X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/kahina_paper1.git/blobdiff_plain/b8c1bf75cd50a698e70c8c9372073050c20380f8..3f15a15c587c4e0d34557bbd791d83705dd519f1:/paper.tex

diff --git a/paper.tex b/paper.tex
index efff5b1..93e111a 100644
--- a/paper.tex
+++ b/paper.tex
@@ -9,6 +9,15 @@
 \usepackage[ruled,vlined]{algorithm2e}
 %\usepackage[french,boxed,linesnumbered]{algorithm2e}
 \usepackage{array,multirow,makecell}
+
+\newcommand{\RC}[2][inline]{%
+  \todo[color=blue!10,#1]{\sffamily\textbf{RC:} #2}\xspace}
+\newcommand{\KG}[2][inline]{%
+  \todo[color=green!10,#1]{\sffamily\textbf{KG:} #2}\xspace}
+\newcommand{\AS}[2][inline]{%
+  \todo[color=orange!10,#1]{\sffamily\textbf{AS:} #2}\xspace}
+
+
 \setcellgapes{1pt}
 \makegapedcells
 \newcolumntype{R}[1]{>{\raggedleft\arraybackslash }b{#1}}
@@ -16,6 +25,8 @@
 \newcolumntype{C}[1]{>{\centering\arraybackslash }b{#1}}
 \modulolinenumbers[5]
 
+
+
 \journal{Journal of \LaTeX\ Templates}
 
 %%%%%%%%%%%%%%%%%%%%%%%
@@ -82,13 +93,13 @@
 
 \begin{abstract}
 Polynomials are mathematical algebraic structures that play a great
-role in science and engineering. Finding roots of high degree
+role in science and engineering. Finding the roots of high degree
 polynomials is computationally demanding. In this paper, we present
 the results of a parallel implementation of the Ehrlich-Aberth
 algorithm for the root finding problem for high degree polynomials on
 GPU architectures. The main result of this
 work is to be able to solve high degree polynomials (up
-to 1,000,000) very efficiently. We also compare the results with a
+to 1,000,000)  efficiently. We also compare the results with a
 sequential implementation and the Durand-Kerner method on full and
 sparse polynomials.
 \end{abstract}
@@ -101,15 +112,15 @@ Polynomial root finding, Iterative methods, Ehrlich-Aberth, Durand-Kerner, GPU
 
 \linenumbers
 
-\section{The problem of finding roots of a polynomial}
-Polynomials are mathematical algebraic structures used in science and engineering to capture physical phenomenons and to express any outcome in the form of a function of some unknown variables. Formally speaking,  a polynomial $p(x)$ of degree \textit{n} having $n$ coefficients in the complex plane \textit{C} is :
+\section{The problem of finding the roots of a polynomial}
+Polynomials are mathematical algebraic structures used in science and engineering to capture physical phenomena and to express any outcome in the form of a function of some unknown variables. Formally speaking,  a polynomial $p(x)$ of degree \textit{n} having $n$ coefficients in the complex plane \textit{C} is :
 %%\begin{center}
 \begin{equation}
      {\Large p(x)=\sum_{i=0}^{n}{a_{i}x^{i}}}.
 \end{equation}
 %%\end{center}
 
-The root finding problem consists in finding the values of all the $n$ values of the variable $x$ for which \textit{p(x)} is nullified. Such values are called zeroes of $p$. If zeros are $\alpha_{i},\textit{i=1,...,n}$ the $p(x)$ can be written as :
+The root finding problem consists in finding the values of all the $n$ values of the variable $x$ for which \textit{p(x)} is nullified. Such values are called zeros of $p$. If zeros are $\alpha_{i},\textit{i=1,...,n}$ the $p(x)$ can be written as :
 \begin{equation}
      {\Large p(x)=a_{n}\prod_{i=1}^{n}(x-\alpha_{i}), a_{0} a_{n}\neq 0}.
 \end{equation}
@@ -127,22 +138,22 @@ root-finding problem into a fixed-point problem by setting :
 $g(x)= f(x)-x$.
 \end{center}
 
-Often it is not be possible to solve such nonlinear equation
-root-finding problems analytically. When this occurs we turn to
+It is often impossible to solve such nonlinear equation
+root-finding problems analytically. When this occurs, we turn to
 numerical methods to approximate the solution. 
 Generally speaking, algorithms for solving problems can be divided into
 two main groups: direct methods and iterative methods.
-\\
-Direct methods exist only for $n \leq 4$, solved in closed form by G. Cardano
-in the mid-16th century. However, N. H. Abel in the early 19th
-century showed that polynomials of degree five or more could not
+
+Direct methods only exist for $n \leq 4$, solved in closed form
+by G. Cardano in the mid-16th century. However, N. H. Abel in the early 19th
+century proved that polynomials of degree five or more could not
 be solved by  direct methods. Since then, mathematicians have
 focussed on numerical (iterative) methods such as the famous
-Newton method, the Bernoulli method of the 18th, and the Graeffe method.
+Newton method, the Bernoulli method of the 18th century, and the Graeffe method.
 
 Later on, with the advent of electronic computers, other methods have
 been developed such as the Jenkins-Traub method, the Larkin method,
-the Muller method, and several methods for simultaneous
+the Muller method, and several other methods for the simultaneous
 approximation of all the roots, starting with the Durand-Kerner (DK)
 method:
 %%\begin{center}
@@ -151,7 +162,7 @@ method:
  DK: z_i^{k+1}=z_{i}^{k}-\frac{P(z_i^{k})}{\prod_{i\neq j}(z_i^{k}-z_j^{k})},   i = 1, . . . , n,
 \end{equation}
 %%\end{center}
-where $z_i^k$ is the $i^{th}$ root of the polynomial $P$ at the
+where $z_i^k$ is the $i^{th}$ root of the polynomial $p$ at the
 iteration $k$.
 
 
@@ -169,37 +180,37 @@ Aberth~\cite{Aberth73} uses a different iteration formula given as:
  EA: z_i^{k+1}=z_i^{k}-\frac{1}{{\frac {P'(z_i^{k})} {P(z_i^{k})}}-{\sum_{i\neq j}\frac{1}{(z_i^{k}-z_j^{k})}}}, i = 1, . . . , n,
 \end{equation}
 %%\end{center}
-where $P'(z)$ is the polynomial derivative of $P$ evaluated in the
+where $p'(z)$ is the polynomial derivative of $p$ evaluated in the
 point $z$.
 
 Aberth, Ehrlich and Farmer-Loizou~\cite{Loizou83} have proved that
 the Ehrlich-Aberth method (EA) has a cubic order of convergence for simple roots whereas the Durand-Kerner has a quadratic order of convergence.
 
 
-Iterative methods raise several problem when implemented e.g.
-specific sizes of numbers must be used to deal with this
-difficulty. Moreover, the convergence time of iterative methods
+Moreover, the convergence times of iterative methods
 drastically increases like the degrees of high polynomials. It is expected that the
-parallelization of these algorithms will improve the convergence
-time.
+parallelization of these algorithms will reduce the execution times.
 
 Many authors have dealt with the parallelization of
 simultaneous methods, i.e. that find all the zeros simultaneously. 
 Freeman~\cite{Freeman89} implemented and compared DK, EA and another method of the fourth order proposed
-by Farmer and Loizou~\cite{Loizou83}, 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 equal to 5.5. Later,
+by Farmer and Loizou~\cite{Loizou83}, on an 8-processor linear
+chain, for polynomials of degree 8. The third method often
+diverges, but the first two methods have speed-ups equal to 5.5. Later,
 Freeman and Bane~\cite{Freemanall90}  considered asynchronous
 algorithms, in which each processor continues to update its
-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.
+approximations even though the latest values of other roots
+have not yet been received from the other processors.  In contrast,
+synchronous algorithms   wait the computation of all roots at a given
+iterations  before making a new one.
 Couturier and al.~\cite{Raphaelall01} proposed two methods of parallelization for
 a shared memory architecture and for distributed memory one. They were able to
-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
-where it takes up to 3300 seconds to obtain the same results, the authors show an interesting speedup.
+compute the roots of sparse polynomials of degree 10,000 in 430 seconds with only 8
+personal computers and 2 communications per iteration. Compared to sequential implementations
+where it takes up to 3,300 seconds to obtain the same results, the
+authors' work experiment show an interesting speedup.
 
-Very few works had been performed since this last work until the appearing of
+Few works have been conducted after those works until the appearance of
 the Compute Unified Device Architecture (CUDA)~\cite{CUDA10}, a
 parallel computing platform and a programming model invented by
 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
@@ -211,7 +222,7 @@ Ghidouche and al~\cite{Kahinall14} proposed an implementation of the
 Durand-Kerner method on GPU. Their main
 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. 
+polynomials of degree 48,000. 
 
 
 In this paper, we focus on the implementation of the Ehrlich-Aberth
@@ -225,15 +236,17 @@ 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
+and experimental study results. Finally, Section~\ref{sec7} concludes
 this paper and gives some hints for future research directions in this
 topic.
 
 \section{Ehrlich-Aberth method}
 \label{sec1}
-A cubically convergent iteration method for finding zeros of
-polynomials was proposed by O. Aberth~\cite{Aberth73}. In the
-following we present the main stages of our implementation the Ehrlich-Aberth method.
+A cubically convergent iteration method to find zeros of
+polynomials was proposed by O. Aberth~\cite{Aberth73}. The
+Ehrlich-Aberth method contains 4 main steps, presented in what
+follows.
+
 %The Aberth method is a purely algebraic derivation. 
 %To illustrate the derivation, we let $w_{i}(z)$ be the product of linear factors 
 
@@ -259,7 +272,7 @@ following we present the main stages of our implementation the Ehrlich-Aberth me
 
 
 \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}
@@ -267,16 +280,16 @@ The initialization of a polynomial p(z) is done by setting each of the $n$ compl
 \end{equation}
 
 
-\subsection{Vector $z^{(0)}$ Initialization}
+\subsection{Vector $Z^{(0)}$ Initialization}
 \label{sec:vec_initialization}
-As for any iterative method, we need to choose $n$ initial guess points $z^{(0)}_{i}, i = 1, . . . , n.$
+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.
 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
 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}.
+circles which relies on the result of~\cite{Ostrowski41}.
 
 \begin{equation}
 \label{eq:radiusR}
@@ -296,16 +309,16 @@ v_{i}=\frac{|\frac{a_{n}}{a_{i}}|^{\frac{1}{n-i}}}{2}.
 %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: 
+Here we give a second form of the iterative function used by the Ehrlich-Aberth method: 
 
 \begin{equation}
 \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
+EA2: z^{k+1}_{i}=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}
+{1-\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}\sum_{j=1,j\neq i}^{j=n}{\frac{1}{(z_{i}^{k}-z_{j}^{k})}}}, i=1,. . . .,n
 \end{equation}
 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
+Ehrlich-Aberth method and find the roots of high degree polynomials. More
 details are given in Section~\ref{sec2}.
 \subsection{Convergence Condition}
 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:
@@ -319,15 +332,15 @@ The convergence condition determines the termination of the algorithm. It consis
 \section{Improving the Ehrlich-Aberth Method for high degree polynomials with exp-log formulation}
 \label{sec2}
 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
-mantissa of floating points representations makes the computation of p(z) wrong when z
+as well as the Durand-Kerner implementation, suffers from overflow problems. This
+situation occurs, for instance, in the case where a polynomial,
+having positive coefficients and a large degree, is computed at a
+point $\xi$ where $|\xi| > 1$, where $|z|$ stands for the modolus of a complex $z$. Indeed, the limited number in the
+mantissa of floating points representations makes the computation of $p(z)$ wrong when z
 is large. For example $(10^{50}) +1+ (- 10^{50})$ will give the wrong result
 of $0$ instead of $1$. Consequently, we can not compute the roots
-for large degrees. This problem was early discussed in
-~\cite{Karimall98} for the Durand-Kerner method, the authors
+for large degrees. This problem was discussed earlier in
+~\cite{Karimall98} for the Durand-Kerner method. The authors
 propose to use the logarithm and the exponential of a complex in order to compute the power at a high exponent.
 
 \begin{equation}
@@ -344,16 +357,15 @@ propose to use the logarithm and the exponential of a complex in order to comput
 %%\end{equation}
 
 Using the logarithm (eq.~\ref{deflncomplex}) and the exponential (eq.~\ref{defexpcomplex}) operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations
-manipulate lower absolute values and the roots for large polynomial's degrees can be looked for successfully~\cite{Karimall98}.
+manipulate lower absolute values and the roots for large polynomial degrees can be looked for successfully~\cite{Karimall98}.
 
-Applying this solution for the Ehrlich-Aberth method we obtain the
-iteration function with exponential and logarithm:
+Applying this solution for the iteration function Eq.~\ref{Eq:Hi} of
+Ehrlich-Aberth method, we obtain the following iteration function with exponential and logarithm:
 %%$$ \exp \bigl(  \ln(p(z)_{k})-ln(\ln(p(z)_{k}^{'}))- \ln(1- \exp(\ln(p(z)_{k})-ln(\ln(p(z)_{k}^{'})+\ln\sum_{i\neq j}^{n}\frac{1}{z_{k}-z_{j}})$$
 \begin{equation}
 \label{Log_H2}
-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),
+EA.EL: z^{k+1}_{i}=z_{i}^{k}-\exp \left(\ln \left(
+p(z_{i}^{k})\right)-\ln\left(p'(z^{k}_{i})\right)- \ln\left(1-Q(z^{k}_{i})\right)\right),
 \end{equation}
 
 where:
@@ -361,14 +373,15 @@ where:
 \begin{equation}
 \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).
+\sum_{i\neq j}^{n}\frac{1}{z^{k}_{i}-z^{k}_{j}}\right)\right)i=1,...,n,
 \end{equation}
 
-This solution is applied when the root except the circle unit, represented by the radius $R$ evaluated in C language as:
+This solution is applied when the root except the circle unit, represented by the radius $R$ evaluated in C language as :
+\begin{equation}
+\label{R.EL}
+R = exp(log(DBL\_MAX)/(2*n) );
+\end{equation}
 
-\begin{verbatim}
-R = exp(log(DBL_MAX)/(2*n) );
-\end{verbatim} 
 
 %\begin{equation}
 
@@ -378,26 +391,27 @@ R = exp(log(DBL_MAX)/(2*n) );
 
 \section{Implementation of simultaneous methods in a parallel computer}
 \label{secStateofArt}   
-The main problem of simultaneous methods is that the necessary
+The main problem of simultaneous methods is that the 
 time needed for convergence is increased when we increase
 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
-iterations as by Miranker
+iterations as in Miranker
 ~\cite{Mirankar68,Mirankar71}, Schedler~\cite{Schedler72} and
-Winogard~\cite{Winogard72}. The second approach aims at reducing the
+Winograd~\cite{Winogard72}. The second approach aims at reducing the
 computation time per iteration, as reported
 in~\cite{Benall68,Jana06,Janall99,Riceall06}. 
 
 There are many schemes for the simultaneous approximation of all roots of a given
 polynomial. Several works on different methods and issues of root
-finding have been reported in~\cite{Azad07, Gemignani07, Kalantari08, Zhancall08, Zhuall08}. However, Durand-Kerner and Ehrlich-Aberth methods are the most practical choices among
+finding have been reported in~\cite{Azad07, Gemignani07, Kalantari08,
+  Zhancall08, Zhuall08}. However, the Durand-Kerner and the Ehrlich-Aberth methods are the most practical choices among
 them~\cite{Bini04}. These two methods have been extensively
-studied for parallelization due to their intrinsics parallelism, i.e. the
-computations involved in both methods has some inherent
+studied for parallelization due to their intrinsic parallelism, i.e. the
+computations involved in both methods have some inherent
 parallelism that can be suitably exploited by SIMD machines.
-Moreover, they have fast rate of convergence (quadratic for the
+Moreover, they have fast a rate of convergence (quadratic for the
 Durand-Kerner and cubic for the Ehrlich-Aberth). Various parallel
 algorithms reported for these methods can be found
 in~\cite{Cosnard90, Freeman89,Freemanall90,Jana99,Janall99}.
@@ -408,11 +422,12 @@ 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
-for the Durand-Kerner method, and Ehrlich-Aberth method on a model of
-Optoelectronic Transpose Interconnection System (OTIS).The
-algorithms are mapped on an OTIS-2D torus using N processors. This
-solution needs N processors to compute N roots, which is not
-practical for solving polynomials with large degrees.
+for the Durand-Kerner method, and the Ehrlich-Aberth method on a model of
+Optoelectronic Transpose Interconnection System (OTIS). The
+algorithms are mapped on an OTIS-2D torus using $N$ processors. This
+solution needs $N$ processors to compute $N$ roots, which is not
+practical for solving large degree polynomials.
+
 %Until very recently, the literature did not mention implementations
 %able to compute the roots of large degree polynomials (higher then
 %1000) and within small or at least tractable times.
@@ -424,7 +439,7 @@ In~\cite{Kahinall14} we already proposed the first implementation
 of a root finding method on GPUs, that of the Durand-Kerner method. The main result showed
 that a parallel CUDA implementation is 10 times as fast as the
 sequential implementation on a single CPU for high degree
-polynomials of 48000.
+polynomials of 48,000.
 %In this paper we present a parallel implementation of Ehrlich-Aberth
 %method on GPUs for sparse and full polynomials with high degree (up
 %to $1,000,000$).
@@ -490,7 +505,7 @@ polynomials of 48000.
 %% texture space, which reside in external DRAM, and are accessed via
 %% read-only caches.
 
-\section{ Implementation of Ehrlich-Aberth method on GPU}
+\section{ Implementation of the Ehrlich-Aberth method on GPU}
 \label{sec5}
 %%\subsection{A CUDA implementation of the Aberth's method }
 %%\subsection{A GPU implementation of the Aberth's method }
@@ -542,23 +557,31 @@ polynomials of 48000.
 \subsection{Parallel implementation with CUDA }
 
 In order to implement the Ehrlich-Aberth method in CUDA, it is
-possible to use the Jacobi scheme or the Gauss Seidel one.  With the
+possible to use the Jacobi scheme or the Gauss-Seidel one.  With the
 Jacobi iteration, at iteration $k+1$ we need all the previous values
-$z^{(k)}_{i}$ to compute the new values $z^{(k+1)}_{i}$, that is :
+$z^{k}_{i}$ to compute the new values $z^{k+1}_{i}$, that is :
 
 \begin{equation}
-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.
+EAJ: z^{k+1}_{i}=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}
+{1-\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}\sum_{j=1,j\neq i}^{j=n}{\frac{1}{(z_{i}^{k}-z_{j}^{k})}}}, i=1,. . . .,n.
 \end{equation}
 
 With the Gauss-Seidel iteration, we have:
+%\begin{equation}
+%\label{eq:Aberth-H-GS}
+%EAGS: z^{k+1}_{i}=\frac{p(z^{k}_{i})}{p'(z^{k}_{i})-p(z^{k}_{i})(\sum^{i-1}_{j=1}\frac{1}{z^{k}_{i}-z^{k+1}_{j}}+\sum^{n}_{j=i+1}\frac{1}{z^{k}_{i}-z^{k}_{j}})}, i=1,...,n.
+%\end{equation}
+
 \begin{equation}
 \label{eq:Aberth-H-GS}
-EAGS: z^{k+1}_{i}=\frac{p(z^{k}_{i})}{p'(z^{k}_{i})-p(z^{k}_{i})(\sum^{i-1}_{j=1}\frac{1}{z^{k}_{i}-z^{k+1}_{j}}+\sum^{n}_{j=i+1}\frac{1}{z^{k}_{i}-z^{k}_{j}})}, i=1,...,n.
+EAGS: z^{k+1}_{i}=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}
+{1-\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}(\sum^{i-1}_{j=1}\frac{1}{z^{k}_{i}-z^{k+1}_{j}}+\sum_{j=i+1}^{j=n}{\frac{1}{(z_{i}^{k}-z_{j}^{k})}})}, i=1,. . . .,n
 \end{equation}
-%%Here a finiched my revision %%
+
 Using Eq.~\ref{eq:Aberth-H-GS} to update the vector solution
 \textit{Z}, we expect the Gauss-Seidel iteration to converge more
-quickly because, just as any Jacobi algorithm (for solving linear systems of equations), it uses the most fresh computed roots $z^{k+1}_{i}$.
+quickly because, just as any Jacobi algorithm (for solving linear
+systems of equations), it uses the freshest computed roots $z^{k+1}_{i}$.
 
 %The $4^{th}$ step of the algorithm checks the convergence condition using Eq.~\ref{eq:Aberth-Conv-Cond}.
 %Both steps 3 and 4 use 1 thread to compute all the $n$ roots on CPU, which is very harmful for performance in case of the large degree polynomials.
@@ -575,42 +598,52 @@ quickly because, just as any Jacobi algorithm (for solving linear systems of equ
 
 %In CUDA programming, all the instructions of the  \verb=for= loop are executed by the GPU as a kernel. A kernel is a function written in CUDA and defined by the  \verb=__global__= qualifier added before a usual \verb=C= function, which instructs the compiler to generate appropriate code to pass it to the CUDA runtime in order to be executed on the GPU. 
 
-Algorithm~\ref{alg2-cuda} shows a sketch of the Ehrlich-Aberth algorithm using CUDA.
+Algorithm~\ref{alg2-cuda} shows a sketch of the Ehrlich-Aberth method using CUDA.
 
+\begin{enumerate}
 \begin{algorithm}[H]
 \label{alg2-cuda}
 %\LinesNumbered
 \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), Pu (the derivative of P), $n$ (Polynomial's degrees), $\Delta z_{max}$ (maximum value of stop condition)}
+\KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (Error tolerance
+  threshold), P (Polynomial to solve), Pu (Derivative of P), $n$ (Polynomial degrees), $\Delta z_{max}$ (Maximum value of stop condition)}
 
-\KwOut {$Z$ (The solution root's vector), $ZPrec$ (the previous solution root's vector)}
+\KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}
 
 \BlankLine
 
-Initialization of the of P\;
-Initialization of the of Pu\;
-Initialization of the solution vector $Z^{0}$\;
-Allocate and copy initial data to the GPU global memory\;
-k=0\;
+\item Initialization of the of P\;
+\item Initialization of the of Pu\;
+\item Initialization of the solution vector $Z^{0}$\;
+\item Allocate and copy initial data to the GPU global memory\;
+\item k=0\;
 \While {$\Delta z_{max} > \epsilon$}{
- Let $\Delta z_{max}=0$\;
-$ kernel\_save(ZPrec,Z)$\;
-k=k+1\;
-$ kernel\_update(Z,P,Pu)$\;
-$kernel\_testConverge(\Delta z_{max},Z,ZPrec)$\;
+\item Let $\Delta z_{max}=0$\;
+\item $ kernel\_save(ZPrec,Z)$\;
+\item  k=k+1\;
+\item $ kernel\_update(Z,P,Pu)$\;
+\item $kernel\_testConverge(\Delta z_{max},Z,ZPrec)$\;
 
 }
-Copy results from GPU memory to CPU memory\;
+\item Copy results from GPU memory to CPU memory\;
 \end{algorithm}
+\end{enumerate}
 ~\\ 
 
-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 access data already present in their memories. Next, all the data-parallel arithmetic operations inside the main loop \verb=(do ... while(...))= are executed as kernels by the GPU. The first kernel named \textit{save} in line 6 of Algorithm~\ref{alg2-cuda} consists in saving the vector of polynomial's root found at the previous time-step in GPU memory, in order to check the convergence of the roots after each iteration (line 8, Algorithm~\ref{alg2-cuda}).
-
-The second kernel executes the iterative function $H$ and updates
-$d\_Z$, according to Algorithm~\ref{alg3-update}. We notice that the
-update kernel is called in two forms, separated with the value of
+After the initialization step, all data of the root finding problem
+must be copied from the CPU memory to the GPU global memory. Next, all
+the data-parallel arithmetic operations inside the main loop
+\verb=(while(...))= are executed as kernels by the GPU. The
+first kernel named \textit{save} in line 7 of
+Algorithm~\ref{alg2-cuda} consists in saving the vector of
+polynomial roots found at the previous time-step in GPU memory, in
+order to check the convergence of the roots after each iteration (line
+10, Algorithm~\ref{alg2-cuda}).
+
+The second kernel executes the iterative function and updates
+$Z$, according to Algorithm~\ref{alg3-update}. We notice that the
+update kernel is called in two forms, according to the value
 \emph{R} which determines the radius beyond which we apply the
 exponential logarithm algorithm. 
 
@@ -620,35 +653,34 @@ exponential logarithm algorithm.
 \caption{Kernel update}
 
 \eIf{$(\left|Z\right|<= R)$}{
-$kernel\_update((Z,P,Pu)$\;}
+$kernel\_update(Z,P,Pu)$\;}
 {
-$kernel\_update\_ExpoLog((Z,P,Pu))$\;
+$kernel\_update\_ExpoLog(Z,P,Pu)$\;
 }
 \end{algorithm}
 
-The first form executes formula the EA function Eq.~\ref{Eq:Hi} 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 the EA.EL
-function Eq.~\ref{Log_H2}
-(with Eq.~\ref{deflncomplex}, Eq.~\ref{defexpcomplex}). The radius $R$ is evaluated as :
-
-$$R = \exp( \log(DBL\_MAX) / (2*n) )$$ where $DBL\_MAX$ stands for the maximum representable double value.
+If the modulus
+of the current complex is less than a given value called the
+radius i.e. ($ |z^{k}_{i}|<= R$), then the classical form of the EA
+function Eq.~\ref{Eq:Hi} is executed else the EA.EL
+function Eq.~\ref{Log_H2} is executed.
+(with Eq.~\ref{deflncomplex}, Eq.~\ref{defexpcomplex}). The radius $R$ is evaluated as in Eq.~\ref{R.EL}.
 
 The last kernel checks the convergence of the roots after each update
-of $Z^{(k)}$, according to formula Eq.~\ref{eq:Aberth-Conv-Cond}. We used the functions of the CUBLAS Library (CUDA Basic Linear Algebra Subroutines) to implement this kernel. 
+of $Z^{k}$, according to formula Eq.~\ref{eq:Aberth-Conv-Cond}. We used the functions of the CUBLAS Library (CUDA Basic Linear Algebra Subroutines) to implement this kernel. 
 
 The kernel terminates its computations when all the roots have
 converged. It should be noticed that, as blocks of threads are
 scheduled automatically by the GPU, we have absolutely no control on
 the order of the blocks. Consequently, our algorithm is executed more
-or less in an asynchronous iteration model, where blocks of roots are
-updated in a non deterministic way. As the Durand-Kerner method has
-been proved to converge with asynchronous iterations, we think it is
-similar with the Ehrlich-Aberth method, but we did not try to prove
+or less with the asynchronous iteration model, where blocks of roots
+are updated in a non deterministic way. As the Durand-Kerner method
+has been proved to converge with asynchronous iterations, we think it
+is similar with the Ehrlich-Aberth method, but we did not try to prove
 this in that paper. Another consequence of that, is that several
-executions of our algorithm with the same polynomial do no give
-necessarily the same result (but roots have the same accuracy) and the
-same number of iterations (even if the variation is not very
+executions of our algorithm with the same polynomial do not
+necessarily give the same result (but roots have the same accuracy)
+and the same number of iterations (even if the variation is not very
 significant).
 
 
@@ -684,14 +716,14 @@ all the coefficients are not null. A full polynomial is defined by:
 %polynomials. The execution time remains the
 %element-key which justifies our work of parallelization.
 For our tests, a CPU Intel(R) Xeon(R) CPU
-E5620@2.40GHz and a GPU K40 (with 6 Go of ram) is used. 
+E5620@2.40GHz and a GPU K40 (with 6 Go of ram) are used. 
 
 
 %\subsection{Comparative study}
 %First, performances of the Ehrlich-Aberth method  of root finding polynomials
 %implemented on CPUs and on GPUs are studied.
 
-We performed a set of experiments on the sequential and the parallel algorithms, for both sparse and full polynomials and different sizes. We took into account the execution times, the  polynomial size and the number of threads per block performed by sum or each experiment on CPU and on GPU.
+We performed a set of experiments on the sequential and the parallel algorithms, for both sparse and full polynomials of different sizes. We took into account the execution times, the  polynomial size and the number of threads per block performed by sum or each experiment on CPU and on GPU.
 
 All experimental results obtained from the simulations are made in
 double precision data, the convergence threshold of the methods is set
@@ -712,14 +744,36 @@ of the methods are given in Section~\ref{sec:vec_initialization}.
   on a CPU with OpenMP (1 core, 4 cores) and 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, 
-In Figure~\ref{fig:01}, 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 2,500 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.
+%%Figure 1 %%show a comparison of execution time between the parallel
+%%and sequential version of the Ehrlich-Aberth algorithm with sparse
+%%polynomial exceed 100000,
+
+In Figure~\ref{fig:01}, we report the execution times of the
+Ehrlich-Aberth method on one core of a Quad-Core Xeon E5620 CPU, on
+four cores on the same machine with \textit{OpenMP} and on a Nvidia
+Tesla K40 GPU.  We chose different sparse polynomials with degrees
+ranging from 100,000 to 1,000,000. We can see that the implementation
+on the GPU is faster than those implemented on the CPU.
+However, the execution time for the
+CPU (4 cores) implementation exceed 5,000s for 250,000 degrees
+polynomials. On the other hand, the GPU implementation for the same
+polynomials do not take more 100s. With the GPU
+we can solve high degree polynomials very quickly up to degree 1,000,000. We can also notice that the GPU implementation are
+ almost 40 times faster then the implementation on the CPU (4 cores).
+
+
+ 
+
+%This reduction of time allows us to compute roots of polynomial of more important degree at the same time than with a CPU.
  
  %We notice that the convergence precision is a round $10^{-7}$ for the both implementation on CPU and GPU. Consequently, we can conclude that Ehrlich-Aberth on GPU are faster and accurately then CPU implementation.
 
 \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 1,024. We took into account the execution time for both sparse and full of 10 different polynomials of size 50,000 and 10 different polynomials of size 500,000 degrees.
+To optimize the performances of an algorithm on a GPU, it is necessary to maximize the use of cores GPU (maximize the number of threads executed in parallel). In fact, it is interesting to see the influence of the number of threads per block on the execution time of Ehrlich-Aberth algorithm. 
+For that, we noticed that the maximum number of threads per block for
+the Nvidia Tesla K40 GPU is 1,024, so we varied the number of threads
+per block from 8 to 1,024. We took into account the execution time for
+10 different sparse and full polynomials of degree 50,000 and of degree 500,000.
 
 \begin{figure}[htbp]
 \centering
@@ -728,15 +782,21 @@ For that, we notice that the maximum number of threads per block for the Nvidia
 \label{fig:02}
 \end{figure}
 
-The figure 2 show that, the best execution time for both sparse and full polynomial are given when the threads number varies between 64 and 256 threads per bloc. We notice that with small polynomials the best number of threads per block is 64, Whereas, the large polynomials the best number of threads per block is 256. However,In the following experiments we specify that the number of thread by block is 256.
+Figure~\ref{fig:02} shows that, the best execution time for both
+sparse and full polynomial are given when the threads number varies
+between 64 and 256 threads per block. We notice that with small
+polynomials the best number of threads per block is 64, whereas for large polynomials the best number of threads per block is
+256. However, in the following experiments we specify that the number
+of threads per block is 256.
+
 
 \subsection{Influence of exp-log solution to compute high degree polynomials}
 
-In this experiment we report the performance of the exp-log solution described in Section~\ref{sec2} to compute very high degrees polynomials.   
+In this experiment we report the performance of the exp-log solution described in Section~\ref{sec2} to compute high degree polynomials.   
 \begin{figure}[htbp]
 \centering
   \includegraphics[width=0.8\textwidth]{figures/sparse_full_explog}
-\caption{The impact of exp.log solution to compute very high degrees of  polynomial.}
+\caption{The impact of exp-log solution to compute  high degree  polynomials}
 \label{fig:03}
 \end{figure}
 
@@ -746,14 +806,14 @@ the Ehrlich-Aberth method using the exp-log solution and the
 execution time of the Ehrlich-Aberth method without this solution,
 with full and sparse polynomials degrees. We can see that the
 execution times for both algorithms are the same with full polynomials
-degrees less than 4,000 and sparse polynomials less than 150,000. We
-also clearly show that the classical version (without log-exp) of
-Ehrlich-Aberth algorithm do not converge after these degree with
-sparse and full polynomials. In counterpart, the new version of
-Ehrlich-Aberth algorithm with the exp-log solution can solve very
+degree inferior to 4,000 and sparse polynomials inferior to 150,000. We
+also clearly show that the classical version (without exp-log) of
+Ehrlich-Aberth algorithm does not converge after these degrees with
+sparse and full polynomials. On the contrary, the new version of
+the Ehrlich-Aberth algorithm with the exp-log solution can solve
 high degree polynomials.
 
-%in fact, when the modulus of the roots are up than \textit{R} given in ~\ref{R},this exceed the limited number in the mantissa of floating points representations and can not compute the iterative function given in ~\ref{eq:Aberth-H-GS} to obtain the root solution, who justify the divergence of the classical Ehrlich-Aberth algorithm. However, applying 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 . 
+%in fact, when the modulus of the roots are up than \textit{R} given in ~\ref{R},this exceed the limited number in the mantissa of floating points representations and can not compute the iterative function given in ~\ref{eq:Aberth-H-GS} to obtain the root solution, who justify the divergence of the classical Ehrlich-Aberth algorithm. However, applying exp-log solution given in ~\ref{sec2} took into account the limit of floating using the iterative function in(Eq.~\ref{Log_H1},Eq.~\ref{Log_H2} and allows to solve a very large polynomials degrees . 
 
 
 
@@ -761,7 +821,7 @@ high degree polynomials.
 \subsection{Comparison of the Durand-Kerner and the Ehrlich-Aberth methods}
 
 In this part, we  compare the Durand-Kerner and the Ehrlich-Aberth
-methods on GPU. We took into account the execution times, the number of iterations and the polynomials size for the both sparse and full polynomials.  
+methods on GPU. We took into account the execution times, the number of iterations and the polynomials size for both sparse and full polynomials.  
 
 \begin{figure}[htbp]
 \centering
@@ -770,18 +830,11 @@ methods on GPU. We took into account the execution times, the number of iteratio
 \label{fig:04}
 \end{figure}
 
-\begin{figure}[htbp]
-\centering
-  \includegraphics[width=0.8\textwidth]{figures/EA_DK1}
-\caption{Execution times of the  Durand-Kerner and the Ehrlich-Aberth methods on GPU}
-\label{fig:0}
-\end{figure}
-
 Figure~\ref{fig:04} shows the execution times of both methods with
 sparse polynomial degrees ranging from 1,000 to 1,000,000. We can see
 that the Ehrlich-Aberth algorithm is faster than Durand-Kerner
-algorithm, with an average of 25 times faster. Then, when degrees of
-polynomial exceed 500,000 the execution times with DK are very long.
+algorithm, being on average 25 times faster. Then, when degrees of
+polynomials exceed 500,000 the execution times with DK are very long.
 
 %with double precision not exceed $10^{-5}$.
 
@@ -793,17 +846,18 @@ polynomial exceed 500,000 the execution times with DK are very long.
 \label{fig:05}
 \end{figure}
 
-Figure~\ref{fig:05} show the evaluation of the number of iteration according
-to degree of polynomial from both EA and DK algorithms, we can see
-that the iteration number of DK is of order 100 while EA is of order
-10. Indeed the computing of the derivative of P (the polynomial to
-resolve) in the iterative function (Eq.~\ref{Eq:Hi}) executed by EA
-allows the algorithm to converge more quickly. In counterpart, the
-DK operator (Eq.~\ref{DK}) needs low operation, consequently low
-execution time per iteration, but it needs more iterations to converge.
+Figure~\ref{fig:05} shows the evaluation of the number of iterations according
+to the degree of polynomials for both EA and DK algorithms. We can see
+that the number of iterations of DK is of order 100 while EA is of order
+10. Indeed the computation of the derivative of P in the iterative function (Eq.~\ref{Eq:Hi}) executed by EA
+allows the algorithm to converge faster. On the contrary, the
+DK operator (Eq.~\ref{DK}) needs low operations, consequently low
+execution times per iteration, but it needs more iterations to converge.
+
+
 
 
- \section{Conclusion and perspective}
+ \section{Conclusion and perspectives}
 \label{sec7}
 In this paper we have presented the parallel implementation
 Ehrlich-Aberth method on GPU for the problem of finding roots
@@ -813,17 +867,18 @@ applied to the iterative function allows to solve high degree
 polynomials.
 
 We have performed many experiments with the Ehrlich-Aberth method in
-GPU. These experiments highlight that this method is very efficient in
-GPU compared to all the other implementations. The improvement with
+GPU. These experiments highlight that this method is more efficient in
+GPU than all the other implementations. The improvement with
 the exponential logarithm solution allows us to solve sparse and full
 high degree polynomials up to 1,000,000 degree. Hence, it may be
-possible to consider to use polynomial root finding methods in other
+possible to consider  using polynomial root finding methods in other
 numerical applications on GPU.
 
 
 In future works, we plan to investigate the possibility of using
-several multiple GPUs simultaneously, either with multi-GPU machine or
-with cluster of GPUs.
+several multiple GPUs simultaneously, either with a multi-GPU machine or
+with a cluster of GPUs. It may also be interesting to study the
+implementation of other root finding polynomial methods on GPU.
 
 
 
