MAJ eq:Aberth-Conv-Cond

[kahina_paper1.git] / paper.tex

diff --git a/paper.tex b/paper.tex
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--- a/paper.tex
+++ b/paper.tex
@@ -1,11 +1,13 @@
 \documentclass[review]{elsarticle}
 
 \usepackage{lineno,hyperref}
 \documentclass[review]{elsarticle}
 
 \usepackage{lineno,hyperref}

-%%\usepackage[utf8]{inputenc}
+\usepackage[utf8]{inputenc}

 %%\usepackage[T1]{fontenc}
 %%\usepackage[french]{babel}
 %%\usepackage[T1]{fontenc}
 %%\usepackage[french]{babel}

+\usepackage{float} 

 \usepackage{amsmath,amsfonts,amssymb}
 \usepackage[ruled,vlined]{algorithm2e}
 \usepackage{amsmath,amsfonts,amssymb}
 \usepackage[ruled,vlined]{algorithm2e}

+%\usepackage[french,boxed,linesnumbered]{algorithm2e}

 \usepackage{array,multirow,makecell}
 \setcellgapes{1pt}
 \makegapedcells
 \usepackage{array,multirow,makecell}
 \setcellgapes{1pt}
 \makegapedcells


@@ -52,39 +54,47 @@
 
 \begin{frontmatter}
 
 
 \begin{frontmatter}
 

-\title{Parallel polynomial root finding  using GPU}
+\title{Efficient high degree polynomial root finding using GPU}

 
 %% Group authors per affiliation:
 
 %% 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:
 
 %% or include affiliations in footnotes:

-\author[mymainaddress]{Ghidouche Kahina\corref{mycorrespondingauthor}}
-%%\ead[url]{kahina.ghidouche@gmail.com}
+\author[mymainaddress]{Kahina Ghidouche}
+%%\ead[url]{kahina.ghidouche@univ-bejaia.dz}

 \cortext[mycorrespondingauthor]{Corresponding author}
 \cortext[mycorrespondingauthor]{Corresponding author}

-\ead{kahina.ghidouche@gmail.com}
+\ead{kahina.ghidouche@univ-bejaia.dz}

 
 

-\author[mysecondaryaddress]{Couturier Raphael\corref{mycorrespondingauthor}}
+\author[mysecondaryaddress]{Raphaël Couturier\corref{mycorrespondingauthor}}

 %%\cortext[mycorrespondingauthor]{Corresponding author}
 \ead{raphael.couturier@univ-fcomte.fr}
 
 %%\cortext[mycorrespondingauthor]{Corresponding author}
 \ead{raphael.couturier@univ-fcomte.fr}
 

-\author[mymainaddress]{Abderrahmane Sider\corref{mycorrespondingauthor}}
+\author[mymainaddress]{Abderrahmane Sider}

 %%\cortext[mycorrespondingauthor]{Corresponding author}
 \ead{ar.sider@univ-bejaia.dz}
 
 %%\cortext[mycorrespondingauthor]{Corresponding author}
 \ead{ar.sider@univ-bejaia.dz}
 

-\address[mymainaddress]{Department of informatics,University of Bejaia,Algeria}
-\address[mysecondaryaddress]{FEMTO-ST Institute, University of Franche-Compté }
+\address[mymainaddress]{Laboratoire LIMED, Faculté des sciences
+  exactes, Université de Bejaia, 06000, Algeria}
+\address[mysecondaryaddress]{FEMTO-ST Institute, University of
+  Bourgogne Franche-Comte, France }

 
 \begin{abstract}
 
 \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. Finding 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
+to 1,000,000) 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}
 \end{abstract}
 
 \begin{keyword}

-root finding of polynomials, high degree, iterative methods, Durant-Kerner, GPU, CUDA, CPU , Parallelization
+Polynomial root finding, Iterative methods, Ehrlich-Aberth, Durand-Kerner, GPU

 \end{keyword}
 
 \end{frontmatter}
 \end{keyword}
 
 \end{frontmatter}


@@ -92,22 +102,27 @@ root finding of polynomials, high degree, iterative methods, Durant-Kerner, GPU,
 \linenumbers
 
 \section{The problem of finding roots of a polynomial}
 \linenumbers
 
 \section{The problem of finding roots of a polynomial}

-Polynomials are algebraic structures used in mathematics that capture physical phenomenons and that express the outcome in the form of a function of some unknown variable. Formally speaking,  a polynomial $p(x)$ of degree \textit{n} having $n$ coefficients in the complex plane \textit{C} and zeros $\alpha_{i},\textit{i=1,...,n}$ 
+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}
 %%\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}
 
 \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 $p$. 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
+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}
 $x=g(x)$
 \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}


@@ -119,22 +134,26 @@ 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
 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
+in the mid-16th century. However, N. H. Abel in the early 19th

 century showed that polynomials of degree five or more could not
 century showed that polynomials of degree five or more could not

-be solved by  directs methods. Since then, mathmathicians have
+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's method, Bernoulli's method of the 18th, and Graeffe's.
+Newton method, the Bernoulli method of the 18th, and the Graeffe 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
+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

 approximation of all the roots, starting with the Durand-Kerner (DK)
 approximation of all the roots, starting with the Durand-Kerner (DK)

-method :
+method:

 %%\begin{center}
 \begin{equation}
 %%\begin{center}
 \begin{equation}

- Z_{i}=Z_{i}-\frac{P(Z_{i})}{\prod_{i\neq j}(z_{i}-z_{j})}
+\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
+iteration $k$.
+

 
 This formula is mentioned for the first time by
 Weiestrass~\cite{Weierstrass03} as part of the fundamental theorem
 
 This formula is mentioned for the first time by
 Weiestrass~\cite{Weierstrass03} as part of the fundamental theorem


@@ -143,15 +162,18 @@ 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}=Z_{i}-\frac{1}{{\frac {P'(Z_{i})} {P(Z_{i})}}-{\sum_{i\neq j}(z_{i}-z_{j})}}.
+\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 $z$.

 
 Aberth, Ehrlich and Farmer-Loizou~\cite{Loizon83} have proved that
 
 Aberth, Ehrlich and Farmer-Loizou~\cite{Loizon83} have proved that

-the Ehrlisch-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.
 
 
 Iterative methods raise several problem when implemented e.g.


@@ -161,75 +183,83 @@ drastically increases like the degrees of high polynomials. It is expected that
 parallelization of these algorithms will improve the convergence
 time.
 
 parallelization of these algorithms will improve the convergence
 time.
 

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

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

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

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

-diverges, but the first two methods have speed-up 5.5
-(speed-up=(Time on one processor)/(Time on p processors)). Later,
+diverges, but the first two methods have speed-up equal to 5.5. Later,

 Freeman and Bane~\cite{Freemanall90}  considered asynchronous
 algorithms, in which each processor continues to update its
 approximations even though the latest values of other $z_i((k))$
 have not been received from the other processors, in contrast with synchronous algorithms where it would wait those values before making a new iteration.
 Freeman and Bane~\cite{Freemanall90}  considered asynchronous
 algorithms, in which each processor continues to update its
 approximations even though the latest values of other $z_i((k))$
 have not been received from the other processors, in contrast with synchronous algorithms where it would wait those values before making a new iteration.

-Couturier et al. ~\cite{Raphaelall01} proposed two methods of parallelisation for
+Couturier and al.~\cite{Raphaelall01} proposed two methods of parallelization for

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

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

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

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

 
 

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

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

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

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

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

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

-result showed that a parallel CUDA implementation is 10 times as fast as
-the sequential implementation on a single CPU for high degree
-polynomials of about 48000. To our knowledge, it is the first time such high degree polynomials are numerically solved.
-
-
-In this paper, we focus on the implementation of the Aberth method for
-high degree polynomials on GPU. The paper is organised as fellows. Initially, we recall the Aberth method in Section.\ref{sec1}. Improvements for the Aberth method are proposed in Section.\ref{sec2}. Related work to the implementation of simultaneous methods using a parallel approach is presented in Section.\ref{secStateofArt}.
-In Section.4 we propose a parallel implementation of the Aberth method on GPU and discuss it. Section 5 presents and investigates our implementation and experimental study results. Finally, Section 6 concludes this paper and gives some hints for future research directions in this topic.  
-
-\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 48000. 
+
+
+In this paper, we focus on the implementation of the Ehrlich-Aberth
+method for high degree polynomials on GPU. We propose an adaptation of
+the exponential logarithm in order to be able to solve sparse and full
+polynomial of degree up to $1,000,000$. The paper is organized as
+follows. Initially, we recall the Ehrlich-Aberth method in Section
+\ref{sec1}. Improvements for the Ehrlich-Aberth method are proposed in
+Section \ref{sec2}. Related work to the implementation of simultaneous
+methods using a parallel approach is presented in Section
+\ref{secStateofArt}.  In Section \ref{sec5} we propose a parallel
+implementation of the Ehrlich-Aberth method on GPU and discuss
+it. Section \ref{sec6} presents and investigates our implementation
+and experimental study results. Finally, Section\ref{sec7} 6 concludes
+this paper and gives some hints for future research directions in this
+topic.
+
+\section{The Sequential Ehrlich-Aberth method}

 \label{sec1}
 A cubically convergent iteration method for finding zeros of
 \label{sec1}
 A cubically convergent iteration method for finding zeros of

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

 
 

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

 
 

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

 
 

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

 
 

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

 
 

-\begin{equation}
-\frac{R_{i}(z)}{R_{i}^{'}(z)}= \frac{p(z)}{p^{'}(z)-p(z)\frac{w_{i}(z)}{w_{i}^{'}(z)}}= \frac{p(z)}{p^{'}(z)-p(z) \sum _{j=1,j \neq i}^{n}\frac{1}{z-x_{i}}}, i=1,2,...,n
-\end{equation}
+%\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.% 

 
 

-Substituting $x_{j}$ for z 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}


@@ -239,12 +269,12 @@ The initialization of a polynomial p(z) is done by setting each of the $n$ compl
 
 \subsection{Vector $z^{(0)}$ Initialization}
 
 
 \subsection{Vector $z^{(0)}$ Initialization}
 

-Like for any iterative method, we need to choose $n$ initial guess points $z^{(0)}_{i}, i = 1, . . . , n.$
+As for any iterative method, we need to choose $n$ initial guess points $z^{(0)}_{i}, i = 1, . . . , n.$

 The initial guess is very important since the number of steps needed by the iterative method to reach
 a given approximation strongly depends on it.
 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
 circles and relies on the result of~\cite{Ostrowski41}.
 
 performed this choice by selecting complex numbers along different
 circles and relies on the result of~\cite{Ostrowski41}.
 


@@ -261,31 +291,35 @@ u_{i}=2.|a_{i}|^{\frac{1}{i}};
 v_{i}=\frac{|\frac{a_{n}}{a_{i}}|^{\frac{1}{n-i}}}{2}.
 \end{equation}
 
 v_{i}=\frac{|\frac{a_{n}}{a_{i}}|^{\frac{1}{n-i}}}{2}.
 \end{equation}
 

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

 
 \begin{equation}
 
 \begin{equation}

-H_{i}(z)=z_{i}-\frac{1}{\frac{p^{'}(z_{i})}{p(z_{i})}-\sum_{j\neq
-i}{\frac{1}{z_{i}-z_{j}}}}
+\label{Eq:Hi}
+EA2: z^{k+1}=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}
+{1-\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}\sum_{j=1,j\neq i}^{j=n}{\frac{1}{(z_{i}^{k}-z_{j}^{k})}}}, i=0,. . . .,n

 \end{equation}
 \end{equation}

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

 \label{sec2}
 \label{sec2}

-The Ehrlisch-Aberth method implementation suffers of overflow problems. This
+With high degree polynomial, the Ehrlich-Aberth method implementation,
+as well as the Durand-Kerner implement, suffers from overflow problems. This

 situation occurs, for instance, in the case where a polynomial
 having positive coefficients and a large degree is computed at a
 point $\xi$ where $|\xi| > 1$, where $|x|$ stands for the modolus of a complex $x$. Indeed, the limited number in the
 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


@@ -312,29 +346,40 @@ propose to use the logarithm and the exponential of a complex in order to comput
 Using the logarithm (eq.~\ref{deflncomplex}) and the exponential (eq.~\ref{defexpcomplex}) operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations
 manipulate lower absolute values and the roots for large polynomial's degrees can be looked for successfully~\cite{Karimall98}.
 
 Using the logarithm (eq.~\ref{deflncomplex}) and the exponential (eq.~\ref{defexpcomplex}) operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations
 manipulate lower absolute values and the roots for large polynomial's degrees can be looked for successfully~\cite{Karimall98}.
 

-Applying this solution for the Aberth method we obtain the
-iteration function with logarithm:
+Applying this solution for the Ehrlich-Aberth method we obtain the
+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}
 %%$$ \exp \bigl(  \ln(p(z)_{k})-ln(\ln(p(z)_{k}^{'}))- \ln(1- \exp(\ln(p(z)_{k})-ln(\ln(p(z)_{k}^{'})+\ln\sum_{i\neq j}^{n}\frac{1}{z_{k}-z_{j}})$$
 \begin{equation}

-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),
+\label{Log_H2}
+EA.EL: z^{k+1}=z_{i}^{k}-\exp \left(\ln \left(
+p(z_{i}^{k})\right)-\ln\left(p'(z^{k}_{i})\right)- \ln
+\left(1-Q(z^{k}_{i})\right)\right),

 \end{equation}
 
 where:
 
 \begin{equation}
 \end{equation}
 
 where:
 
 \begin{equation}

-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).
+\label{Log_H1}
+Q(z^{k}_{i})=\exp\left( \ln (p(z^{k}_{i}))-\ln(p'(z^{k}_{i}))+\ln \left(
+\sum_{k\neq j}^{n}\frac{1}{z^{k}_{i}-z^{k}_{j}}\right)\right).

 \end{equation}
 
 \end{equation}
 

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

 
 

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

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

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

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


@@ -346,15 +391,15 @@ 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
 
 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 Ehrlisch-Aberth 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 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
 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
 parallelism that can be suitably exploited by SIMD machines.
 Moreover, they have fast rate of convergence (quadratic for the

-Durand-Kerner and cubic for the Ehrlisch-Aberth). Various parallel
+Durand-Kerner and cubic for the Ehrlich-Aberth). Various parallel

 algorithms reported for these methods can be found
 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
 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


@@ -362,7 +407,7 @@ 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 Ehrlisch-Aberth method on a 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 needs N processors to compute N roots, which is not
 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


@@ -379,7 +424,9 @@ polynomials of 48000. In this paper we present a parallel implementation of Ehli
 GPUs, which details are discussed in the sequel.
 
 
 GPUs, which details are discussed in the sequel.
 
 

-\section {A CUDA parallel Ehrlisch-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
 
 \subsection{Background on the GPU architecture}
 A GPU is viewed as an accelerator for the data-parallel and


@@ -437,23 +484,22 @@ provides two read-only memory spaces, the constant space and the
 texture space, which reside in external DRAM, and are accessed via
 read-only caches.
 
 texture space, which reside in external DRAM, and are accessed via
 read-only caches.
 

-\subsection{ The implementation of Aberth method on GPU}
+\section{ The implementation of 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} :
+\subsection{A sequential Ehrlich-Aberth algorithm}
+The main steps of Ehrlich-Aberth method are shown in Algorithm.~\ref{alg1-seq} :

   
 \begin{algorithm}[H]
 \label{alg1-seq}
 %\LinesNumbered
   
 \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)}
+\caption{A sequential algorithm to find roots with the Ehrlich-Aberth method}

 
 

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

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


@@ -470,7 +516,7 @@ $Z\left[j\right]=H\left(j,Z\right)$\;
 
 \For{$i \gets 0 $ \KwTo $n-1$}{
 $c=\frac{\left|Z\left[i\right]-ZPrec\left[i\right]\right|}{Z\left[i\right]}$\;
 
 \For{$i \gets 0 $ \KwTo $n-1$}{
 $c=\frac{\left|Z\left[i\right]-ZPrec\left[i\right]\right|}{Z\left[i\right]}$\;

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

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


@@ -478,61 +524,36 @@ $\Delta z_{max}$=c\;}
 
 ~\\ 
 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.
 
 ~\\ 
 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 $z^{(k+1)}_{i}$, taht is :
+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}
 
 \begin{equation}

-H(i,z^{k+1})=\frac{p(z^{(k)}_{i})}{p'(z^{(k)}_{i})-p(z^{(k)}_{i})\sum^{n}_{j=1 j\neq i}\frac{1}{z^{(k)}_{i}-z^{(k)}_{j}}}, i=1,...,n.
+EAJ: z^{k+1}_{i}=\frac{p(z^{k}_{i})}{p'(z^{k}_{i})-p(z^{k}_{i})\sum^{n}_{j=1 j\neq i}\frac{1}{z^{k}_{i}-z^{k}_{j}}}, i=1,...,n.

 \end{equation}
 
 \end{equation}
 

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

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

-H(i,z^{k+1})=\frac{p(z^{(k)}_{i})}{p'(z^{(k)}_{i})-p(z^{(k)}_{i})(\sum^{i-1}_{j=1}\frac{1}{z^{(k)}_{i}-z^{(k+1)}_{j}}+\sum^{n}_{j=i+1}\frac{1}{z^{(k)}_{i}-z^{(k)}_{j}})}, i=1,...,n.
+EAGS: z^{k+1}_{i}=\frac{p(z^{k}_{i})}{p'(z^{k}_{i})-p(z^{k}_{i})(\sum^{i-1}_{j=1}\frac{1}{z^{k}_{i}-z^{k+1}_{j}}+\sum^{n}_{j=i+1}\frac{1}{z^{k}_{i}-z^{k}_{j}})}, i=1,...,n.

 \end{equation}
 \end{equation}

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

 
 The $4^{th}$ step of the algorithm checks the convergence condition using Equation.~\ref{eq:Aberth-Conv-Cond}.
 Both steps 3 and 4 use 1 thread to compute all the $n$ roots on CPU, which is very harmful for performance in case of the large degree polynomials.
 
 
 The $4^{th}$ step of the algorithm checks the convergence condition using Equation.~\ref{eq:Aberth-Conv-Cond}.
 Both steps 3 and 4 use 1 thread to compute all the $n$ roots on CPU, which is very harmful for performance in case of the large degree polynomials.
 

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

 
 

-\subsubsection{A Parallel implementation with CUDA }
+\subsection{A Parallel implementation with CUDA }

 On the CPU,  both steps 3 and 4 contain the loop \verb=for= and a single thread executes all the instructions in the loop $n$ times. In this subsection, we explain how the GPU architecture can compute this loop and reduce the execution time.
 In the GPU, the schduler assigns the execution of this loop to a group of threads organised as a grid of blocks with block containing a number of threads. All threads within a block are executed concurrently in parallel. The instructions run on the GPU are grouped in special function called kernels. It's up to the programmer, to describe the execution context, that is the size of the Grid, the number of blocks and the number of threads per block upon the call of a given kernel, according to a special syntax defined by CUDA.
 
 On the CPU,  both steps 3 and 4 contain the loop \verb=for= and a single thread executes all the instructions in the loop $n$ times. In this subsection, we explain how the GPU architecture can compute this loop and reduce the execution time.
 In the GPU, the schduler assigns the execution of this loop to a group of threads organised as a grid of blocks with block containing a number of threads. All threads within a block are executed concurrently in parallel. The instructions run on the GPU are grouped in special function called kernels. It's up to the programmer, to describe the execution context, that is the size of the Grid, the number of blocks and the number of threads per block upon the call of a given kernel, according to a special syntax defined by CUDA.
 

-Let N be the number of threads executed in parallel,  Equation.~\ref{eq:T-global} becomes then : 
-
-\begin{equation}
-T=\left[\frac{n}{N}\left(T_{i}(n)+T_{j}\right)+O(n)\right].K.
-\end{equation}
-
-In theory, total execution time $T$ on GPU is speed up $N$ times as  $T$ on CPU. We will see at what extent this is true in the experimental study hereafter. 
-~\\
-~\\
-In CUDA programming, all the instructions of the  \verb=for= loop are executed by the GPU as a kernel. A kernel is a function written in CUDA and defined by the  \verb=__global__= qualifier added before a usual ``C`` function, which instructs the compiler to generate appropriate code to pass it to the CUDA runtime in order to be executed on the GPU. 
+In CUDA programming, all the instructions of the  \verb=for= loop are executed by the GPU as a kernel. A kernel is a function written in CUDA and defined by the  \verb=__global__= qualifier added before a usual \verb=C= function, which instructs the compiler to generate appropriate code to pass it to the CUDA runtime in order to be executed on the GPU. 

 
 

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

 
 \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)}


@@ -561,7 +582,7 @@ The second kernel executes the iterative function $H$ and updates $z^{k}$, accor
 \begin{algorithm}[H]
 \label{alg3-update}
 %\LinesNumbered
 \begin{algorithm}[H]
 \label{alg3-update}
 %\LinesNumbered

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

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


@@ -579,34 +600,27 @@ The last kernel verifies the convergence of the roots after each update of $Z^{(
 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)
 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)

-\subsection{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:
+\section{Experimental study}
+\label{sec6}
+%\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 polynomial for which the roots are distributed on 2 distinct circles :

 \begin{equation}
 \begin{equation}

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


@@ -614,75 +628,103 @@ 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
 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. 
+\subsection{The execution time in seconds of Ehrlich-Aberth algorithm on CPU OpenMP (1 core, 4 cores) vs. on a Tesla GPU}
+
+
+%\begin{figure}[H]
+%\centering
+ % \includegraphics[width=0.8\textwidth]{figures/Compar_EA_algorithm_CPU_GPU}
+%\caption{The execution time in seconds of Ehrlich-Aberth algorithm on CPU core vs. on a Tesla GPU}
+%\label{fig:01}
+%\end{figure}
+
+\begin{figure}[H]
+\centering
+  \includegraphics[width=0.8\textwidth]{figures/openMP-GPU}
+\caption{The execution time in seconds of Ehrlich-Aberth algorithm on CPU OpenMP (1 core, 4 cores) vs. on a Tesla GPU}
+\label{fig:01}
+\end{figure}
+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 respectively the execution time of the Ehrlich-Aberth method implemented initially on one core of the Quad-Core Xeon E5620 CPU than on four cores of the same machine with \textit{OpenMP} platform and the execution time of the same method implemented on one Nvidia Tesla K40c GPU, with sparse polynomial degrees ranging from 100,000 to 1,000,000. We can see that the method implemented on the GPU are faster than those implemented on the CPU (4 cores). This is due to the GPU ability to compute the data-parallel functions faster than its CPU counterpart. However, the execution time for the CPU(4 cores) implementation exceed 5,000 s for 250,000 degrees polynomials, in counterpart  the GPU implementation for the same polynomials not reach 100 s, more than again, with an execution time under to 2500 s CPU (4 cores) implementation can resolve polynomials degrees of only 200,000, whereas GPU implementation can resolve polynomials more than 1,000,000 degrees. We can also notice that the GPU implementation are almost 47 faster then those implementation on the CPU(4 cores). However the CPU(4 cores) implementation are almost 4 faster then his implementation on CPU (1 core). Furthermore, we verify that the number of iterations and the convergence precision is the same for the both CPU and GPU implementation. %This reduction of time allows us to compute roots of polynomial of more important degree at the same time than with a CPU.

  
  

-\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}
-
-
-\begin{figure}[htbp]
+ %We notice that the convergence precision is a round $10^{-7}$ for the both implementation on CPU and GPU. Consequently, we can conclude that Ehrlich-Aberth on GPU are faster and accurately then CPU implementation.
+
+\subsection{Influence of the number of threads on the execution times of different polynomials (sparse and full)}
+To optimize the performances of an algorithm on a GPU, it is necessary to maximize the use of cores GPU (maximize the number of threads executed in parallel) and to optimize the use of the various memoirs GPU. In fact, it is interesting to see the influence of the number of threads per block on the execution time of Ehrlich-Aberth algorithm. 
+For that, we notice that the maximum number of threads per block for the Nvidia Tesla K40 GPU is 1024, so we varied the number of threads per block from 8 to 1024. We took into account the execution time for both sparse and full of 10 different polynomials of size 50000 and 10 different polynomials of size 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}
 
 \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 when the threads number varies between 64 and 256 threads per bloc. We notice that with small polynomials the best number of threads per block is 64, Whereas, the large polynomials the best number of threads per block is 256. However,In the following experiments we specify that the number of thread by block is 256.
+
+\subsection{The impact of exp-log solution to compute very high degrees of  polynomial}
+
+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 exp.log solution and the execution time of the Ehrlich-Aberth algorithm without applying exp.log solution, with full and sparse polynomials degrees. We can see that the execution time for the both algorithms are the same while the full polynomials degrees are less than 4000 and full polynomials are less than 150,000. After,we show clearly that the classical version of Ehrlich-Aberth algorithm (without applying log.exp) stop to converge and can not solving any polynomial sparse or full. In counterpart, the new version of Ehrlich-Aberth algorithm (applying log.exp solution) can solve very high and large full polynomial exceed 100,000 degrees.
+
+in fact, when the modulus of the roots are up than \textit{R} given in ~\ref{R},this exceed the limited number in the mantissa of floating points representations and can not compute the iterative function given in ~\ref{eq:Aberth-H-GS} to obtain the root solution, who justify the divergence of the classical Ehrlich-Aberth algorithm. However, applying log.exp solution given in ~\ref{sec2} took into account the limit of floating using the iterative function in(Eq.~\ref{Log_H1},Eq.~\ref{Log_H2} and allows to solve a very large polynomials degrees . 
+
+
+%\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. 
+
+ 
+\subsection{A comparative study between Ehrlich-Aberth algorithm and Durand-kerner algorithm}
+In this part, we are interesting to compare the simultaneous methods, Ehrlich-Aberth and Durand-Kerner in parallel computer using GPU. We took into account the execution time, the number of iteration and the polynomial's size. for the both sparse and full polynomials.  
+
+\begin{figure}[H]
+\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}
+
+%\subsubsection{The execution time of Ehrlich-Aberth algorithm on OpenMP(1 core, 4 cores) vs. on a Tesla GPU}
+
+
+

 
 
 
 

-\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}

 
 

+\section{Conclusion and perspective}

 
 

+\label{sec7}

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