, 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 . 
+
+
+%\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]
+\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}
+\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 $10^{-5}$.
+
+\begin{figure}[H]
+\centering
+  \includegraphics[width=0.8\textwidth]{figures/EA_DK_nbr}
+\caption{The iteration number of Ehrlich-Aberth versus Durand-Kerner algorithm}
+\label{fig:01}
+\end{figure}
 
 \bibliography{mybibfile}
 
