X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/kahina_paper2.git/blobdiff_plain/cebad47898c16e7229d945e3a8fe361d18316cb4..HEAD:/paper.tex?ds=sidebyside

diff --git a/paper.tex b/paper.tex
index 9733c3d..e90eac4 100644
--- a/paper.tex
+++ b/paper.tex
@@ -7,6 +7,9 @@
 \usepackage[utf8]{inputenc}
 \usepackage[T1]{fontenc}
 \usepackage[textsize=footnotesize]{todonotes}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{float} 
 \newcommand{\LZK}[2][inline]{%
   \todo[color=red!10,#1]{\sffamily\textbf{LZK:} #2}\xspace}
 \newcommand{\RC}[2][inline]{%
@@ -21,7 +24,7 @@
 
 \title{Two parallel implementations of Ehrlich-Aberth algorithm for root-finding of polynomials on multiple GPUs with OpenMP and MPI}
 
-\author{\IEEEauthorblockN{Kahina Guidouche, Abderrahmane Sider }
+\author{\IEEEauthorblockN{Kahina Ghidouche, Abderrahmane Sider }
   \IEEEauthorblockA{Laboratoire LIMED\\
     Faculté des sciences exactes\\
     Université de Bejaia, 06000, Algeria\\
@@ -36,22 +39,22 @@ Email: zianekhodja.lilia@gmail.com\\ raphael.couturier@univ-fcomte.fr}}
 \maketitle
 
 \begin{abstract}
-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.
+Finding the roots of polynomials is a very important part of solving
+real-life problems but the higher the degree of the polynomials is,
+the less easy it becomes. 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 use 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 the roots of polynomials of degree up-to
+1.4 million. Moreover, other experiments show it is possible to find the
+roots of polynomials of degree up-to 5 million.
 \end{abstract}
 
-% no keywords
-\LZK{Faut pas mettre des keywords?}
-
+\begin{IEEEkeywords}
+  root finding method, Ehrlich-Aberth method, GPU, MPI, OpenMP
+\end{IEEEkeywords}
 
 \IEEEpeerreviewmaketitle
 
@@ -61,55 +64,84 @@ of polynomials of degree up-to 5 millions.
 \section{Introduction}
 
 
-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:
+Finding the roots of polynomials of very high degrees arises in many complex problems of 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}
-p(x) = \displaystyle\sum^n_{i=0}{a_ix^i},a_n\neq 0, 
+p(x) = \displaystyle\sum^n_{i=0}{\alpha_ix^i},\alpha_n\neq 0, 
 \end{equation}
-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 :
+where $\{\alpha_i\}_{0\leq i\leq n}$ are complex coefficients and $n$ is a high integer number. If $\alpha_n\neq0$ 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}
- p(x)=a_n\displaystyle\prod_{i=1}^n(x-z_i), a_n\neq 0.
+ p(x)=\alpha_n\displaystyle\prod_{i=1}^n(x-z_i), \alpha_n\neq 0.
 \end{equation}
 
-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}.
-
-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.
-
-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.}
+Most of the numerical methods that deal with the polynomial
+root-finding problems are simultaneous methods, \textit{i.e.} the
+iterative methods to find simultaneous approximations of the $n$
+polynomial roots. These methods start from the initial approximation
+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
+the Durand-Kerner method~\cite{Durand60,Kerner66} and the Ehrlich-Aberth method~\cite{Ehrlich67,Aberth73}.
+
+
+The convergence time of simultaneous methods drastically increases
+with the increasing of the polynomial's degree. The great challenge
+with simultaneous methods is to parallelize them and to improve their
+convergence. Many authors have proposed parallel simultaneous
+methods~\cite{Freeman89,Loizou83,Freemanall90,bini96,cs01:nj,Couturier02},
+using several paradigms of parallelization (synchronous or
+asynchronous computations, mechanism of shared or distributed memory,
+etc). However, so far until now, only polynomials not exceeding
+degrees of less than 100,000 have been solved.
+
+%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 and al.~\cite{cs01:nj} 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. The authors showed an interesting
+%speedup that is 20 times as fast as the sequential implementation. 
+
+The recent advent of the Compute Unified Device Architecture
+(CUDA)~\cite{CUDA15}, a programming
+model and a parallel computing architecture developed by NVIDIA, has revived parallel programming interest in
+this problem. Indeed, the computing power of GPUs (Graphics Processing
+Units) has exceeded that of traditional  CPUs processors, which makes
+it very appealing to the research community to investigate new
+parallel implementations for a whole set of scientific problems in the
+reasonable hope to solve bigger instances of well known
+computationally demanding issues such as the one beforehand. However,
+CUDA provides an efficient massive data computing model which is
+suited to GPU architectures. 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. 
+
+In this paper we propose the parallelization of the Ehrlich-Aberth
+(EA) method which has a much better cubic convergence rate than the
+quadratic rate of the Durand-Kerner method that has already been investigated in \cite{Kahinall14}. In the other hand,  EA is suitable to be implemented in parallel computers according to the data-parallel paradigm. In this model, computing elements carry computations on the data they are assigned and communicate with other computing elements in order to get fresh data or to synchronize. Classically, two parallel programming paradigms OpenMP and MPI are used to code such solutions. But in our case, computing elements are CUDA multi-GPU platforms. This architectural setting poses new programming challenges but offers also new opportunities to efficiently solve huge problems, otherwise considered intractable until recently. To the best of our knowledge, our CUDA-MPI and CUDA-OpenMP codes are the first implementations of EA method with multiple GPUs for finding roots of polynomials. Our major contributions include:
  \begin{itemize}
 
-\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. 
+\item The parallel implementation of EA 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 EA 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 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.
+\item
+  Our method is efficient to compute the roots of sparse and full
+  polynomials of degree up to 5 million.
  \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.}
+
 
 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. 
 
@@ -119,22 +151,61 @@ The paper is organized as follows. In Section~\ref{sec2} we present three differ
  
 \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.
+Our objective consists in implementing a root-finding algorithm of
+polynomials on multiple GPUs. To this end, it is essential to know how
+to manage the CUDA contexts of different GPUs. A direct method to control 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}
-
-
-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. 
+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  a relatively 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} 
-
-
-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.
+MPI (Message Passing Interface) is a portable message passing style of
+the parallel programming designed specifically for 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 a way that 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 the most used HPC programming model to solve large scale and complex applications.
  
 \subsection{CUDA}
-
-
-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.
+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. 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 latter. 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 with CUDA programming,
+it is necessary to design carefully the arrangement of the thread
+blocks in order to ensure a low latency and a proper use of the shared
+memory. As for the global memory accesses, it should also be minimized.
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -152,7 +223,17 @@ 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. 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. 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 Ehrlich-Aberth method will converge to the roots of the considered polynomial. 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 $\epsilon$.
+This method contains 4 steps. The first step consists in the
+initializing the polynomial. 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. 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. 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}
@@ -160,14 +241,13 @@ This method contains 4 steps. The first step consists in the initializing the po
 \end{equation}
 
 \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.
+With high degree polynomials, the EA 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.
  
-
 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
+we propose a new version of the EA 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 $i=1,\dots,n$:
+100,000 and to reach a degree up to more than 1,000,000. The reformulation of the iteration~(\ref{Eq:EA1}) of the EA method with exponential and logarithm operators is defined as follows, for $i=1,\dots,n$:
 
 \begin{equation}
 \label{Log_H2}
@@ -181,185 +261,204 @@ where:
 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}
 
-
-
-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}. 
+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}. In practice, the  exponential and
+logarithm mode is used when a root is outside the circle unit represented by the radius $R$ evaluated in C language with:
+\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 and $n$ is the degree of the polynomial.
 
 
 \subsection{The Ehrlich-Aberth parallel implementation on CUDA}
-
-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 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.}
-
-
-
+%\KG{
+  The algorithm ~\ref{alg1-cuda} shows sketch of the Ehrlich-Aberth method using CUDA.
+The first steps consist in the initialization of the input data like, the polynomial P,derivative of P and the vector solution Z. Then, all data of the root finding problem
+must be copied from the CPU memory to the GPU global memory,because
+the GPUs only work on the data filled in their memories.
+Next, all the data-parallel arithmetic operations inside the main loop
+\verb=(while(...))= are executed as kernels by the GPU. The
+first kernel named \textit{Kernelsave} in line 5 of Algorithm~\ref{alg1-cuda} consists in saving the vector of polynomial roots found at the previous time-step in GPU memory, in
+order to check the convergence of the roots after each iteration (line
+7, Algorithm~\ref{alg1-cuda}). Then the new roots with the
+new iterations are computed using the EA method with a Gauss-Seidel
+iteration mode in order to use the latest updated roots (line
+6). This improves the convergence. This kernel is, in practice, very
+long since it performs all the operations with complex numbers with
+the normal mode of the EA method as in Eq.~\ref{Eq:EA1} but also with the logarithm-exponential one as in Eq.(~\ref{Log_H1},~\ref{Log_H2}). The last kernel checks the convergence of the roots after each update of $Z^{k}$, according to formula Eq.~\ref{eq:Aberth-Conv-Cond} line (7). We used the functions of the CUBLAS Library (CUDA Basic Linear Algebra Subroutines) to implement this kernel. 
+
+The algorithm terminates its computations when all the roots have
+converged.
+%}
+
+
+ %The code is organized as kernels which are parts of codes that are run
+%on GPU devices. Algorithm~\ref{alg1-cuda} describes the CUDA
+%implementation of the Ehrlich-Aberth on a GPU.  This algorithms starts
+%by initializing the polynomial and its derivative (line 1). The
+%initialization of the roots is performed (line 2). Data are transferred
+%%from the CPU to the GPU (after the allocation of the required memory on
+%the GPU) (line 3). Then at each iteration, if the error is greater
+%%than the threshold, the following operations are performed. The previous
+%roots are saved using a kernel (line 5). Then the new roots with the
+%new iterations are computed using the EA method with a Gauss-Seidel
+%iteration mode in order to use the latest updated roots (line
+%6). This improves the convergence. This kernel is, in practice, very
+%long since it performs all the operations with complex numbers with
+%the normal mode of the EA method but also with the
+%logarithm-exponential one. Then the error is computed with a final
+%kernel (line 7). Finally when the EA method has converged, the roots
+%are transferred from the GPU to the CPU.
 \begin{algorithm}[htpb]
+\label{alg1-cuda}
 \LinesNumbered
 \SetAlgoNoLine
 \caption{Finding roots of polynomials with the Ehrlich-Aberth method on a GPU}
-\KwIn{$n$ (polynomial's degree), $\epsilon$ (tolerance threshold)}
+\KwIn{ $\epsilon$ (tolerance threshold)}
 \KwOut{$Z$ (solution vector of roots)}
 Initialize the polynomial $P$ and its derivative $P'$\;
 Set the initial values of vector $Z$\;
 Copy $P$, $P'$ and $Z$ from CPU to GPU\;
-\While{$\Delta Z_{max} > \epsilon$}{
-  $Z^{prev}$ = KernelSave($Z,n$)\;
-  $Z$ = KernelUpdate($P,P',Z,n$)\;
-  $\Delta Z_{max}$ = KernelComputeError($Z,Z^{prev},n$)\;
+\While{$error > \epsilon$}{
+  $Z^{prev}$ = KernelSave($Z$)\;
+  $Z$ = KernelUpdate($P,P',Z$)\;
+  $error$ = KernelComputeError($Z,Z^{prev}$)\;
 }
 Copy $Z$ from GPU to CPU\;
-\label{alg1-cuda}
-\RC{La ligne avec TestConvergence ca fait une ligne de plus.\LZK{Oui j'ai hésité à l'ajouter. On peut faire le test dans la condition de while mais quelle est la valeur initiale de $\Delta Z_{max}$?! Ou bien on s'en fiche?}}
 \end{algorithm}
+\ \\
+This figure shows the second kernel code
+\begin{figure}[htbp]
+\centering
+\includegraphics[angle=+0,width=0.5\textwidth]{code}
+\caption{The Kernel Update code}
+\label{fig:00}
+\end{figure}
 
+%We noticed that the code is executed by a large number of GPU threads organized as grid of to dimension (Number of block per grid (NbBlock), number of threads per block(Nbthread)), the Nbthread is fixed initially, the NbBlock is computed as fallow: 
+%$ NbBlocks= \frac{N+Nbthreads-1}{Nbthreads} where N: the number of root$
+%the  such that each thread in grid is in charge of the computation of one root.
  
-\section{The EA algorithm on Multiple GPUs}
+The development of this code is a rather long task due to the
+development of all the kernels that compute the parts ported on the
+GPU.  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 section the GPU parallel implementation of the
+Ehrlich-Aberth method with OpenMP and MPI is presented.
+
+\section{The Ehrlich-Aberth algorithm on multiple GPUs}
 \label{sec4}
-\subsection{an OpenMP-CUDA approach}
+\KG{we remind that to manage the CUDA contexts of different GPUs, We investigate two parallel paradigms: OpenMP and MPI. In this section we present the both \textit{OpenMP-CUDA} approach and the \textit{MPI-CUDA} approach} used to implement the Ehrlich-Aberth algorithm on multiple GPUs.
+\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.
+OpenMP and CUDA programming model. This algorithm is presented in
+Algorithm~\ref{alg2-cuda-openmp}.  All the data are shared with OpenMP
+among all the OpenMP threads. The shared data are the solution vector
+$Z$, the polynomial to solve $P$, its derivative $P'$, and the error
+vector $error$. 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 for updating its own part of the vector $Z$. This part is
+called $Z_{loc}$ 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.
+followings steps. First, roots are shared with OpenMP and the
+computation of the local size for each GPU is performed (line 4). Each
+thread starts by copying all the previous roots inside its GPU (line
+5). At each iteration, the following operations are performed. First
+the vector $Z$ is transferred from the CPU to the GPU (line 7).  Each
+GPU copies the previous roots (line 8) and it computes an iteration of
+the EA method on its own roots (line 9).  For that all the other roots
+are used. The local error is computed on the new roots (line 10) and
+the maximum of the local errors is computed on all OpenMP threads (line 11). At
+the end of an iteration, the updated roots are copied from the GPU to
+the CPU (line 12) and each CPU directly updates its own roots in the shared
+memory arrays containing all the roots.
 
 
 
 \begin{algorithm}[htpb]
+\label{alg2-cuda-openmp}
 \LinesNumbered
 \SetAlgoNoLine
 \caption{Finding roots of polynomials with the Ehrlich-Aberth method on multiple GPUs using OpenMP}
-\KwIn{$n$ (polynomial's degree), $\epsilon$ (tolerance threshold), $ngpu$ (number of GPUs)}
+\KwIn{ $\epsilon$ (tolerance threshold)}
 \KwOut{$Z$ (solution vector of roots)}
 Initialize the polynomial $P$ and its derivative $P'$\;
 Set the initial values of vector $Z$\;
-Start of a parallel part with OpenMP ($Z$, $\Delta Z_{max}$, $P$, $P'$ are shared variables)\;
-$id_{gpu}$ = cudaGetDevice()\;
-$n_{loc}$ = $n/ngpu$ (local size)\;
-$offset$ = $id_{gpu}\times n_{loc}$ (local offset)\;
+Start of a parallel part with OpenMP ($Z$, $error$, $P$, $P'$ are shared variables)\;
+Determine the local part of the OpenMP thread\;
 Copy $P$, $P'$ from CPU to GPU\;
-\While{$max > \epsilon$}{
+\While{$error > \epsilon$}{
   Copy $Z$ from CPU to GPU\;
-  $Z^{prev}$ = KernelSave($Z,n$)\;
-  $Z[offset]$ = KernelUpdate($P,P',Z,n_{loc}$)\;
-  $\Delta Z_{max}[id_{gpu}]$ = KernelComputeError($Z[offset],Z^{prev}[offset],n_{loc}$)\;
-  Copy $Z[offset]$ from GPU to $Z$ in CPU\;
-  $max$ = MaxFunction($\Delta Z_{max},ngpu$)\; 
+  $Z^{prev}_{loc}$ = KernelSave($Z_{loc}$)\;
+  $Z_{loc}$ = KernelUpdate($P,P',Z$)\;
+  $error_{loc}$ = KernelComputeError($Z_{loc},Z^{prev}_{loc}$)\;
+  $error = max(error_{loc})$\;
+  Copy $Z_{loc}$ from GPU to $Z$ in CPU\;
 }
-\label{alg2-cuda-openmp}
-\RC{Je l'ai rajouté. Bon sinon le n\_loc ne remplace pas
-  vraiment un offset et une taille mais bon... et là il y a 4 lignes
-  pour la convergence, c'est bcp ... Zloc, Zmax, max et
-  testconvergence. On pourrait faire mieux\LZK{Modifié, c'est bon!}}
 \end{algorithm}
 
 
 
 
 
-\subsection{a MPI-CUDA approach}
-
-Our parallel implementation of EA to find root of polynomials using a
-CUDA-MPI approach follows a similar computing approach to the one used
-in CUDA-OpenMP. Each process is responsible to compute its own part of
+\subsection{A MPI-CUDA approach}
+Our parallel implementation of EA to find the roots of polynomials using a
+CUDA-MPI approach follows a similar approach to the one used in
+CUDA-OpenMP. Each processor is responsible for computing its own part of
 roots using all the roots computed by other processors at the previous
 iteration. The difference between both approaches lies in the way
-processes communicate and exchange data. With MPI processors need to
-send and receive data explicitely. So in
-Algorithm~\ref{alg2-cuda-mpi}, after the initialization all the
+processors communicate and exchange data. With MPI, processors need to
+send and receive data explicitly. So in
+Algorithm~\ref{alg2-cuda-mpi}, after the initialization phase all the
 processors have the same $Z$ vector. Then they need to compute the
-parameters used by the $MPI\_AlltoAll$ routines (line 4). In practise,
-each processor needs to compute its offset and its local size. Then
-processors need to allocate memory on their GPU (line 5). At the
-beginning of each iteration, a processor starts by transfering the
-whole vector Z from the CPU to the GPU (line 7). Then only the local
-part of $Z^{prev}$ is saved (line 8). After that, a processor is able
-to compute its own roots (line 9). Next, the local error can be
-computed (ligne 10) and the global error (line 11). Then the local
-roots are transfered from the GPU memory to the CPU memory (line 12)
-before being exchanged between all processors (linge 13) in order to
-give to all processors the last version of the roots. If the
-convergence is not statisfied, an new iteration is executed.
-
-
+parameters used by the $MPI\_AlltoAll$ routines (line 4). In practice,
+each processor needs to compute its offset and its local
+size. Processors need to allocate memory on their GPU and need to copy
+their data on the GPU (line 5). At the beginning of each iteration, a
+processor starts by transferring the whole vector $Z$ from the CPU to the
+GPU (line 7). Only the local part of $Z^{prev}$ is saved (line
+8). After that, a processor is able to compute an updated version of
+its own roots (line 9) with the EA method. The local error is computed
+(line 10) and the global error is also computed using $MPI\_Reduce$ (line 11). Then
+the local roots are transferred from the GPU memory to the CPU memory
+(line 12) before being exchanged between all processors (line 13) in
+order to give to all processors the last version of the roots (with
+the $MPI\_AlltoAll$ routine). If the convergence is not satisfied, a
+new iteration is executed.
 
 \begin{algorithm}[htpb]
 \label{alg2-cuda-mpi}
 \LinesNumbered
-\caption{CUDA-MPI Algorithm to find roots with the Ehrlich-Aberth method}
+\SetAlgoNoLine
+\caption{Finding roots of polynomials with the Ehrlich-Aberth method on multiple GPUs using MPI}
 
-\KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (Error tolerance
-  threshold), P (Polynomial to solve), P' (Derivative of P), $n$ (Polynomial degrees), $\Delta z$ ( error of stop condition)}
+\KwIn{  $\epsilon$ (tolerance threshold)}
 
-\KwOut {$Z$ (Solution root's vector)}
+\KwOut {$Z$ (solution vector of roots)}
 
 \BlankLine
-Initialization of P\;
-Initialization of Pu\;
-Initialization of the solution vector $Z^{0}$\;
+Initialize the polynomial $P$ and its derivative $P'$\;
+Set the initial values of vector $Z$\;
+Determine the local part of the MPI process\;
 Computation of the parameters for the $MPI\_AlltoAll$\;
-Allocate memory to GPU\;
+Copy $P$, $P'$ from CPU to GPU\;
 \While {$error > \epsilon$}{
-copy Z from CPU to GPU\;
-$Z^{Prev}_{loc}=kernel\_save(Z_{loc})$\;
-$Z_{loc}=kernel\_update(Z,P,P')$\;
-$\Delta z=kernel\_testConv(Z_{loc},Z^{prev}_{loc})$\;
-$error=MPI\_Reduce(\Delta z)$\;
-Copy $Z_{loc}$ from GPU to CPU\;
-$Z=MPI\_AlltoAll(Z_{loc})$\;
+  Copy $Z$ from CPU to GPU\;
+  $Z^{prev}_{loc}$ = KernelSave($Z_{loc}$)\;
+  $Z_{loc}$ = KernelUpdate($P,P',Z$)\;
+  $error_{loc}$ = KernelComputeError($Z_{loc},Z^{prev}_{loc}$)\;
+  $error=MPI\_Reduce(error_{loc})$\;
+  Copy $Z_{loc}$ from GPU to CPU\;
+  $Z=MPI\_AlltoAll(Z_{loc})$\;
 }
-\RC{A uniformiser avec les autres algos, mais les grandes lignes sont là}
 \end{algorithm}
 
 
@@ -368,201 +467,129 @@ $Z=MPI\_AlltoAll(Z_{loc})$\;
 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}
-	\forall \alpha_{1} \alpha_{2} \in C,\forall n_{1},n_{2} \in N^{*}; P(z)= (z^{n_{1}}-\alpha_{1})(z^{n_{2}}-\alpha_{2})
+	\forall \alpha_{1} \alpha_{2} \in \mathbb{C},\forall n_{1},n_{2} \in \mathbb{N}^{*}; p(z)= (z^{n_{1}}-\alpha_{1})(z^{n_{2}}-\alpha_{2})
 \end{equation}\noindent
 {\it A full polynomial} is, in contrast, a polynomial for which all the coefficients are not null. A full polynomial is defined by:
-%%\begin{equation}
-	%%\forall \alpha_{i} \in C,\forall n_{i}\in N^{*}; P(z)= \sum^{n}_{i=1}(z^{n^{i}}.a_{i})
-%%\end{equation}
 
 \begin{equation}
-     {\Large \forall a_{i} \in C, i\in N;  p(x)=\sum^{n}_{i=0} a_{i}.x^{i}} 
+     {\Large \forall \alpha_{i} \in \mathbb{C}, i\in \mathbb{N};  p(x)=\sum^{n}_{i=0} \alpha_{i}.x^{i}} 
 \end{equation}
 
-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
+\KG{For our tests, a CPU Intel(R) Xeon(R) CPU
+X5650@2.40GHz and 4 GPUs cards Tesla Kepler K40,are used with CUDA version 7.5}.
+ 
+ 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
+degrees.  All experimental results obtained are performed with double
+precision floating-point 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}.
+the methods are given by the Guggenheimer method~\cite{Gugg86}.
 
+\subsection{Evaluation of the multi-GPUs approaches}
+In this part, we evaluate the performances of the CUDA-OpenMP and
+CUDA-MPI approaches of the EA algorithm on different GPU platforms
+composed each of 1, 2, 3 or 4 GPUs. In this experiments we report the
+experimental results of the EA algorithms to find the roots of different sparse and full polynomials of high degrees ranging from 100,000 to 1,400,000. Figures~\ref{fig:01} and~\ref{fig:02} show the execution times to solve, respectively, sparse and full polynomials with the CUDA-OpenMP algorithm, and Figures~\ref{fig:03} and~\ref{fig:04} show those to solve, respectively, sparse and full polynomials with the CUDA-MPI algorithm.
 
-\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-GPUs with (2,3,4) GPUs, for different sparse polynomial degrees ranging from 100,000 to 1,400,000.
+All these figures show that the CUDA-OpenMP and the CUDA-MPI approaches of the EA algorithm, compared to the single GPU version, are efficient and scale well with multiple GPUs. Both approaches allow us to solve sparse and full polynomials of very high degrees. Using 4 GPUs allows us to achieve a quasi-linear speedup.
 
 \begin{figure}[htbp]
 \centering
-  \includegraphics[angle=-90,width=0.5\textwidth]{Sparse_omp}
-\caption{Execution time in seconds of the Ehrlich-Aberth method to
-  solve sparse polynomials on multiple GPUs with CUDA-OpenMP.}
+\includegraphics[angle=-90,width=0.5\textwidth]{Sparse_omp}
+\caption{Execution times in seconds of the Ehrlich-Aberth method to solve sparse polynomials on multiple GPUs with CUDA-OpenMP.}
 \label{fig:01}
 \end{figure}
 
-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 of the EA method to solve full polynomials on multiple GPUs}
-
-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 time in seconds of the Ehrlich-Aberth method to
-  solve full polynomials on multiple GPUs with CUDA-OpenMP.}
+\includegraphics[angle=-90,width=0.5\textwidth]{Full_omp}
+\caption{Execution times in seconds of the Ehrlich-Aberth method to solve full polynomials on multiple GPUs with CUDA-OpenMP.}
 \label{fig:02}
 \end{figure}
 
-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{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 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 time in seconds of the Ehrlich-Aberth method to
-  solve sparse polynomials on multiple GPUs with CUDA-MPI.}
+ \caption{Execution times in seconds of the Ehrlich-Aberth method to solve sparse polynomials on multiple GPUs with CUDA-MPI.}
 \label{fig:03}
-\end{figure}
-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}
+ \end{figure}
 
 \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  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.
+ \centering
+  \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}
 
-\subsection{Comparison of  the CUDA-OpenMP and the CUDA-MPI approaches}
 
-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.
+\subsection{Comparison between the CUDA-OpenMP and the CUDA-MPI approaches}
+In the previous section we saw that both approaches are very efficient to reduce the execution times to solve sparse and full polynomials. In this section we try to compare these two approaches. In this experiment three sparse polynomials and three full polynomials of degrees 200,000, 800,000 and 1,400,000 are investigated. Figures~\ref{fig:05} and~\ref{fig:06} show the comparison between CUDA-OpenMP and CUDA-MPI algorithms of the EA method to solve sparse and full polynomials, respectively. 
 
-\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}
+\caption{Execution times  to solve sparse polynomials of three distinct degrees on multiple GPUs using OpenMP and MPI 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}
+\caption{Execution times  to solve full polynomials of three distinct degrees on multiple GPUs using OpenMP and MPI with the Ehrlich-Aberth method}
 \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.
 
-\subsubsection{Solving sparse and full polynomials of the same size with CUDA-MPI}
+In Figure~\ref{fig:05} there is one curve for CUDA-OpenMP and another one for CUDA-MPI for each polynomial investigated. We can see that the results are quite similar between OpenMP and MPI for the polynomial degree of 200K. For the degree of 800K, the MPI version is a little bit slower than the OpenMP version but for the degree of 1,4 million, there is a slight advantage for the MPI version. In Figure~\ref{fig:06}, we can see that when it comes to full polynomials, both approaches are almost equivalent.
+
+
+\subsection{Solving sparse and full polynomials of the same degree on multiple GPUs}
+In this experiment we compare the execution times of the EA algorithm
+according to the number of GPUs to solve sparse and full polynomials
+on multiple GPUs using OpenMP or MPI approaches. We chose three sparse
+and three full polynomials of degrees 200,000, 800,000 and
+1,400,000. Figures~\ref{fig:07} and~\ref{fig:08} show the execution
+times to solve sparse and full polynomials of the same degrees with
+the CUDA-OpenMP version and the CUDA-MPI version, respectively.
 
-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.}
+ \includegraphics[angle=-90,width=0.5\textwidth]{OMP}
+\caption{Execution times to solve sparse and full polynomials of three distinct degrees on multiple GPUs using OpenMP.}
 \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}
+ \includegraphics[angle=-90,width=0.5\textwidth]{MPI}
+\caption{Execution times to solve sparse and full polynomials of three distinct degrees on multiple GPUs using MPI.}
 \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. 
+In Figure~\ref{fig:07} the execution times of the CUDA-OpenMP version to solve sparse polynomials are very low compared to those to solve full polynomials. With sparse polynomials the number of monomials is reduced, consequently the number of operations is reduced and the execution time decreases. Figure~\ref{fig:08} shows the impact of sparsity on the efficiency of the CUDA-MPI approach. We can see that the impact follows the same pattern, a difference in execution times 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.
+These experiments report the execution times of the EA method for sparse and full polynomials of high degrees ranging from 1,000,000 to 5,000,000. In Figure~\ref{fig:09} we can see that both approaches (CUDA-OpenMP and CUDA-MPI) 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.
+
 \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}
+ \caption{Execution times in seconds of the Ehrlich-Aberth method to solve sparse and full polynomials of high degree on 4 GPUs for degrees 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.
  
 
 \section{Conclusion}
 \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 this paper, we have presented parallel implementations of the Ehrlich-Aberth algorithm to solve full and sparse polynomials, on a 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 or 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 a single GPU. 
 
-
-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).
+Our next objective is to extend the model presented here to clusters of GPU nodes, with a three-level scheme: inter-node communications via MPI processes (distributed memory), management of multi-GPU nodes by OpenMP threads (shared memory). Actual platforms may probably also contain purely multi-core nodes without any GPU. This heterogeneous setting may lead to the integration of load balancing algorithms so as to allow an optimal use of hardware resources. 
 
 
 \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.
-
+This  paper  is   partially  funded  by  the  Labex   ACTION  program  (contract
+ANR-11-LABX-01-01). 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.
 
 
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