X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/kahina_paper2.git/blobdiff_plain/7620333f88cf490eac21e8a1c1e1d29a6934533f..6690e4dc2d54dee46bca4d1c54bf8f1444659d71:/paper.tex?ds=inline

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@@ -178,7 +178,7 @@
 
 % *** SPECIALIZED LIST PACKAGES ***
 %
-\usepackage{algorithmic}
+
 % algorithmic.sty was written by Peter Williams and Rogerio Brito.
 % This package provides an algorithmic environment fo describing algorithms.
 % You can use the algorithmic environment in-text or within a figure
@@ -316,6 +316,27 @@
 % argument is your BibTeX string definitions and bibliography database(s)
 %\bibliography{IEEEabrv,../bib/paper}
 %\bibliographystyle{elsarticle-num}
+
+
+
+
+\usepackage{amsfonts}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage[textsize=footnotesize]{todonotes}
+\newcommand{\LZK}[2][inline]{%
+  \todo[color=red!10,#1]{\sffamily\textbf{LZK:} #2}\xspace}
+\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}
+
+
+
+
+
 \begin{document}
 %
 % paper title
@@ -324,29 +345,22 @@
 % not capitalized unless they are the first or last word of the title.
 % Linebreaks \\ can be used within to get better formatting as desired.
 % Do not put math or special symbols in the title.
-\title{A parallel implementation of Ehrlich-Aberth algorithm  for root finding of polynomials
-on Multi-GPU with OpenMP/MPI}
+\title{Two parallel implementations of Ehrlich-Aberth algorithm for root-finding of polynomials on multiple GPUs with OpenMP and MPI}
 
 
 % author names and affiliations
 % use a multiple column layout for up to three different
 % affiliations
-\author{\IEEEauthorblockN{Michael Shell}
-\IEEEauthorblockA{School of Electrical and\\Computer Engineering\\
-Georgia Institute of Technology\\
-Atlanta, Georgia 30332--0250\\
-Email: http://www.michaelshell.org/contact.html}
+\author{\IEEEauthorblockN{Kahina Guidouche, Abderrahmane Sider }
+  \IEEEauthorblockA{Laboratoire LIMED\\
+    Faculté des sciences exactes\\
+    Université de Bejaia, 06000, Algeria\\
+Email: \{kahina.ghidouche,ar.sider\}@univ-bejaia.dz}
 \and
-\IEEEauthorblockN{Homer Simpson}
-\IEEEauthorblockA{Twentieth Century Fox\\
-Springfield, USA\\
-Email: homer@thesimpsons.com}
-\and
-\IEEEauthorblockN{James Kirk\\ and Montgomery Scott}
-\IEEEauthorblockA{Starfleet Academy\\
-San Francisco, California 96678--2391\\
-Telephone: (800) 555--1212\\
-Fax: (888) 555--1212}}
+\IEEEauthorblockN{Lilia Ziane Khodja, Raphaël Couturier}
+\IEEEauthorblockA{FEMTO-ST Institute\\
+  University of   Bourgogne Franche-Comte, France\\
+Email: zianekhodja.lilia@gmail.com\\ raphael.couturier@univ-fcomte.fr}}
 
 % conference papers do not typically use \thanks and this command
 % is locked out in conference mode. If really needed, such as for
@@ -385,10 +399,21 @@ Fax: (888) 555--1212}}
 % As a general rule, do not put math, special symbols or citations
 % in the abstract
 \begin{abstract}
-The abstract goes here.
+Finding roots of polynomials is a very important part of solving
+real-life problems but it is not so easy for polynomials of high
+degrees. In this paper, we present two different parallel algorithms
+of the Ehrlich-Aberth method to find roots of sparse and fully defined
+polynomials of high degrees. Both algorithms are based on CUDA
+technology to be implemented on multi-GPU computing platforms but each
+using different parallel paradigms: OpenMP or MPI. The experiments
+show a quasi-linear speedup by using up-to 4 GPU devices compared to 1
+GPU to find roots of polynomials of degree up-to 1.4
+million. Moreover, other experiments show it is possible to find roots
+of polynomials of degree up-to 5 millions.
 \end{abstract}
 
 % no keywords
+\LZK{Faut pas mettre des keywords?}
 
 
 
@@ -404,66 +429,110 @@ The abstract goes here.
 \IEEEpeerreviewmaketitle
 
 
-
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Introduction}
-Polynomials are mathematical algebraic structures that play an important role in science and engineering by capturing physical phenomena and by expressing any outcome as 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}
+%Polynomials are mathematical algebraic structures that play an important role in science and engineering by capturing physical phenomena and expressing any outcome as a function of some unknown variables. Formally speaking, a polynomial $p(x)$ of degree $n$ having $n$ coefficients in the complex plane $\mathbb{C}$ is:
+%\begin{equation}
+%p(x)=\sum_{i=0}^{n}{a_ix^i}.
+%\end{equation}
+%\LZK{Dans ce cas le polynôme a $n+1$ coefficients et non pas $n$!}
 
-The root finding problem consists in finding the values of all the $n$ different values of the variable $x$ for which \textit{p(x)} is null. Such values are called zeros of $p$. If zeros are $\alpha_{i},\textit{i=1,...,n}$ then $p(x)$ 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 issue of finding the roots of polynomials of very high degrees arises in many complex problems in various fields, such as algebra, biology, finance, physics or climatology [1]. In algebra for example, finding eigenvalues or eigenvectors of any real/complex matrix amounts to that of finding the roots of the so-called characteristic polynomial.
 
-The problem of finding the roots of polynomials can be encountered in numerous applications. Most of the numerical methods that deal with this problem are simultaneous ones, i.e that find concurrently all of $n$ zeroes. These methods start from the initial approximations of all the roots of the polynomial and give a sequence of approximations that converge to the roots of the polynomial. The first method of this group is Durand-Kerner method:
+Finding roots of polynomials of very high degrees arises in many complex problems in various domains such as algebra, biology or physics. A polynomial $p(x)$ in $\mathbb{C}$ in one variable $x$ is an algebraic expression in $x$ of the form:
 \begin{equation}
-\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,
+p(x) = \displaystyle\sum^n_{i=0}{a_ix^i},a_n\neq 0, 
 \end{equation}
-%%\end{center}
-where $z_i^k$ is the $i^{th}$ root of the polynomial $p$ at the
-iteration $k$.
-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:
-%%\begin{center}
+where $\{a_i\}_{0\leq i\leq n}$ are complex coefficients and $n$ is a high integer number. If $a_n\neq0$ then $n$ is called the degree of the polynomial. The root-finding problem consists in finding the $n$ different values of the unknown variable $x$ for which $p(x)=0$. Such values are called roots of $p(x)$. Let $\{z_i\}_{1\leq i\leq n}$ be the roots of polynomial $p(x)$, then $p(x)$ can be written as :
 \begin{equation}
-\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,
+ p(x)=a_n\displaystyle\prod_{i=1}^n(x-z_i), a_n\neq 0.
 \end{equation}
+%\LZK{Pourquoi $a_0a_n\neq 0$ ?: $a_0$ pour la premiere equation et $a_n$ pour la deuxieme equation }
+
+%The problem of finding the roots of polynomials can be encountered in numerous applications. \LZK{A mon avis on peut supprimer cette phrase}
+Most of the numerical methods that deal with the polynomial root-finding problem are simultaneous methods, \textit{i.e.} the iterative methods to find simultaneous approximations of the $n$ polynomial roots. These methods start from the initial approximations of all $n$ polynomial roots and give a sequence of approximations that converge to the roots of the polynomial. Two examples of well-known simultaneous methods for root-finding problem of polynomials are  Durand-Kerner method~\cite{Durand60,Kerner66} and Ehrlich-Aberth method~\cite{Ehrlich67,Aberth73}.
+%\LZK{Pouvez-vous donner des références pour les deux méthodes?, c'est fait}
+
+%The first method of this group is Durand-Kerner method:
+%\begin{equation}
+%\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, \ldots, n,
+%\end{equation}
+%where $z_i^k$ is the $i^{th}$ root of the polynomial $p$ at the iteration $k$. Another method discovered by Borsch-Supan~\cite{ Borch-Supan63} and also described by Ehrlich~\cite{Ehrlich67} and Aberth~\cite{Aberth73} uses a different iteration form as follows:
+%%\begin{center}
+%\begin{equation}
+%\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, \ldots, n,
+%\end{equation}
 %%\end{center}
-where $p'(z)$ is the polynomial derivative of $p$ evaluated in the
-point $z$.
+%where $p'(z)$ is the polynomial derivative of $p$ evaluated in the point $z$.
 
 %Aberth, Ehrlich and Farmer-Loizou~\cite{Loizou83} have proved that
 %the Ehrlich-Aberth method (EA) has a cubic order of convergence for simple roots whereas the Durand-Kerner has a quadratic order of %convergence.
 
-The main problem of the simultaneous methods is that the necessary time needed for the convergence is increased with the increasing of the degree of the polynomial. Many authors have treated the problem of implementing  simultaneous methods in parallel. 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, Freeman and Bane~\cite{Freemanall90} considered asynchronous algorithms, in which each processor continues to update its approximations even though the latest values of other $z^{k}_{i}$ 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 and al.~\cite{Raphaelall01} proposed two methods of parallelization for a shared memory architecture with \textit{OpenMP} and for distributed memory one with \textit{MPI}. They were able to compute the roots of sparse polynomials of degree 10,000 in 116 seconds with \textit{OpenMP} and 135 seconds with \textit{MPI} only by using 8 personal computers and 2 communications per iteration. Comparing to the sequential implementation where it takes up to 3,300 seconds to obtain the same results, the authors show an interesting speedup.
-
-Very few work had been performed since then until the appearing of the Compute Unified Device Architecture (CUDA)~\cite{CUDA10}, a parallel computing platform and a programming model invented by NVIDIA. The computing 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. 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 48,000.
-
-Finding polynomial roots rapidly and accurately is the main objective of our work. In this paper we propose the parallelization of Ehrlich-Aberth method using parallel programming paradigms (OpenMP, MPI) on GPUs. We consider two architectures: shared memory with OpenMP API and distributed memory MPI API. The first approach is based on threads from the same system process, with each thread attached to one GPU and after the various memory allocations, each thread launches its part of computations. To do this we must first load on the GPU required data and after the computations are carried, repatriate the result on the host. The second approach i.e distributed memory with MPI relies on the MPI library which is often used for parallel programming~\cite{Peter96} in
-cluster systems because it is a message-passing programming language. Each GPU is attached to one MPI process, and a loop is in charge of the distribution of tasks between the MPI processes. This solution can be used on one GPU, or executed on a distributed cluster of GPUs, employing the Message Passing Interface (MPI) to communicate between separate CUDA cards. This solution permits scaling of the problem size to larger classes than would be possible on a single device and demonstrates the performance which users might expect from future
-HPC architectures where accelerators are deployed. 
- 
-This paper is organized as follows, in section 2 we recall the Ehrlich-Aberth method. In section 3 we present EA algorithm on single GPU. In section 4 we propose the EA algorithm implementation on Multi-GPU for (OpenMP-CUDA) approach and (MPI-CUDA) approach. In section 5 we present our experiments and discus it. Finally, Section~\ref{sec6} concludes this paper and gives some hints for future research directions in this topic.
+The main problem of the simultaneous methods is that the necessary
+time needed for the convergence increases with the increasing of the
+polynomial's degree. Many authors have treated the problem of
+implementing  simultaneous methods in
+parallel. Freeman~\cite{Freeman89} implemented and compared
+Durand-Kerner method, Ehrlich-Aberth method and another method of the
+fourth order of convergence proposed by Farmer and
+Loizou~\cite{Loizou83} on a 8-processor linear chain, for polynomials
+of degree up-to 8. The method of Farmer and Loizou~\cite{Loizou83}
+often diverges, but the first two methods (Durand-Kerner and
+Ehrlich-Aberth methods) have a speed-up equals 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 approximations $z^{k}_{i}$ 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 parallelization for a shared memory architecture with OpenMP and
+for a distributed memory one with MPI. They are able to compute the
+roots of sparse polynomials of degree 10,000 in 116 seconds with
+OpenMP and 135 seconds with MPI only by using 8 personal computers and
+2 communications per iteration. The authors showed an interesting
+speedup comparing to the sequential implementation which takes up-to
+3,300 seconds to obtain same results. 
+\RC{si on donne des temps faut donner le proc, comme c'est vieux à mon avis faut supprimer ca, votre avis?} 
+\LZK{Supprimons ces détails et mettons une référence s'il y en a une}
+
+Very few work had been performed since then until the appearing of the Compute Unified Device Architecture (CUDA)~\cite{CUDA15}, a parallel computing platform and a programming model invented by NVIDIA. The computing power of GPUs (Graphics Processing Units) has exceeded that of traditional processors CPUs. However, CUDA adopts a totally new computing architecture to use the hardware resources provided by the GPU in order to offer a stronger computing ability to the massive data computing. Ghidouche et al.~\cite{Kahinall14} proposed an implementation of the Durand-Kerner method on a single GPU. Their main results showed that a parallel CUDA implementation is about 10 times faster than the sequential implementation on a single CPU for sparse polynomials of degree 48,000.
+
+%Finding polynomial roots rapidly and accurately is the main objective of our work. In this paper we propose the parallelization of Ehrlich-Aberth method using two parallel programming paradigms OpenMP and MPI on multi-GPU platforms. We consider two architectures: shared memory and distributed memory computers. The first parallel algorithm is implemented on shared memory computers by using OpenMP API. It is based on threads created from the same system process, such that each thread is attached to one GPU. In this case the communications between GPUs are done by OpenMP threads through shared memory. The second parallel algorithm uses the MPI API, such that each GPU is attached and managed by a MPI process. The GPUs exchange their data by message-passing communications. This latter approach is more used on distributed memory clusters to solve very complex problems that are too large for traditional supercomputers, which are very expensive to build and run.
+%\LZK{Cette partie est réécrite. \\ Sinon qu'est ce qui a été fait pour l'accuracy dans ce papier (Finding polynomial roots rapidly and accurately is the main objective of our work.)?}
+%\LZK{Les contributions ne sont pas définies !!}
+
+%In this paper we propose the parallelization of Ehrlich-Aberth method using two parallel programming paradigms OpenMP and MPI on CUDA multi-GPU platforms. Our CUDA-MPI and CUDA-OpenMP codes are the first implementations of Ehrlich-Aberth method with multiple GPUs for finding roots of polynomials. Our major contributions include:
+%\LZK{Pourquoi la méthode Ehrlich-Aberth et pas autres? the Ehrlich-Aberth have very good convergence  and it is suitable to be implemented in parallel computers.}
+In this paper we propose the parallelization of Ehrlich-Aberth method which has a good convergence and it is suitable to be implemented in parallel computers. We use two parallel programming paradigms OpenMP and MPI on CUDA multi-GPU platforms. Our CUDA-MPI and CUDA-OpenMP codes are the first implementations of Ehrlich-Aberth method with multiple GPUs for finding roots of polynomials. Our major contributions include:
+\LZK{J'ai ajouté une phrase pour justifier notre choix de la méthode Ehrlich-Aberth. A revérifier.}
+ \begin{itemize}
+ %\item An improvements for the Ehrlich-Aberth method using the exponential logarithm in order to be able to solve sparse and full polynomial of degree up to 1, 000, 000.\RC{j'ai envie de virer ca, car c'est pas la nouveauté dans ce papier}
+ %\item A parallel implementation of Ehrlich-Aberth method on single GPU with CUDA.\RC{idem}
+\item The parallel implementation of Ehrlich-Aberth algorithm on a multi-GPU platform with a shared memory using OpenMP API. It is based on threads created from the same system process, such that each thread is attached to one GPU. In this case the communications between GPUs are done by OpenMP threads through shared memory.
+\item The parallel implementation of Ehrlich-Aberth algorithm on a multi-GPU platform with a distributed memory using MPI API, such that each GPU is attached and managed by a MPI process. The GPUs exchange their data by message-passing communications. 
+ \end{itemize}
+This latter approach is more used on clusters to solve very complex problems that are too large for traditional supercomputers, which are very expensive to build and run.
+\LZK{Pas d'autres contributions possibles? J'ai supprimé les deux premiers points proposés précédemment.}
+
+%This paper is organized as follows. In Section~\ref{sec2} we recall the Ehrlich-Aberth method. In section~\ref{sec3} we present EA algorithm on single GPU. In section~\ref{sec4} we propose the EA algorithm implementation on Multi-GPU for (OpenMP-CUDA) approach and (MPI-CUDA) approach. In sectioné\ref{sec5} we present our experiments and discus it. Finally, Section~\ref{sec6} concludes this paper and gives some hints for future research directions in this topic.}
+
+The paper is organized as follows. In Section~\ref{sec2} we present three different parallel programming models OpenMP, MPI and CUDA. In Section~\ref{sec3} we present the implementation of the Ehrlich-Aberth algorithm on a single GPU. In Section~\ref{sec4} we present the parallel implementations of the Ehrlich-Aberth algorithm on multiple GPUs using the OpenMP and MPI approaches. In section~\ref{sec5} we present our experiments and discuss them. Finally, Section~\ref{sec6} concludes this paper and gives some hints for future research directions in this topic. 
+%\LZK{A revoir toute cette organization: je viens de la revoir}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  
- 
-\section{Parallel Programmings Model}
+\section{Parallel programming models}
+\label{sec2}
+Our objective consists in implementing a root-finding algorithm of polynomials on multiple GPUs. To this end, it is primordial to know how to manage CUDA contexts of different GPUs. A direct method for controlling the various GPUs is to use as many threads or processes as GPU devices. We investigate two parallel paradigms: OpenMP and MPI. In this case, the GPU indices are defined according to the identifiers of the OpenMP threads or the ranks of the MPI processes. In this section we present the parallel programming models: OpenMP, MPI and CUDA.
  
 \subsection{OpenMP}
-Open Multi-Processing (OpenMP) is a shared memory architecture API that provides multi thread capacity~\cite{openmp13}. OpenMP is
-a portable approach for parallel programming on shared memory systems based on compiler directives, that can be included in order
-to parallelize a loop. In this way, a set of loops can be distributed along the different threads that will access to different data allocated in local shared memory. One of the advantages of OpenMP is its global view of application memory address space that allows relatively fast development of parallel applications with easier maintenance. However, it is often difficult to get high rates of performance in large scale applications. Although usage of OpenMP  threads and managed data explicitly done with MPI can be considered, this approcache undermines the advantages of OpenMP.
+%Open Multi-Processing (OpenMP) is a shared memory architecture API that provides multi thread capacity~\cite{openmp13}. OpenMP is a portable approach for parallel programming on shared memory systems based on compiler directives, that can be included in order to parallelize a loop. In this way, a set of loops can be distributed along the different threads that will access to different data allocated in local shared memory. One of the advantages of OpenMP is its global view of application memory address space that allows relatively fast development of parallel applications with easier maintenance. However, it is often difficult to get high rates of performance in large scale applications. Although usage of OpenMP  threads and managed data explicitly done with MPI can be considered, this approcache undermines the advantages of OpenMP.
 
-%\subsection{OpenMP} %L'article en Français Programmation multiGPU – OpenMP versus MPI
+%\subsection{OpenMP} 
 %OpenMP is a shared memory programming API based on threads from
 %the same system process. Designed for multiprocessor shared memory UMA or
 %NUMA [10], it relies on the execution model SPMD ( Single Program, Multiple Data Stream )
@@ -473,29 +542,31 @@ to parallelize a loop. In this way, a set of loops can be distributed along the
 %Sequential natively. Threads share some or all of the available memory and can
 %have private memory areas [6].
 
+OpenMP (Open Multi-processing) is an application programming interface for parallel programming~\cite{openmp13}. It is a portable approach based on the multithreading designed for shared memory computers, where a master thread forks a number of slave threads which execute blocks of code in parallel. An OpenMP program alternates sequential regions and parallel regions of code, where the sequential regions are executed by the master thread and the parallel ones may be executed by multiple threads. During the execution of an OpenMP program the threads communicate their data (read and modified) in the shared memory. One advantage of OpenMP is the global view of the memory address space of an application. This allows relatively a fast development of parallel applications with easier maintenance. However, it is often difficult to get high rates of performances in large scale-applications. 
+
 \subsection{MPI} 
-The MPI (Message Passing Interface) library allows to create computer programs that run on a distributed memory architecture. The various processes have their own environment of execution and execute their code in a asynchronous way, according to the MIMD model  (Multiple Instruction streams, Multiple Data streams); they communicate and synchronise by exchanging messages~\cite{Peter96}. MPI messages are explicitly sent, while the exchanges are implicit within the framework of a multi-thread programming environment like OpenMP or Pthreads.
+%The MPI (Message Passing Interface) library allows to create computer programs that run on a distributed memory architecture. The various processes have their own environment of execution and execute their code in a asynchronous way, according to the MIMD model  (Multiple Instruction streams, Multiple Data streams); they communicate and synchronize by exchanging messages~\cite{Peter96}. MPI messages are explicitly sent, while the exchanges are implicit within the framework of a multi-thread programming environment like OpenMP or Pthreads.
+
+MPI (Message Passing Interface) is a portable message passing style of the parallel programming designed especially for the distributed memory architectures~\cite{Peter96}. In most MPI implementations, a computation contains a fixed set of processes created at the initialization of the program in such way one process is created per processor. The processes synchronize their computations and communicate by sending/receiving messages to/from other processes. In this case, the data are explicitly exchanged by message passing while the data exchanges are implicit in a multithread programming model like OpenMP and Pthreads. However in the MPI programming model, the processes may either execute different programs referred to as multiple program multiple data (MPMD) or every process executes the same program (SPMD). The MPI approach is one of most used HPC programming model to solve large scale and complex applications.
  
-\subsection{CUDA}%L'article en anglais Multi-GPU and multi-CPU accelerated FDTD scheme for vibroacoustic applications
-CUDA (an acronym for Compute Unified Device Architecture) is a parallel computing architecture developed by NVIDIA~\cite{CUDA10}. The
-unit of execution in CUDA is called a thread. Each thread executes a kernel by the streaming processors in parallel. In CUDA,
-a group of threads that are executed together is called a thread block, and the computational grid consists of a grid of thread
-blocks. Additionally, a thread block can use the shared memory on a single multiprocessor while the grid executes a single
-CUDA program logically in parallel. Thus in CUDA programming, it is necessary to design carefully the arrangement of the thread
-blocks in order to ensure low latency and a proper usage of shared memory, since it can be shared only in a thread block
-scope. The effective bandwidth of each memory space depends on the memory access pattern. Since the global memory has lower
-bandwidth than the shared memory, the global memory accesses should be minimized.
+\subsection{CUDA}
+%CUDA (is an acronym of the Compute Unified Device Architecture) is a parallel computing architecture developed by NVIDIA~\cite{CUDA10}.The unit of execution in CUDA is called a thread. Each thread executes a kernel by the streaming processors in parallel. In CUDA, a group of threads that are executed together is called a thread block, and the computational grid consists of a grid of thread blocks. Additionally, a thread block can use the shared memory on a single multiprocessor while the grid executes a single CUDA program logically in parallel. Thus in CUDA programming, it is necessary to design carefully the arrangement of the thread blocks in order to ensure low latency and a proper usage of shared memory, since it can be shared only in a thread block scope. The effective bandwidth of each memory space depends on the memory access pattern. Since the global memory has lower bandwidth than the shared memory, the global memory accesses should be minimized.
+
+CUDA (Compute Unified Device Architecture) is a parallel computing architecture developed by NVIDIA~\cite{CUDA15} for GPUs. It provides a high level GPGPU-based programming model to program GPUs for general purpose computations and non-graphic applications. The GPU is viewed as an accelerator such that data-parallel operations of a CUDA program running on a CPU are off-loaded onto GPU and executed by this later. The data-parallel operations executed by GPUs are called kernels. The same kernel is executed in parallel by a large number of threads organized in grids of thread blocks, such that each GPU multiprocessor executes one or more thread blocks in SIMD fashion (Single Instruction, Multiple Data) and in turn each core of the multiprocessor executes one or more threads within a block. Threads within a block can cooperate by sharing data through a fast shared memory and coordinate their execution through synchronization points. In contrast, within a grid of thread blocks, there is no synchronization at all between blocks. The GPU only works on data filled in the global memory and the final results of the kernel executions must be transferred out of the GPU. In the GPU, the global memory has lower bandwidth than the shared memory associated to each multiprocessor. Thus in the CUDA programming, it is necessary to design carefully the arrangement of the thread blocks in order to ensure low latency and a proper usage of the shared memory, and the global memory accesses should be minimized.
 
+%We introduced three paradigms of parallel programming. Our objective consists in implementing a root finding polynomial algorithm on multiple GPUs. To this end, it is primordial to know how to manage CUDA contexts of different GPUs. A direct method for controlling the various GPUs is to use as many threads or processes as GPU devices. We can choose the GPU index based on the identifier of OpenMP thread or the rank of the MPI process. Both approaches will be investigated.
 
-We introduced three paradigms of parallel programming. Our objective consist to implement an algorithm of root finding polynomial on multiple GPUs. It primordial to know how to manage CUDA contexts of different GPUs. A direct method for controlling the various GPU is to use as many threads or processes as GPU devices. We can choose the GPU index based on the identifier of OpenMP thread or the rank of the MPI process. Both approaches will be investigated.
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-\section{The EA algorithm on single GPU}
-\subsection{the EA method}
+\section{The Ehrlich-Aberth algorithm on a GPU}
+\label{sec3}
 
-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.
+\subsection{The Ehrlich-Aberth method}
+%A cubically convergent iteration method to find zeros of
+%polynomials was proposed by O. Aberth~\cite{Aberth73}. The
+%Ehrlich-Aberth (EA is short) 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 
@@ -521,59 +592,59 @@ follows.
 %Substituting $x_{j}$ for $z_{j}$ we obtain the Aberth iteration method.% 
 
 
-\subsubsection{Polynomials Initialization}
-The initialization of a polynomial $p(z)$ is done by setting each of the $n$ complex coefficients $a_{i}$:
+%\subsubsection{Polynomials Initialization}
+%The initialization of a polynomial $p(z)$ is done by setting each of the $n$ complex coefficients %$a_{i}$:
 
-\begin{equation}
-\label{eq:SimplePolynome}
-  p(z)=\sum{a_{i}z^{n-i}} , a_{n} \neq 0,a_{0}=1, a_{i}\subset C
-\end{equation}
+%\begin{equation}
+%\label{eq:SimplePolynome}
+%  p(z)=\sum{a_{i}z^{n-i}} , a_{n} \neq 0,a_{0}=1, a_{i}\subset C
+%\end{equation}
 
 
-\subsubsection{Vector $Z^{(0)}$ Initialization}
-\label{sec:vec_initialization}
-As for any iterative method, we need to choose $n$ initial guess points $z^{0}_{i}, i = 1, . . . , n.$
-The initial guess is very important since the number of steps needed by the iterative method to reach
-a given approximation strongly depends on it.
-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 which relies on the result of~\cite{Ostrowski41}.
+%\subsubsection{Vector $Z^{(0)}$ Initialization}
+%\label{sec:vec_initialization}
+%As for any iterative method, we need to choose $n$ initial guess points $z^{0}_{i}, i = 1, . . . , %n.$
+%The initial guess is very important since the number of steps needed by the iterative method to %reach
+%a given approximation strongly depends on it.
+%In~\cite{Aberth73} the Ehrlich-Aberth iteration is started by selecting $n$
+%equi-distant 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 which relies on the result of~\cite{Ostrowski41}.
 
-\begin{equation}
-\label{eq:radiusR}
+%\begin{equation}
+%\label{eq:radiusR}
 %%\begin{align}
-\sigma_{0}=\frac{u+v}{2};u=\frac{\sum_{i=1}^{n}u_{i}}{n.max_{i=1}^{n}u_{i}};
-v=\frac{\sum_{i=0}^{n-1}v_{i}}{n.min_{i=0}^{n-1}v_{i}};\\
+%\sigma_{0}=\frac{u+v}{2};u=\frac{\sum_{i=1}^{n}u_{i}}{n.max_{i=1}^{n}u_{i}};
+%v=\frac{\sum_{i=0}^{n-1}v_{i}}{n.min_{i=0}^{n-1}v_{i}};\\
 %%\end{align}
-\end{equation}
-Where:
-\begin{equation}
-u_{i}=2.|a_{i}|^{\frac{1}{i}};
-v_{i}=\frac{|\frac{a_{n}}{a_{i}}|^{\frac{1}{n-i}}}{2}.
-\end{equation}
+%\end{equation}
+%Where:
+%\begin{equation}
+%u_{i}=2.|a_{i}|^{\frac{1}{i}};
+%v_{i}=\frac{|\frac{a_{n}}{a_{i}}|^{\frac{1}{n-i}}}{2}.
+%\end{equation}
 
-\subsubsection{Iterative Function}
-The operator used by the Aberth method is corresponding to the
-following equation~\ref{Eq:EA1} which will enable the convergence towards
-polynomial solutions, provided all the roots are distinct.
+%\subsubsection{Iterative Function}
+%The operator used by the Aberth method  corresponds to the
+%equation~\ref{Eq:EA1}, it enables the convergence towards
+%the polynomials zeros, provided all the roots are distinct.
 
 %Here we give a second form of the iterative function used by the Ehrlich-Aberth method: 
 
-\begin{equation}
-\label{Eq:EA1}
-EA: 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}
+%\begin{equation}
+%\label{Eq:EA-1}
+%EA: 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}
 
-\subsubsection{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:
+%\subsubsection{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:
 
-\begin{equation}
-\label{eq:Aberth-Conv-Cond}
-\forall i \in [1,n];\vert\frac{z_{i}^{k}-z_{i}^{k-1}}{z_{i}^{k}}\vert<\xi
-\end{equation}
+%\begin{equation}
+%\label{eq:AAberth-Conv-Cond}
+%\forall i \in [1,n];\vert\frac{z_{i}^{k}-z_{i}^{k-1}}{z_{i}^{k}}\vert<\xi
+%\end{equation}
 
 
 %\begin{figure}[htbp]
@@ -583,49 +654,135 @@ The convergence condition determines the termination of the algorithm. It consis
 %\label{fig:03}
 %\end{figure}
 
-%the Ehrlich-Aberth method is an iterative  method, contain 4 steps, start from the initial approximations of all the
-%roots of the polynomial,the second step initialize the solution vector $Z$ using the Guggenheimer method to assure the distinction of the initial vector roots, than in step 3 we apply the the iterative function based on the Newton's method and Weiestrass operator[...,...], wich will make it possible to converge to the roots solution, provided that all the root are different. At the end of each application of the iterative function, a stop condition is verified consists in stopping the iterative process when the whole of the modules of the roots
-%are lower than a fixed value $ε$ 
-
-
-\subsection{EA parallel implementation on CUDA}
-Like any parallel code, a GPU parallel implementation first
-requires to determine the sequential tasks and the
-parallelizable parts of the sequential version of the
-program/algorithm. In our case, all the operations that are easy
-to execute in parallel must be made by the GPU to accelerate
-the execution of the application, like the step 3 and step 4. On the other hand, all the
-sequential operations and the operations that have data
-dependencies between threads or recursive computations must
-be executed by only one CUDA or CPU thread (step 1 and step 2). Initially we specifies the organization of threads in parallel, need to specify the dimension of the grid Dimgrid: the number of block per grid and block by DimBlock: the number of threads per block required to process a certain task. 
-
-we create the kernel, for step 3 we have two kernels, the
-first named \textit{save} is used to save vector $Z^{K-1}$ and the kernel
-\textit{update} is used to update the $Z^{K}$ vector. In step 4 a kernel is
-created to test the convergence of the method. In order to
-compute function H, we have two possibilities: either to use
-the Jacobi method, or the Gauss-Seidel method which uses the
-most recent computed roots. It is well known that the Gauss-
-Seidel mode converges more quickly. So, we used the Gauss-Seidel mode of iteration. To
-parallelize the code, we created kernels and many functions to
-be executed on the GPU for all the operations dealing with the
+%the Ehrlich-Aberth method is an iterative  method, contain 4 steps, start from the initial approximations of all the roots of the polynomial,the second step initialize the solution vector $Z$ using the Guggenheimer method to assure the distinction of the initial vector roots, than in step 3 we apply the the iterative function based on the Newton's method and Weiestrass operator~\cite{,}, witch will make it possible to converge to the roots solution, provided that all the root are different.
+
+The Ehrlich-Aberth method is a simultaneous method~\cite{Aberth73} using the following iteration
+\begin{equation}
+\label{Eq:EA1}
+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,\ldots,n
+\end{equation}
+
+This method contains 4 steps. The first step consists in the
+initializing the polynomial.\LZK{Pas compris?? \RC{changé}}.
+The second step initializes the solution vector $Z$ using the
+Guggenheimer method~\cite{Gugg86} to ensure that initial roots are all
+distinct from each other. \LZK{Quelle est la différence entre la 1st
+  step et la 2nd step? Que veut dire " to ensure the distinction of
+  the initial vector roots"? \RC{reformulé}} 
+In step 3, the iterative function based on the Newton's
+method~\cite{newt70} and Weiestrass operator~\cite{Weierstrass03} is
+applied. In our case, the  Ehrlich-Aberth is applied as in (\ref{Eq:EA1}).
+Iterations of the EA method will converge to the roots of the
+considered polynomial.\LZK{On ne peut pas expliquer un peu plus
+  comment? Donner des formules comment elle se base sur la méthode de
+  Newton et de l'opérateur de  Weiestrass? \RC{amélioré}}
+\LZK{Elle est où la 4th step??}
+\LZK{Conclusion: Méthode mal présentée et j'ai presque rien compris!
+  \RC{après} }
+
+
+In order to stop the iterative function, a stop condition is applied,
+this is the 4th step. This condition checks that all the root modules
+are lower than a fixed value $\epsilon$.
+
+\begin{equation}
+\label{eq:Aberth-Conv-Cond}
+\forall i\in[1,n],~\vert\frac{z_i^k-z_i^{k-1}}{z_i^k}\vert<\epsilon
+\end{equation}
+
+\LZK{On ne dit pas plutôt "the relative errors" à la place de "root
+  modules"? Raph nous confirmera quelle critère d'arrêt a
+  utilisé. \RC{normalement c'est bon, l'erreur est calculée avec le
+    module de chaque racine}}
+
+\subsection{Improving Ehrlich-Aberth method}
+With high degree polynomials, the Ehrlich-Aberth method suffers from floating point overflows due to the mantissa of floating points representations. This induces errors in the computation of $p(z)$ when $z$ is large.
+ 
+%Experimentally, it is very difficult to solve polynomials with the Ehrlich-Aberth method and have roots which except the circle of unit, represented by the radius $r$ evaluated as: 
+
+%\begin{equation}
+%\label{R.EL}
+%R = exp(log(DBL\_MAX)/(2*n) );
+%\end{equation}
+
+
+
+% where \verb=DBL_MAX= stands for the maximum representable \verb=double= value.
+ 
+In order to solve this problem, we propose to modify the iterative
+function by using the logarithm and the exponential of a complex and
+we propose a new version of the Ehrlich-Aberth method.  This method
+allows us to exceed the computation of the polynomials of degree
+100,000 and to reach a degree up to more than 1,000,000. The reformulation of the iteration~(\ref{Eq:EA1}) of the Ehrlich-Aberth method with exponential and logarithm is defined as follows, for $i=1,\dots,n$:
+
+\begin{equation}
+\label{Log_H2}
+z^{k+1}_i = z_i^k - \exp(\ln(p(z_i^k)) - \ln(p'(z^k_i)) - \ln(1-Q(z^k_i))),
+\end{equation}
+
+where:
+
+\begin{equation}
+\label{Log_H1}
+Q(z^k_i) = \exp(\ln(p(z^k_i)) - \ln(p'(z^k_i)) + \ln(\sum_{i\neq j}^n\frac{1}{z^k_i-z^k_j})).
+\end{equation}
+
+
+%We propose to use the logarithm and the exponential of a complex in order to compute the power at a high exponent. 
+Using the logarithm  and the exponential operators, we can replace any
+multiplications and divisions with additions and
+subtractions. Consequently, computations manipulate lower values in absolute
+values~\cite{Karimall98}. \LZK{Je n'ai pas compris cette dernière
+  phrase? \RC{changé : on veut dire on manipule des valeurs plus petites en valeur absolues}}
+
+%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. Using the logarithm  and the exponential operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations manipulate lower absolute values and the roots for large polynomial degrees can be looked for successfully~\cite{Karimall98}.
+
+\subsection{The Ehrlich-Aberth parallel implementation on CUDA}
+%We introduced three paradigms of parallel programming.
+
+Our objective consists in implementing a root finding polynomial
+algorithm on multiple GPUs. To this end, it is primordial to know how
+to manage CUDA contexts of different GPUs. A direct method for
+controlling the various GPUs is to use as many threads or processes as
+GPU devices. We can choose the GPU index based on the identifier of
+OpenMP thread or the rank of the MPI process. Both approaches will be
+investigated. \LZK{Répétition! Le même texte est déjà écrit comme
+  intro dans la section II. Sinon ici on parle seulement de
+  l'implémentation cuda sans mpi et openmp! \RC{Je suis d'accord à
+    revoir après, quand les 2 parties suivantes seront plus stables}}
+
+
+
+
+Like any parallel code, a GPU parallel implementation first requires to determine the sequential code and the data-parallel operations of a algorithm. In fact, all the operations that are easy to execute in parallel must be made by the GPU to accelerate the execution, like the steps 3 and 4. On the other hand, all the sequential operations and the operations that have data dependencies between CUDA threads or recursive computations must be executed by only one CUDA thread or a CPU thread (the steps 1 and 2).\LZK{La méthode est déjà mal présentée, dans ce cas c'est encore plus difficile de comprendre que représentent ces différentes étapes!} Initially, we specify the organization of parallel threads by specifying the dimension of the grid \verb+Dimgrid+, the number of blocks per grid \verb+DimBlock+ and the number of threads per block.
+
+The code is organized as kernels which are parts of code that are run on GPU devices. For step 3, there are two kernels, the first is named \textit{save} is used to save vector $Z^{K-1}$ and the second one is
+named \textit{update} and is used to update the $Z^{K}$ vector. For
+step 4, a kernel tests the convergence of the method. In order to
+compute the function H, we have two possibilities: either to use the
+Jacobi mode, or the Gauss-Seidel mode of iterating which uses the most
+recent computed roots. It is well known that the Gauss-Seidel mode
+converges more quickly. So, we use Gauss-Seidel iterations. To
+parallelize the code, we create kernels and many functions to be
+executed on the GPU for all the operations dealing with the
 computation on complex numbers and the evaluation of the
-polynomials. As said previously, we managed both functions
-of evaluation of a polynomial: the normal method, based on
-the method of Horner and the method based on the logarithm
-of the polynomial. All these methods were rather long to
-implement, as the development of corresponding kernels with
-CUDA is longer than on a CPU host. This comes in particular
-from the fact that it is very difficult to debug CUDA running
-threads like threads on a CPU host. In the following paragraph
-Algorithm 1 shows the GPU parallel implementation of Ehrlich-Aberth method.
-
-Algorithm~\ref{alg2-cuda} shows a sketch of the Ehrlich-Aberth method using CUDA.
-
-\begin{enumerate}
+polynomials. As said previously, we manage both functions of
+evaluation: the normal method, based on the method of
+Horner and the method based on the logarithm of the polynomial. All
+these methods were rather long to implement, as the development of
+corresponding kernels with CUDA is longer than on a CPU host. This
+comes in particular from the fact that it is very difficult to debug
+CUDA running threads like threads on a CPU host. In the following
+paragraph Algorithm~\ref{alg1-cuda} shows the GPU parallel
+implementation of Ehrlich-Aberth method.
+\LZK{Vaut mieux expliquer l'implémentation en faisant référence à l'algo séquentiel que de parler des différentes steps.}
+
 \begin{algorithm}[htpb]
 \label{alg1-cuda}
-%\LinesNumbered
+\LinesNumbered
+\SetAlgoNoLine
 \caption{CUDA Algorithm to find roots with the Ehrlich-Aberth method}
 
 \KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (Error tolerance
@@ -635,32 +792,48 @@ Algorithm~\ref{alg2-cuda} shows a sketch of the Ehrlich-Aberth method using CUDA
 
 %\BlankLine
 
-\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\;
+Initialization of P\;
+Initialization of Pu\;
+Initialization of the solution vector $Z^{0}$\;
+Allocate and copy initial data to the GPU global memory\;
 \While {$\Delta z_{max} > \epsilon$}{
-\item Let $\Delta z_{max}=0$\;
-\item $ kernel\_save(ZPrec,Z)$\;
-\item  k=k+1\;
-\item $ kernel\_update(Z,P,Pu)$\;
-\item $kernel\_testConverge(\Delta z_{max},Z,ZPrec)$\;
+   $ ZPres=kernel\_save(Z)$\;
+   $ Z=kernel\_update(Z,P,Pu)$\;
+ $\Delta z_{max}=kernel\_testConv(Z,ZPrec)$\;
 
 }
-\item Copy results from GPU memory to CPU memory\;
+Copy results from GPU memory to CPU memory\;
 \end{algorithm}
-\end{enumerate}
-~\\ 
-
-
+\RC{Si l'algo vous convient, il faudrait le détailler précisément}
  
-\section{The EA algorithm on Multi-GPU}
-
-\subsection{MGPU (OpenMP-CUDA) approach}
-Our OpenMP-CUDA implementation of EA algorithm is based on the hybrid OpenMP and CUDA programming model. It works
-as follows.
-Based on the metadata, a shared memory is used to make data evenly shared among OpenMP threads. The shared data are the solution vector $Z$, the polynomial to solve $P$. vector of error of stop condition $\Delta z$. Let(T\_omp) number of OpenMP threads is equal to the number of GPUs, each threads OpenMP checks one GPU,  and control a part of the shared memory, that is a part of the vector Z  like: $(n/num\_gpu)$ roots, n: the polynomial's degrees, $num\_gpu$ the number of GPUs. Each OpenMP thread copies its data from host memory to GPU’s device memory.Than every GPU will have a grid of computation organized with its performances and the size of data of which it checks and compute kernels. %In principle a grid is set by two parameter DimGrid, the number of block per grid, DimBloc: the number of threads per block. The following schema  shows the architecture of (CUDA,OpenMP).
+\section{The EA algorithm on Multiple GPUs}
+\label{sec4}
+\subsection{an OpenMP-CUDA approach}
+Our OpenMP-CUDA implementation of EA algorithm is based on the hybrid
+OpenMP and CUDA programming model.  All the data are shared with
+OpenMP amoung all the OpenMP threads. The shared data are the solution
+vector $Z$, the polynomial to solve $P$, and the error vector $\Delta
+z$. The number of OpenMP threads is equal to the number of GPUs, each
+OpenMP thread binds to one GPU, and it controls a part of the shared
+memory. More precisely each OpenMP thread will be responsible to
+update its owns part of the vector Z. This part is call $Z_{loc}$ in
+the following. Then all GPUs will have a grid of computation organized
+according to the device performance and the size of data on which it
+runs the computation kernels.
+
+To compute one iteration of the EA method each GPU performs the
+followings steps. First roots are shared with OpenMP and the
+computation of the local size for each GPU is performed (lines 5-7 in
+Algo\ref{alg2-cuda-openmp}). Each thread starts by copying all the
+previous roots inside its GPU (line 9). Then each GPU will copy the
+previous roots (line 10) and it will compute an iteration of the EA
+method on its own roots (line 11).  For that all the other roots are
+used. The convergence is checked on the new roots (line 12). At the end
+of an iteration, the updated roots are copied from the GPU to the
+CPU (line 14) by direcly updating its own roots in the shared memory
+arrays containing all the roots.
+
+%In principle a grid is set by two parameter DimGrid, the number of block per grid, DimBloc: the number of threads per block. The following schema  shows the architecture of (CUDA,OpenMP).
 
 %\begin{figure}[htbp]
 %\centering
@@ -670,58 +843,69 @@ Based on the metadata, a shared memory is used to make data evenly shared among
 %\end{figure}
 %Each thread OpenMP compute the kernels on GPUs,than after each iteration they copy out the data from GPU memory to CPU shared memory. The kernels are re-runs is up to the roots converge sufficiently. Here are below the corresponding algorithm:
 
-$num\_gpus$ thread OpenMP are created using \verb=omp_set_num_threads();=function (line,Algorithm \ref{alg2-cuda-openmp}), the shared memory is created using \verb=#pragma omp parallel shared()= OpenMP function (line 5,Algorithm\ref{alg2-cuda-openmp}), than each OpenMP threads allocate and copy initial data from CPU memory to the GPU global memories, execute the kernels on GPU, and compute only his portion of roots indicated with variable \textit{index} initialized in (line 5, Algorithm \ref{alg2-cuda-openmp}), used as input data in the $kernel\_update$ (line 10, Algorithm \ref{alg2-cuda-openmp}). After each iteration, OpenMP threads synchronize using \verb=#pragma omp barrier;= to recuperate all values of vector $\Delta z$, to compute the maximum stop condition in vector $\Delta z$(line 12, Algorithm \ref{alg2-cuda-openmp}).Finally,they copy the results from GPU memories to CPU memory. The OpenMP threads execute kernels until the roots converge sufficiently.  
-\begin{enumerate}
-\begin{algorithm}[htpb]
+%% \RC{Surement à virer ou réécrire pour etre compris sans algo}
+%% $num\_gpus$ OpenMP threads  are created using
+%% \verb=omp_set_num_threads();=function (step $3$, Algorithm
+%% \ref{alg2-cuda-openmp}), the shared memory is created using
+%% \verb=#pragma omp parallel shared()= OpenMP function (line $5$,
+%% Algorithm\ref{alg2-cuda-openmp}), then each OpenMP thread allocates
+%% memory and copies initial data from CPU memory to GPU global memory,
+%% executes the kernels on GPU, but computes only his portion of roots
+%% indicated with variable \textit{index} initialized in (line 5,
+%% Algorithm \ref{alg2-cuda-openmp}), used as input data in the
+%% $kernel\_update$ (line 10, Algorithm \ref{alg2-cuda-openmp}). After
+%% each iteration, all OpenMP threads synchronize using
+%% \verb=#pragma omp barrier;= to gather all the correct values of
+%% $\Delta z$, thus allowing the computation the maximum stop condition
+%% on vector $\Delta z$ (line 12, Algorithm
+%% \ref{alg2-cuda-openmp}). Finally, threads copy the results from GPU
+%% memories to CPU memory. The OpenMP threads execute kernels until the
+%% roots sufficiently converge.
+
+
+\begin{algorithm}[h]
 \label{alg2-cuda-openmp}
-%\LinesNumbered
+\LinesNumbered
+\SetAlgoNoLine
 \caption{CUDA-OpenMP 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 (Derivative of P), $n$ (Polynomial degrees), $\Delta z$ ( Vector of errors of stop condition), $num_gpus$ (number of OpenMP threads/ number of GPUs), $Size$ (number of roots)}
+  threshold), P (Polynomial to solve), Pu (Derivative of P), $n$ (Polynomial degree), $\Delta z$ ( Vector of errors for stop condition), $num\_gpus$ (number of OpenMP threads/ Number of GPUs), $Size$ (number of roots)}
 
-\KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}
+\KwOut {$Z$ ( Root's vector), $ZPrec$ (Previous root's vector)}
 
 \BlankLine
 
-\item Initialization of the of P\;
-\item Initialization of the of Pu\;
-\item Initialization of the solution vector $Z^{0}$\;
-\verb=omp_set_num_threads(num_gpus);=
-\verb=#pragma omp parallel shared(Z,$\Delta$ z,P);=
-\verb=cudaGetDevice(gpu_id);=
-\item Allocate and copy initial data from CPU memory to the GPU global memories\;
-\item index= $Size/num\_gpus$\;
-\item k=0\;
+Initialization of P\;
+Initialization of Pu\;
+Initialization of the solution vector $Z^{0}$\;
+Start of a parallel part with OpenMP (Z, $\Delta z$, P are shared variables)\;
+gpu\_id=cudaGetDevice()\; 
+Allocate memory on GPU\;
+Compute local size and offet according to gpu\_id\;
 \While {$error > \epsilon$}{
-\item Let $\Delta z=0$\;
-\item $ kernel\_save(ZPrec,Z)$\;
-\item  k=k+1\;
-\item $ kernel\_update(Z,P,Pu,index)$\;
-\item $kernel\_testConverge(\Delta z[gpu\_id],Z,ZPrec)$\;
-%\verb=#pragma omp barrier;=
-\item error= Max($\Delta z$)\;
+  copy Z from CPU to GPU\;
+$ ZPrec_{loc}=kernel\_save(Z_{loc})$\;
+$ Z_{loc}=kernel\_update(Z,P,Pu)$\;
+$\Delta z[gpu\_id] = kernel\_testConv(Z_{loc},ZPrec_{loc})$\;
+$  error= Max(\Delta z)$\;
+  copy $Z_{loc}$ from GPU to Z in CPU
 }
-
-\item Copy results from GPU memories to CPU memory\;
 \end{algorithm}
-\end{enumerate}
-~\\ 
 
 
 
-\subsection{Multi-GPU (MPI-CUDA) approach}
+\subsection{an MPI-CUDA approach}
 %\begin{figure}[htbp]
 %\centering
  % \includegraphics[angle=-90,width=0.2\textwidth]{MPI-CUDA}
 %\caption{The MPI-CUDA architecture }
 %\label{fig:03}
 %\end{figure}
-Our parallel implementation of the Ehrlich-Aberth method to find root polynomial using (CUDA-MPI) approach, splits input data of the polynomial to solve between MPI processes. From Algorithm 3, the input data are the polynomial to solve $P$, the solution vector $Z$, the previous solution vector $zPrev$, and the Value of errors of stop condition $\Delta z$. Let $p$ denote the number of MPI processes on and $n$ the size of the polynomial to be solved. The algorithm performs a simple data partitioning by creating $p$ portions, of at most $⌈n/p⌉$ roots to find per MPI process, for each element mentioned above. Consequently, each MPI process $k$ will have its own solution vector $Z_{k}$,polynomial to be solved $p_{k}$, the error of stop condition $\Delta z_{k}$, Than each MPI processes compute only $⌈n/p⌉$ roots.
+Our parallel implementation of EA to find root of polynomials using a CUDA-MPI approach is a data parallel approach. It splits input data of the polynomial to solve among MPI processes. In Algorithm \ref{alg2-cuda-mpi}, input data are the polynomial to solve $P$, the solution vector $Z$, the previous solution vector $ZPrev$, and the value of errors of stop condition $\Delta z$. Let $p$ denote the number of MPI processes on and $n$ the degree of the polynomial to be solved. The algorithm performs a simple data partitioning by creating $p$ portions, of at most $\lceil n/p \rceil$ roots to find per MPI process, for each $Z$ and $ZPrec$. Consequently, each MPI process of rank $k$ will have its own solution vector $Z_{k}$ and $ZPrec$, the error related to the stop condition $\Delta z_{k}$, enabling each MPI process to compute $\lceil n/p \rceil$ roots.
 
-Since a GPU works only on data of its memory, all local input data, $Z_{k}, p_{k}$ and $\Delta z_{k}$, must be transferred from CPU memories to the corresponding GPU memories. Afterward, the same EA algorithm (Algorithm 1) is run by all processes but on different sub-polynomial root $ p(x)_{k}=\sum_{i=k(\frac{n}{p})}^{k+1(\frac{n}{p})} a_{i}x^{i}, k=1,...,p$.  Each processes MPI execute the  loop \verb=(While(...)...do)= contain the kernels. Than each process MPI  compute only his portion of roots indicated with variable \textit{index} initialized in (line 5, Algorithm \ref{alg2-cuda-mpi}), used as input data in the $kernel\_update$ (line 10, Algorithm \ref{alg2-cuda-mpi}). After each iteration, MPI processes synchronize using \verb=MPI_Allreduce= function, in order to compute the maximum error stops condition $\Delta z_{k}$ computed by each process MPI line (line, Algorithm\ref{alg2-cuda-mpi}), and  copy the values of new roots computed from GPU memories to CPU memories, than communicate  her results to the neighboring processes,using \verb=MPI_Alltoallv=. If maximum stop condition $error > \epsilon$ the processes stay to execute the loop \verb= while(...)...do= until all the roots converge sufficiently.
+Since a GPU works only on data already allocated in its memory, all local input data, $Z_{k}$, $ZPrec$ and $\Delta z_{k}$, must be transferred from CPU memories to the corresponding GPU memories. Afterwards, the same EA algorithm (Algorithm \ref{alg1-cuda}) is run by all processes but on different polynomial subset of roots $ p(x)_{k}=\sum_{i=1}^{n} a_{i}x^{i}, k=1,...,p$.  Each MPI process executes the  loop \verb=(While(...)...do)= containing the CUDA kernels but each MPI process  computes only its own portion of the roots according to the rule ``''owner computes``''. The local range of roots is indicated with the \textit{index} variable initialized at (line 5, Algorithm \ref{alg2-cuda-mpi}), and passed as an input variable to $kernel\_update$ (line 10, Algorithm \ref{alg2-cuda-mpi}). After each iteration, MPI processes synchronize  (\verb=MPI_Allreduce= function) by a reduction on $\Delta z_{k}$ in order to compute the maximum error related to the stop condition.   Finally, processes copy the values of new computed roots  from GPU memories to CPU memories, then communicate their results to other processes with \verb=MPI_Alltoall= broadcast. If the stop condition is not verified ($error > \epsilon$) then processes stay withing the loop \verb= while(...)...do= until all the roots sufficiently converge.
 
-\begin{enumerate}
 \begin{algorithm}[htpb]
 \label{alg2-cuda-mpi}
 %\LinesNumbered
@@ -733,30 +917,25 @@ Since a GPU works only on data of its memory, all local input data, $Z_{k}, p_{k
 \KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}
 
 \BlankLine
-\item Initialization of the P\;
-\item Initialization of the Pu\;
-\item Initialization of the solution vector $Z^{0}$\;
-\item Allocate and copy initial data from CPU memories to the GPU global memories\;
-\item $index= Size/num_gpus$\;
-\item k=0\;
+Initialization of P\;
+Initialization of Pu\;
+Initialization of the solution vector $Z^{0}$\;
+Distribution of Z\;
+Allocate memory to GPU\;
 \While {$error > \epsilon$}{
-\item Let $\Delta z=0$\;
-\item $ kernel\_save(ZPrec,Z)$\;
-\item  k=k+1\;
-\item $ kernel\_update(Z,P,Pu,index)$\;
-\item $kernel\_testConverge(\Delta z,Z,ZPrec)$\;
-\item ComputeMaxError($\Delta z$,error)\;
-\item Copy results from GPU memories to CPU memories\;
-\item Send $Z[id]$ to all neighboring processes\;
-\item Receive $Z[j]$ from neighboring process j\;
-
-
+copy Z from CPU to GPU\;
+$ZPrec_{loc}=kernel\_save(Z_{loc})$\;
+$Z_{loc}=kernel\_update(Z,P,Pu)$\;
+$\Delta z=kernel\_testConv(Z_{loc},ZPrec_{loc})$\;
+$error=MPI\_Reduce(\Delta z)$\;
+Copy $Z_{loc}$ from GPU to CPU\;
+$Z=MPI\_AlltoAll(Z_{loc})$\;
 }
 \end{algorithm}
-\end{enumerate}
-~\\ 
 
-\section{experiments}
+
+\section{Experiments}
+\label{sec5}
 We study two categories of polynomials: sparse polynomials and full polynomials.\\
 {\it A sparse polynomial} is a polynomial for which only some coefficients are not null. In this paper, we consider sparse polynomials for which the roots are distributed on 2 distinct circles:
 \begin{equation}
@@ -770,102 +949,164 @@ We study two categories of polynomials: sparse polynomials and full polynomials.
 \begin{equation}
      {\Large \forall a_{i} \in C, i\in N;  p(x)=\sum^{n}_{i=0} a_{i}.x^{i}} 
 \end{equation}
-For our tests, a CPU Intel(R) Xeon(R) CPU E5620@2.40GHz and a GPU K40 (with 6 Go of ram) are used. 
-
-We performed a set of experiments on single GPU and Multi-GPU using (OpenMP/MPI) to find roots polynomials with EA algorithm, for both sparse and full polynomials of different sizes. We took into account the execution times and the  polynomial size performed by sum or each experiment.
-All experimental results obtained from the simulations are made in
-double precision data, the convergence threshold of the methods is set
-to $10^{-7}$.
-%Since we were more interested in the comparison of the
-%performance behaviors of Ehrlich-Aberth and Durand-Kerner methods on
-%CPUs versus on GPUs.
-The initialization values of the vector solution
-of the methods are given in %Section~\ref{sec:vec_initialization}.
-
-\subsection{Test with Multi-GPU (CUDA OpenMP) approach}
-
-In this part we performed  a set of experiments on Multi-GPU (CUDA OpenMP) approach for full and sparse polynomials of different degrees, compare it with Single GPU (CUDA).
- \subsubsection{Execution times in seconds of the Ehrlich-Aberth method for solving sparse polynomials on GPUs using shared memory paradigm with OpenMP}
+
+For our test, 4 cards GPU tesla Kepler K40 are used.  In order to
+evaluate both the GPU and Multi-GPU approaches, we performed a set of
+experiments on a single GPU and multiple GPUs using OpenMP or MPI with
+the EA algorithm, for both sparse and full polynomials of different
+sizes.  All experimental results obtained are perfomed with double
+precision float data and the convergence threshold of the EA method is
+set to $10^{-7}$.  The initialization values of the vector solution of
+the methods are given by Guggenheimer method~\cite{Gugg86}.
+
+
+\subsection{Evaluation of the CUDA-OpenMP approach}
+
+Here we report some experiments witt full and sparse polynomials of
+different degrees with multiple GPUs.
+\subsubsection{Execution times of the EA method to solve sparse polynomials on multiple GPUs}
  
- In this experiments we report the execution time of the EA algorithm, on single GPU and Multi-GPU with (2,3,4) GPUs, for different sparse polynomial degrees ranging from 100,000 to 1,400,000
+In this experiments we report the execution time of the EA algorithm, on single GPU and multi-GPUs with (2,3,4) GPUs, for different sparse polynomial degrees ranging from 100,000 to 1,400,000.
 
 \begin{figure}[htbp]
 \centering
   \includegraphics[angle=-90,width=0.5\textwidth]{Sparse_omp}
-\caption{Execution times in seconds of the Ehrlich-Aberth method for solving sparse polynomials on GPUs using shared memory paradigm with OpenMP}
+\caption{Execution time in seconds of the Ehrlich-Aberth method to
+  solve sparse polynomials on multiple GPUs with CUDA-OpenMP.}
 \label{fig:01}
 \end{figure}
 
-This figure~\ref{fig:01} shows that (CUDA OpenMP) Multi-GPU approach reduce the execution time up to the scale 100 whereas single GPU is of scale 1000 for polynomial who exceed 1,000,000. It shows the advantage to use OpenMP parallel paradigm  to connect the performances of several GPUs and solve a  polynomial of high degrees.   
+Figure~\ref{fig:01} shows that the CUDA-OpenMP approach scales well
+with multiple GPUs. This version allows us to solve sparse polynomials
+of very high degrees.
 
-\subsubsection{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials on GPUs using shared memory paradigm with OpenMP}
+\subsubsection{Execution times of the EA method to solve full polynomials on multiple GPUs}
 
-This experiments shows the execution time of the EA algorithm, on single GPU (CUDA) and Multi-GPU (CUDA OpenMP) approach for full polynomials of degrees ranging from 100,000 to 1,400,000
+These experiments show the execution times of the EA algorithm, on a single GPU and on multiple GPUs using the CUDA OpenMP approach for full polynomials of degrees ranging from 100,000 to 1,400,000.
 
 \begin{figure}[htbp]
 \centering
   \includegraphics[angle=-90,width=0.5\textwidth]{Full_omp}
-\caption{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials on GPUs using shared memory paradigm with OpenMP}
-\label{fig:03}
+\caption{Execution time in seconds of the Ehrlich-Aberth method to
+  solve full polynomials on multiple GPUs with CUDA-OpenMP.}
+\label{fig:02}
 \end{figure}
 
-The second test with full polynomial shows a very important saving of time, for a polynomial of degrees 1,4M (CUDA OpenMP) approach with 4 GPUs compute and solve it 4 times as fast as single GPU. We notice that curves are positioned one below the other one, more the number of used GPUs increases more the execution time decreases.
+In Figure~\ref{fig:02}, we can observe that with full polynomials the EA version with
+CUDA-OpenMP scales also well. Using 4 GPUs allows us to achieve a
+quasi-linear speedup.
 
-\subsection{Test with Multi-GPU (CUDA MPI) approach}
-In this part we perform a set of experiment to compare Multi-GPU (CUDA MPI) approach with single GPU, for solving full and sparse polynomials of degrees ranging from 100,000 to 1,400,000.
+\subsection{Evaluation of the CUDA-MPI approach}
+In this part we perform some experiments to evaluate the CUDA-MPI
+approach to solve full and sparse polynomials of degrees ranging from
+100,000 to 1,400,000.
 
-\subsubsection{Execution times in seconds of the Ehrlich-Aberth method for solving sparse polynomials on GPUs using distributed memory paradigm with MPI}
+\subsubsection{Execution times of the EA method to solve sparse polynomials on multiple GPUs}
 
 \begin{figure}[htbp]
 \centering
   \includegraphics[angle=-90,width=0.5\textwidth]{Sparse_mpi}
-\caption{Execution times in seconds of the Ehrlich-Aberth method for solving sparse polynomials on GPUs using distributed memory paradigm with MPI}
-\label{fig:02}
+\caption{Execution time in seconds of the Ehrlich-Aberth method to
+  solve sparse polynomials on multiple GPUs with CUDA-MPI.}
+\label{fig:03}
 \end{figure}
-~\\
-This figure shows 4 curves of execution time of EA algorithm, a curve with single GPU, 3 curves with Multi-GPUs (2, 3, 4) GPUs. We see clearly that the curve with single GPU is above the other curves, which shows consumption in execution time compared to the Multi-GPU. We can see the approach Multi-GPU (CUDA MPI) reduces the execution time up to the scale 100 for polynomial of degrees more than 1,000,000 whereas single GPU is of the scale 1000.
-\\
-\subsubsection{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials on GPUs using distributed memory paradigm with MPI}
+Figure~\ref{fig:03} shows the execution times of te EA algorithm,
+for a single GPU, and multiple GPUs (2, 3, 4) with the CUDA-MPI approach.
+
+\subsubsection{Execution time of the Ehrlich-Aberth method for solving full polynomials on multiple GPUs using the Multi-GPU appraoch}
 
 \begin{figure}[htbp]
 \centering
-  \includegraphics[angle=-90,width=0.5\textwidth]{Full_mpi}
-\caption{Execution times in seconds of the Ehrlich-Aberth method for full polynomials on GPUs using distributed memory paradigm with MPI}
+ \includegraphics[angle=-90,width=0.5\textwidth]{Full_mpi}
+\caption{Execution times in seconds of the Ehrlich-Aberth method for
+  full polynomials on  multiple GPUs with CUDA-MPI.}
 \label{fig:04}
 \end{figure}
 
+In Figure~\ref{fig:04}, we can also observe that the CUDA-MPI approach
+is also efficient to solve full polynimails on multiple GPUs.
 
-this figure shows the execution time of the algorithm EA, on single GPU and Multi-GPUS with (2, 3, 4) GPUs for full polynomials. With (CUDA-MPI) approach we notice that the three curves are distinct from each other, more we use GPUs more the execution time decreases, on the other hand the curve with single GPU is well above the other curves.
-This is due to the use of parallelization MPI paradigm that divides the polynomial into sub polynomials assigned to each GPU. unlike the single GPU which solves all the polynomial on a single GPU, consequently it engenders more execution time.
+\subsection{Comparison of  the CUDA-OpenMP and the CUDA-MPI approaches}
 
-%\begin{figure}[htbp]
-%\centering
- % \includegraphics[angle=-90,width=0.5\textwidth]{Sparse}
-%\caption{Comparaison between MPI and OpenMP versions of the Ehrlich-Aberth method for solving sparse plynomials on GPUs}
-%\label{fig:05}
-%\end{figure}
+In the previuos section we saw that both approches are very effecient
+to  reduce the execution times the  sparse and full polynomials. In
+this section we try to compare these two approaches.
 
-%\begin{figure}[htbp]
-%\centering
- % \includegraphics[angle=-90,width=0.5\textwidth]{Full}
-%\caption{Comparaison between MPI and OpenMP versions of the Ehrlich-Aberth method for solving full polynomials on GPUs}
-%\label{fig:06}
-%\end{figure}
+\subsubsection{Solving sparse polynomials}
+In this experiment three sparse polynomials of size 200K, 800K and 1,4M are investigated.
+\begin{figure}[htbp]
+\centering
+ \includegraphics[angle=-90,width=0.5\textwidth]{Sparse}
+\caption{Execution times  to solvs sparse polynomials of three
+  distinct sizes on multiple GPUs using MPI and OpenMP with the
+  Ehrlich-Aberth method}
+\label{fig:05}
+\end{figure}
+In Figure~\ref{fig:05} there is one curve for CUDA-MPI and another one
+for CUDA-OpenMP. We can see that the results are quite similar between
+OpenMP and MPI for the polynomials size of 200K. For the size of 800K,
+the MPI version is a little bit slower than the OpenMP approach but for
+the 1,4 millions size, there is a slight advantage for the MPI
+version.
+
+\subsubsection{Solving full polynomials}
+\begin{figure}[htbp]
+\centering
+ \includegraphics[angle=-90,width=0.5\textwidth]{Full}
+\caption{Execution time for solving full polynomials of three distinct sizes on multiple GPUs using MPI and OpenMP approaches using Ehrlich-Aberth}
+\label{fig:06}
+\end{figure}
+In Figure~\ref{fig:06}, we can see that when it comes to full polynomials, both approaches are almost equivalent.
 
-%\begin{figure}[htbp]
-%\centering
- % \includegraphics[angle=-90,width=0.5\textwidth]{MPI}
-%\caption{Comparaison of execution times of the Ehrlich-Aberth method for solving sparse and full polynomials on GPUs with distributed memory paradigm using MPI}
-%\label{fig:07}
-%\end{figure}
+\subsubsection{Solving sparse and full polynomials of the same size with CUDA-MPI}
 
-%\begin{figure}[htbp]
-%\centering
- % \includegraphics[angle=-90,width=0.5\textwidth]{OMP}
-%\caption{Comparaison of execution times of the Ehrlich-Aberth method for solving sparse and full polynomials on GPUs with shared memory paradigm using OpenMP}
-%\label{fig:08}
-%\end{figure}
+In this experiment we compare the execution time of the EA algorithm
+according to the number of GPUs to solve sparse and full
+polynomials on multiples GPUs using MPI. We chose three sparse and full
+polynomials of size 200K, 800K and 1,4M.
+\begin{figure}[htbp]
+\centering
+ \includegraphics[angle=-90,width=0.5\textwidth]{MPI}
+\caption{Execution times to solve sparse and full polynomials of three distinct sizes on multiple GPUs using MPI.}
+\label{fig:07}
+\end{figure}
+In Figure~\ref{fig:07} we can see that CUDA-MPI can solve sparse and
+full polynomials of high degrees, the execution times with sparse
+polynomial are very low compared to full polynomials. With sparse
+polynomials the number of monomials is reduced, consequently the number
+of operations is reduced and the execution time decreases.
+
+\subsubsection{Solving sparse and full polynomials of the same size with CUDA-OpenMP}
+
+\begin{figure}[htbp]
+\centering
+ \includegraphics[angle=-90,width=0.5\textwidth]{OMP}
+\caption{Execution time for solving sparse and full polynomials of three distinct sizes on multiple GPUs using OpenMP}
+\label{fig:08}
+\end{figure}
+
+Figure ~\ref{fig:08} shows the impact of sparsity on the effectiveness of the CUDA-OpenMP approach. We can see that the impact follows the same pattern, a difference in execution time in favor of the sparse polynomials. 
 
+\subsection{Scalability of the EA method on multiple GPUs to solve very high degree polynomials}
+These experiments report the execution times of the EA method for
+sparse and full polynomials ranging from 1,000,000 to 5,000,000.
+\begin{figure}[htbp]
+\centering
+ \includegraphics[angle=-90,width=0.5\textwidth]{big}
+ \caption{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials of high degree on 4 GPUs for sizes ranging from 1M to 5M}
+\label{fig:09}
+\end{figure}
+In Figure~\ref{fig:09} we can see that both approaches are scalable
+and can solve very high degree polynomials. In addition, with full polynomial as well as sparse ones, both
+approaches give very similar results.
+
+%SIDER JE viens de virer \c ca For sparse polynomials here are a noticeable difference in favour of MPI when the degree is
+%above 4 millions. Between 1  and 3 millions, OpenMP is more effecient.
+%Under 1 million, OpenMPI and MPI are almost equivalent.
+
+%SIDER : il faut une explication sur les différences ici aussi.
+ 
+%for sparse and full polynomials
 % An example of a floating figure using the graphicx package.
 % Note that \label must occur AFTER (or within) \caption.
 % For figures, \caption should occur after the \includegraphics.
@@ -964,12 +1205,26 @@ This is due to the use of parallelization MPI paradigm that divides the polynomi
 
 
 \section{Conclusion}
-In this paper, we have presented a parallel implementation of Ehrlich-Aberth algorithm for solving full and sparse polynomials, on single GPU with CUDA and Multi-GPUs using two parallel paradigm, shared memory with OpenMP, distributed memory with MPI.(CUDA-OpenMP) approach and (CUDA-MPI) approach, 
-We have performed many experiments with the Ehrlich-Aberth method in single GPU, Multi-GPU with (CUDA-OpenMP) approach, Multi-GPU with (CUDA-MPI) approach for sparse and full polynomials. the experiments show that, using parallel programming model like (OpenMP, MPI) can effectively manage multiple graphics cards to work together to solve the same problem and accelerate parallel applications, like (CUDA MPI) approach with 4 GPUs can solve a polynomial of 1,000,000 4 speed up than on single GPU.
-
-
-In future, we will evaluate our parallel implementation of Ehrlich-Aberth algorithm on other parallel programming model 
-
+\label{sec6}
+In this paper, we have presented a parallel implementation of
+Ehrlich-Aberth algorithm to solve full and sparse polynomials, on
+single GPU with CUDA and on multiple GPUs using two parallel
+paradigms: shared memory with OpenMP and distributed memory with
+MPI. These architectures were addressed by a CUDA-OpenMP approach and
+CUDA-MPI approach, respectively.  Experiments show that, using
+parallel programming model like (OpenMP, MPI). We can efficiently
+manage multiple graphics cards to solve the same
+problem and accelerate the parallel execution with 4 GPUs and solve a
+polynomial of degree up to 5,000,000, four times faster than on single
+GPU. 
+
+
+%In future, we will evaluate our parallel implementation of Ehrlich-Aberth algorithm on other parallel programming model 
+
+Our next objective is to extend the model presented here with clusters
+of GPU nodes, with a three-level scheme: inter-node communication via
+MPI processes (distributed memory), management of multi-GPU node by
+OpenMP threads (shared memory).
 
 %present a communication approach between multiple GPUs. The comparison between MPI and OpenMP as GPUs controllers shows that these
 %solutions can effectively manage multiple graphics cards to work together
@@ -987,8 +1242,10 @@ In future, we will evaluate our parallel implementation of Ehrlich-Aberth algori
 % use section* for acknowledgment
 \section*{Acknowledgment}
 
+Computations have been performed on the supercomputer facilities of
+the Mésocentre de calcul de Franche-Comté. We also would like to thank
+Nvidia for hardware donation under CUDA Research Center 2014.
 
-The authors would like to thank...