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@@ -326,6 +326,8 @@
 \usepackage[textsize=footnotesize]{todonotes}
 \newcommand{\LZK}[2][inline]{%
   \todo[color=red!10,#1]{\sffamily\textbf{LZK:} #2}\xspace}
 \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}

 
 
 
 
 
 


@@ -339,8 +341,7 @@
 % 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.
 % 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{Two parallel implementations of Ehrlich-Aberth algorithm for root finding of polynomials
-on  multiple GPUs with OpenMP and 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
 
 
 % author names and affiliations


@@ -400,8 +401,17 @@ Fax: (888) 555--1212}}
 % As a general rule, do not put math, special symbols or citations
 % in the abstract
 \begin{abstract}
 % As a general rule, do not put math, special symbols or citations
 % in the abstract
 \begin{abstract}

-\LZK{J'ai un peu modifié l'abstract. Sinon à revoir pour le degré max des polynômes après les tests de raph.}
-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 to find roots of polynomials of degree up-to 1.4 billion. To our knowledge, this is the first paper to present this technology mix to solve such a highly demanding problem in parallel programming. \LZK{Je n'ai pas bien saisi la dernière phrase.}
+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
 \end{abstract}
 
 % no keywords


@@ -423,55 +433,99 @@ Finding roots of polynomials is a very important part of solving real-life probl
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Introduction}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Introduction}

-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 !}
+%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 $n$ different values of the unknown variable $x$ for which $p(x)=0$. Such values are called zeros or roots of $p$. If zeros are $\{\alpha_{i}\}_{1\leq i\leq n}$, then $p(x)$ can be written as :
-\begin{equation}
- 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. \LZK{A mon avis on peut supprimer cette phrase}
-Most of the numerical methods that deal with the polynomial root-finding problem are simultaneous ones, \textit{i.e.} the iterative methods to find simultaneous approximations of the $n$ polynomial zeros. 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. 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}
 \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,
+p(x) = \displaystyle\sum^n_{i=0}{a_ix^i},a_n\neq 0. 

 \end{equation}
 \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}
+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}
 \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,
+ p(x)=a_n\displaystyle\prod_{i=1}^n(x-z_i), a_n\neq 0.

 \end{equation}
 \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}
 %%\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.
 
 
 %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 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 DK, EA 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 (DK and EA) 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 $z^{k}_{i}$ 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 \textit{OpenMP} and for a distributed memory one with \textit{MPI}. They are 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. \LZK{je suppose que c'est pour la version mpi (only by using 8 personal computers and 2 communications per iteration). A t on utilisé le même nombre de procs pour les deux versions openmp et mpi} The authors showed an interesting speedup comparing to the sequential implementation that takes up-to 3,300 seconds to obtain same results.
-
-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 Units) has exceeded that of 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 and 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. We consider two architectures: shared-memory computers with OpenMP API and distributed-memory computers with 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.
+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. \RC{si on donne des temps faut donner
+  le proc, comme c'est vieux à mon avis faut supprimer ca, votre avis?} The authors showed an interesting
+speedup comparing to the sequential implementation which takes up-to
+3,300 seconds to obtain same results. 
+\LZK{``only by using 8 personal computers and 2 communications per iteration''. Pour MPI? et Pour OpenMP: Rep: c'est MPI seulement}
+
+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.}
+ \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.
+ \item A parallel implementation of Ehrlich-Aberth method on single GPU with CUDA.
+\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. 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.
+ \end{itemize}
+\LZK{Pas d'autres contributions possibles?: j'ai rajouté 2}

 
 

-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.)?}
- 

 %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.}
 
 %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.
-\LZK{A revoir toute cette organization}
+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 Multi-GPU using the OpenMP and MPI approaches. In section\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. 
+\LZK{A revoir toute cette organization: je viens de la revoir}

 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  
 \section{Parallel programming models}
 \label{sec2}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  
 \section{Parallel programming models}
 \label{sec2}

-In this section we present three different parallel programming models: OpenMP, MPI and CUDA.
+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.
  
 \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.


@@ -486,36 +540,31 @@ In this section we present three different parallel programming models: OpenMP,
 %Sequential natively. Threads share some or all of the available memory and can
 %have private memory areas [6].
 
 %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 shared memory 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.
-\LZK{Cette partie est réécrite. A relire et à améliorer si possible.}  
+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 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{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 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.

-\LZK{Cette partie est réécrite. A relire et à améliorer si possible.}

  
 \subsection{CUDA}
  
 \subsection{CUDA}

-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.
+%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 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.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 
 

-\section{The EA algorithm on a single GPU}
+\section{The Ehrlich-Aberth algorithm on a GPU}

 \label{sec3}
 \label{sec3}

-\subsection{The EA 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.
+\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 
 
 %The Aberth method is a purely algebraic derivation. 
 %To illustrate the derivation, we let $w_{i}(z)$ be the product of linear factors 


@@ -541,74 +590,141 @@ follows.
 %Substituting $x_{j}$ for $z_{j}$ we obtain the Aberth iteration method.% 
 
 
 %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-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}.
+%\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}
 %%\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{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  corresponds to the
-equation~\ref{Eq:EA1}, it enables the convergence towards
-the polynomials zeros, 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: 
 
 
 %Here we give a second form of the iterative function used by the Ehrlich-Aberth method: 
 

+%\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:
+
+%\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]
+%\centering
+ % \includegraphics[angle=-90,width=0.5\textwidth]{EA-Algorithm}
+%\caption{The Ehrlich-Aberth algorithm on single GPU}
+%\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~\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{} using the following iteration

 \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: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}
 

-\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:
+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~\cite{Gugg86} 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~\cite{newt70} and Weiestrass operator~\cite{Weierstrass03}, 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 $\xi$. 

 
 \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:Aberth-Conv-Cond}
 \forall i \in [1,n];\vert\frac{z_{i}^{k}-z_{i}^{k-1}}{z_{i}^{k}}\vert<\xi
 \end{equation}

+\subsection{Improving Ehrlich-Aberth method}
+With high degree polynomial, the Ehrlich-Aberth method suffer from overflow because the limited number in the mantissa of floating points representations, which makes the computation of $p(z)$ wrong when z is large.
+ 
+Experimentally, it is very difficult to solve polynomials with 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}
+
+
+%\begin{equation}
+
+%R = \exp( \log(DBL\_MAX) / (2*n) )
+%\end{equation}
+ where \verb=DBL_MAX= stands for the maximum representable \verb=double= value.
+ 
+In order to hold into account the limit of size of floats, we propose to modifying the iterative function and compute the logarithm of:
+
+\begin{equation}
+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}
+
+This method allows, indeed, to exceed the computation of the polynomials of degree 100,000 and to reach a degree upper to 1,000,000. For that purpose, it is necessary to use the logarithm and the exponential of a complex. The iterative  function of Ehrlich-Aberth method with exponential and logarithm is given as following:
+
+\begin{equation}
+\label{Log_H2}
+EA.EL: z^{k+1}_{i}=z_{i}^{k}-\exp \left(\ln \left(
+p(z_{i}^{k})\right)-\ln\left(p'(z^{k}_{i})\right)- \ln\left(1-Q(z^{k}_{i})\right)\right),
+\end{equation}
+
+where:
+
+\begin{equation}
+\label{Log_H1}
+Q(z^{k}_{i})=\exp\left( \ln (p(z^{k}_{i}))-\ln(p'(z^{k}_{i}))+\ln \left(
+\sum_{i\neq j}^{n}\frac{1}{z^{k}_{i}-z^{k}_{j}}\right)\right)i=1,...,n,
+\end{equation}
+
+
+%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 absolute values and the roots for large polynomial degrees can be looked for successfully~\cite{Karimall98}.
+
+%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{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.

 
 

-%\begin{figure}[htbp]
-%\centering
- % \includegraphics[angle=-90,width=0.5\textwidth]{EA-Algorithm}
-%\caption{The Ehrlich-Aberth algorithm on single GPU}
-%\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
 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


@@ -619,7 +735,7 @@ 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 specify the organization of parallel threads, by specifying the dimension of the grid Dimgrid, the number of blocks per grid DimBlock and the number of threads per block. 
 
 dependencies between threads or recursive computations must
 be executed by only one CUDA or CPU thread (step 1 and step 2). Initially, we specify the organization of parallel threads, by specifying the dimension of the grid Dimgrid, the number of blocks per grid DimBlock and the number of threads per block. 
 

-The code is organzed  by what is named kernels, portions o code that are run on GPU devices. For step 3, there are two kernels, the
+The code is organized  by what is named kernels, portions code that are run on GPU devices. For step 3, there are two kernels, the

 first named \textit{save} is used to save vector $Z^{K-1}$ and the seconde 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
 first named \textit{save} is used to save vector $Z^{K-1}$ and the seconde 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


@@ -658,11 +774,11 @@ Algorithm~\ref{alg1-cuda} shows the GPU parallel implementation of Ehrlich-Abert
 \item Initialization of the solution vector $Z^{0}$\;
 \item Allocate and copy initial data to the GPU global memory\;
 \item k=0\;
 \item Initialization of the solution vector $Z^{0}$\;
 \item Allocate and copy initial data to the GPU global memory\;
 \item k=0\;

-\While {$\Delta z_{max} > \epsilon$}{
-\item Let $\Delta z_{max}=0$\;
-\item $ kernel\_save(ZPrec,Z)$\;
-\item  k=k+1\;
-\item $ kernel\_update(Z,P,Pu)$\;
+\item \While {$\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 $kernel\_testConverge(\Delta z_{max},Z,ZPrec)$\;
 
 }
 \item $kernel\_testConverge(\Delta z_{max},Z,ZPrec)$\;
 
 }


@@ -673,11 +789,11 @@ Algorithm~\ref{alg1-cuda} shows the GPU parallel implementation of Ehrlich-Abert
 
 
  
 
 
  

-\section{The EA algorithm on Multi-GPU}
+\section{The EA algorithm on Multiple GPUs}

 \label{sec4}
 \label{sec4}

-\subsection{MGPU : an OpenMP-CUDA approach}
+\subsection{M-GPU : an OpenMP-CUDA approach}

 Our OpenMP-CUDA implementation of EA algorithm is based on the hybrid OpenMP and CUDA programming model. It works as follows.
 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 $P$, and  the error vector $\Delta z$. Let (T\_omp) the number of OpenMP threads be equal to the number of GPUs, each OpenMP thread binds to one GPU,  and controls a part of the shared memory, that is a part of the vector Z , that is $(n/num\_gpu)$ roots where $n$ is the polynomial's degree and $num\_gpu$ the total number of available GPUs. Each OpenMP thread copies its data from host memory to GPU’s device memory.Then every GPU will have a grid of computation organized according to the device performance and the size of data on which it runs the computation 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).
+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 $P$, and  the error vector $\Delta z$. Let (T\_omp) the number of OpenMP threads be equal to the number of GPUs, each OpenMP thread binds to one GPU,  and controls a part of the shared memory, that is a part of the vector Z , that is $(n/num\_gpu)$ roots where $n$ is the polynomial's degree and $num\_gpu$ the total number of available GPUs. Each OpenMP thread copies its data from host memory to GPU’s device memory. Then every GPU will have a grid of computation organized according to the device performance and the size of data on which it runs the computation 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).

 
 %\begin{figure}[htbp]
 %\centering
 
 %\begin{figure}[htbp]
 %\centering


@@ -789,8 +905,8 @@ We study two categories of polynomials: sparse polynomials and full polynomials.
 For our tests, a CPU Intel(R) Xeon(R) CPU E5620@2.40GHz and a GPU K40 (with 6 Go of ram) are used. 
 %SIDER : Une meilleure présentation de l'architecture est à faire ici.
 
 For our tests, a CPU Intel(R) Xeon(R) CPU E5620@2.40GHz and a GPU K40 (with 6 Go of ram) are used. 
 %SIDER : Une meilleure présentation de l'architecture est à faire ici.
 

-In order to evaluate both the MGPU and Multi-GPU approaches, we performed a set of experiments on a single GPU and multiple GPUs using OpenMP or MPI by EA algorithm, for both sparse and full polynomials of different sizes.
-All experimental results obtained are made in double precision data, the convergence threshold of the methods is set to $10^{-7}$.
+In order to evaluate both the M-GPU and Multi-GPU approaches, we performed a set of experiments on a single GPU and multiple GPUs using OpenMP or MPI by EA algorithm, for both sparse and full polynomials of different sizes.
+All experimental results obtained are made in double precision data whereas the convergence threshold of the EA method 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.
 %Since we were more interested in the comparison of the
 %performance behaviors of Ehrlich-Aberth and Durand-Kerner methods on
 %CPUs versus on GPUs.


@@ -799,93 +915,122 @@ of the methods are given in %Section~\ref{sec:vec_initialization}.
 
 \subsection{Evaluating the M-GPU (CUDA-OpenMP) approach}
 
 
 \subsection{Evaluating the M-GPU (CUDA-OpenMP) approach}
 

-We report here  the results of the set of experiments with M-GPU approach for full and sparse polynomials of different degrees, and we compare it with a Single GPU execution.
-\subsubsection{Execution times in seconds of the EA method for solving sparse polynomials on GPUs using shared memory paradigm with OpenMP}
+We report here  the results of the set of experiments with the M-GPU approach for full and sparse polynomials of different degrees, and we compare it with a Single GPU execution.
+\subsubsection{Execution time of the EA method for solving sparse polynomials on multiple GPUs using the M-GPU approach}

  
 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.
 
 \begin{figure}[htbp]
 \centering
   \includegraphics[angle=-90,width=0.5\textwidth]{Sparse_omp}
  
 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.
 
 \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 for solving sparse polynomials on multiple GPUs using the M-GPU approach}

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

-This figure~\ref{fig:01} shows that the (CUDA-OpenMP) Multi-GPU approach reduces the execution time by a factor up to 100 w.r.t the single GPU apparaoch and a by a factor of 1000 for polynomials  exceeding degree 1,000,000. It shows the advantage to use the OpenMP parallel paradigm to gather the capabilities of several GPUs and solve polynomials of very high degrees.   
+This figure~\ref{fig:01} shows that the (CUDA-OpenMP) M-GPU approach reduces the execution time by a factor up to 100 w.r.t the single GPU approach and a by a factor of 1000 for polynomials  exceeding degree 1,000,000. It shows the advantage to use the OpenMP parallel paradigm to gather the capabilities of several GPUs and solve 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 time in seconds of the Ehrlich-Aberth method for solving full polynomials on multiple GPUs using the M-GPU approach}

 
 The experiments shows the execution time 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}
 
 The experiments shows the execution time 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 multiple GPUs using shared memory paradigm with OpenMP}
+\caption{Execution time in seconds of the Ehrlich-Aberth method for solving full polynomials on multiple GPUs using the M-GPU appraoch}

 \label{fig:03}
 \end{figure}
 
 Results with full polynomials show very important savings in execution time. For a polynomial of degree 1,4 million, the CUDA-OpenMP approach with 4 GPUs solves it 4 times as fast as single GPU, thus achieving a quasi-linear speedup.
 
 \label{fig:03}
 \end{figure}
 
 Results with full polynomials show very important savings in execution time. For a polynomial of degree 1,4 million, the CUDA-OpenMP approach with 4 GPUs solves it 4 times as fast as single GPU, thus achieving a quasi-linear speedup.
 

-\subsection{Evaluting the Multi-GPU (CUDA-MPI) approach}
-In this part we perform a set of experiments 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{Evaluating the Multi-GPU (CUDA-MPI) approach}
+In this part we perform a set of experiments to compare the Multi-GPU (CUDA MPI) approach with a single GPU, for solving 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 time of the Ehrlich-Aberth method for solving sparse polynomials on multiple GPUs using the Multi-GPU approach}

 
 \begin{figure}[htbp]
 \centering
   \includegraphics[angle=-90,width=0.5\textwidth]{Sparse_mpi}
 
 \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}
+\caption{Execution time in seconds of the Ehrlich-Aberth method for solving sparse polynomials on multiple GPUs using the Multi-GPU approach}

 \label{fig:02}
 \end{figure}
 ~\\
 \label{fig:02}
 \end{figure}
 ~\\

-This figure shows 4 curves of execution time of EA algorithm, a curve with single GPU, 3 curves with multiple GPUs (2, 3, 4). We can clearly see 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 also that the CUDA-MPI approach reduces the execution time by a factor of 100 for polynomials of degree more than 1,000,000 whereas a single GPU is of the scale 1000.
+Figure~\ref{fig:02} shows execution time of EA algorithm, for a single GPU, and multiple GPUs (2, 3, 4) on respectively 2, 3 and four MPI nodes. We can clearly see that the curve for a single GPU is above the other curves, which shows overtime in execution time compared to the Multi-GPU approach. We can see also that the CUDA-MPI approach reduces the execution time by a factor of 10 for polynomials of degree more than 1,000,000. For example, at degree 1000000, the execution time with a single GPU amounted to 10 thousand seconds, while with 4 GPUs, it is lowered to about just one thousand seconds which makes it for a tenfold speedup.

 %%SIDER : Je n'ai pas reformuler car je n'ai pas compris la phrase, merci de l'ecrire ici en fran\cais.
 \\cette figure montre 4 courbes de temps d'exécution pour l'algorithme EA, une courbe avec un seul GPU, 3 courbes pour multiple GPUs(2, 3, 4), on peut constaté clairement que la courbe à un seul GPU est au-dessus des autres courbes, vue sa consomation en temps d'exècution. On peut voir aussi qu'avec l'approche Multi-GPU (CUDA-MPI) reduit le temps d'exècution jusqu'à l'echelle 100 pour le polynômes qui dépasse 1,000,000 tandis que Single GPU est de l'echelle 1000.
 %%SIDER : Je n'ai pas reformuler car je n'ai pas compris la phrase, merci de l'ecrire ici en fran\cais.
 \\cette figure montre 4 courbes de temps d'exécution pour l'algorithme EA, une courbe avec un seul GPU, 3 courbes pour multiple GPUs(2, 3, 4), on peut constaté clairement que la courbe à un seul GPU est au-dessus des autres courbes, vue sa consomation en temps d'exècution. On peut voir aussi qu'avec l'approche Multi-GPU (CUDA-MPI) reduit le temps d'exècution jusqu'à l'echelle 100 pour le polynômes qui dépasse 1,000,000 tandis que Single GPU est de l'echelle 1000.

-\subsubsection{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials on GPUs using distributed memory paradigm with MPI}
+
+\subsubsection{Execution time of the Ehrlich-Aberth method for solving full polynomials on multiple GPUs using the Multi-GPU appraoch}

 
 \begin{figure}[htbp]
 \centering
 
 \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 GPUs using the Multi-GPU}

 \label{fig:04}
 \end{figure}
 \label{fig:04}
 \end{figure}

-%SIDER : Corriger le point de la courbe 3-GPUs qui correpsond à un degré de 600000

 
 

-Figure \ref{fig:04} shows the execution time of the algorithm on single GPU and on multipe GPUs with (2, 3, 4) GPUs for full polynomials. With the 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 a single GPU is well above the other curves.
+
+       Figure \ref{fig:04} shows execution time for a single GPU, and multiple GPUs (2, 3, 4) on respectively 2, 3 and four MPI nodes. With the 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 for a single GPU is well above the other curves.

 
 This is due to the use of MPI parallel paradigm that divides the problem computations and assigns portions to each GPU. But unlike the single GPU which carries all the computations on a single GPU, data communications are introduced, consequently engendering more execution time. But experiments show that execution time is still highly reduced.
 
 
 
 
 This is due to the use of MPI parallel paradigm that divides the problem computations and assigns portions to each GPU. But unlike the single GPU which carries all the computations on a single GPU, data communications are introduced, consequently engendering more execution time. But experiments show that execution time is still highly reduced.
 
 
 

+\subsection{Comparing  the CUDA-OpenMP approach and the CUDA-MPI approach}

 
 

-%\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 effective in reducing execution time for sparse as well as full polynomials. At this stage, the interesting question is which approach is better. In the fellowing, we present appropriate experiments comparing the two Multi-GPU approaches to answer the question.

 
 

-%\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 time  for solving sparse polynomials of three distinct sizes on multiple GPUs using MPI and OpenMP approaches using Ehrlich-Aberth}
+\label{fig:05}
+\end{figure}
+In Figure~\ref{fig:05} there two curves for each polynomial size : one for the MPI-CUDA and another for the OpenMP. We can see that the results are similar between OpenMP and MPI for the polynomials size of 200K. For the size of 800K, the MPI version is a little slower than the OpenMP approach but for for the 1,4M size, there is a slight advantage for the MPI version.

 
 

-%\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 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]{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}
+\subsubsection{Solving sparse and full polynomials of the same size with CUDA-MPI}
+In this experiment we compare the execution time of the EA algorithm according to the number of GPUs for solving sparse and full polynomials on Multi-GPU 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 time for solving 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 time with sparse polynomial are very low comparing to full polynomials. with sparse polynomials the number of monomial are reduce, consequently the number of operation are reduce than the execution time decrease. 

 
 

+\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 fellows the same pattern, a difference in execution time in favor of the sparse polynomials. 
+%SIDER : il faut une explication ici. je ne vois pas de prime abords, qu'est-ce qui engendre cette différence, car quelques soient les coefficients nulls ou non nulls, c'est toutes les racines qui sont calculées qu'elles soient similaires ou non (degrés de multiplicité).
+\subsection{Scalability of the EA method on Multi-GPU to solve very high degree polynomials}
+These experiments report the execution time according to the degrees of polynomials ranging from 1,000,000 to 5,000,000 for both approaches with sparse and full  polynomials. 
+\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. With full polynomial both approaches give interestingly very similar results. For the sparse case however, there are a noticeable difference in favour of MPI when the degree is above 4M. Between 1M and 3M, the OMP approach is more effective and under 1M degree, OMP and MPI approaches are almost equivalent.
+
+%SIDER : il faut une explication sur les différences ici aussi.
+ 
+%for sparse and full polynomials

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 % Note that \label must occur AFTER (or within) \caption.
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@@ -984,7 +1129,7 @@ This is due to the use of MPI parallel paradigm that divides the problem computa
 
 
 \section{Conclusion}
 
 
 \section{Conclusion}

-\label{sec5}
+\label{sec6}

 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 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. 
 The experiments show that, using parallel programming model like (OpenMP, MPI), we can efficiently manage multiple graphics cards to work together to solve the same problem and accelerate the parallel execution with 4 GPUs and solve a polynomial of degree 1,000,000, four times faster than on single GPU, that is a quasi-linear speedup.
 
 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 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. 
 The experiments show that, using parallel programming model like (OpenMP, MPI), we can efficiently manage multiple graphics cards to work together to solve the same problem and accelerate the parallel execution with 4 GPUs and solve a polynomial of degree 1,000,000, four times faster than on single GPU, that is a quasi-linear speedup.