MAJ de la section III: EA method

[kahina_paper2.git] / paper.tex

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+++ b/paper.tex
@@ -422,20 +422,26 @@ 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 et non pas $n$!}
+%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 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 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}
 
 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.
+ p(x)=\sum_{i=0}^{n}{a_ix^i}=a_n\prod_{i=1}^n(x-\alpha_i), a_0 a_n\neq 0.

 \end{equation}
 \end{equation}

-\LZK{C'est $a_n\neq 0$ (polynôme de degré $n$) et non pas $a_0 a_n\neq 0$, non?}
-\LZK{Est-ce $\alpha_i$ sont les $z_i$ définis dans la suite du papier?}
+%\LZK{C'est $a_n\neq 0$ (polynôme de degré $n$) et non pas $a_0 a_n\neq 0$, non?}
+%\LZK{Est-ce $\alpha_i$ sont les $z_i$ définis dans la suite du papier?}

 
 

-The problem of finding the roots of polynomials can be encountered in numerous applications. \LZK{A mon avis on peut supprimer cette phrase}
+%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:
 \begin{equation}
 \label{DK}
 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:
 \begin{equation}
 \label{DK}


@@ -456,17 +462,25 @@ where $p'(z)$ is the polynomial derivative of $p$ evaluated in the point $z$.
 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.
 
 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.
+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 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.
+%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 !!}

 
 

-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.
+In this paper we propose the parallelization of Ehrlich-Aberth method using two parallel programming paradigms OpenMP and MPI on multi-GPU platforms. Our (CUDA MPI)and (CUDA OpenMP) codes is the first implementation of Ehrlich-Aberth algorithm with multiple GPUs for finding roots polynomial. Our major contributions include:
+ 
+\begin{itemize}
+\item The parallel implementation of EA algorithm on multi-GPU platform with 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.
+ 
+\item The parallel implementation of EA algorithm on multi-GPU platform with 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.
+ \end{itemize}

 \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 !!}
 \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 !!}

- 
-%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. In Section~\ref{sec3} we present 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.

 
 

-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}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \LZK{A revoir toute cette organization}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


@@ -505,14 +519,17 @@ CUDA (Compute Unified Device Architecture) is a parallel computing architecture
 
 %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 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.

 
 %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 


@@ -538,59 +555,59 @@ 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: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:
+%\subsubsection{Convergence Condition}
+%The convergence condition determines the termination of the algorithm. It consists in stopping the %iterative function  when the roots are sufficiently stable. We consider that the method converges %sufficiently when:

 
 

-\begin{equation}
-\label{eq:Aberth-Conv-Cond}
-\forall i \in [1,n];\vert\frac{z_{i}^{k}-z_{i}^{k-1}}{z_{i}^{k}}\vert<\xi
-\end{equation}
+%\begin{equation}
+%\label{eq: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{figure}[htbp]
 
 
 %\begin{figure}[htbp]


@@ -600,10 +617,20 @@ The convergence condition determines the termination of the algorithm. It consis
 %\label{fig:03}
 %\end{figure}
 
 %\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 $ε$ 
+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{,}, wich will make it possible to converge to the roots solution, provided that all the root are different.

 
 

+\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}
+
+ 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}

 
 \subsection{EA 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.
 
 \subsection{EA 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.


@@ -828,7 +855,7 @@ The experiments shows the execution time of the EA algorithm, on a single GPU an
 
 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.
 
 
 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}
+\subsection{Evaluating 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.
 
 \subsubsection{Execution times in seconds of the Ehrlich-Aberth method for solving sparse polynomials on GPUs using distributed memory paradigm with MPI}
 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.
 
 \subsubsection{Execution times in seconds of the Ehrlich-Aberth method for solving sparse polynomials on GPUs using distributed memory paradigm with MPI}


@@ -843,6 +870,7 @@ In this part we perform a set of experiments to compare Multi-GPU (CUDA MPI) app
 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.
 %%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.
 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.
 %%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}
 
 \begin{figure}[htbp]
 \subsubsection{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials on GPUs using distributed memory paradigm with MPI}
 
 \begin{figure}[htbp]


@@ -859,35 +887,59 @@ This is due to the use of MPI parallel paradigm that divides the problem computa
 
 
 
 
 
 

+\subsection{Comparative  between (CUDA-OpenMP) approach and (CUDA-MPI) approach}
+In this part we present some experiment comparing the two Multi-GPU approach (OpenMP versus MPI) for solving sparse polynomial, full polynomials than we compare the execution time of the Ehrlich-Aberth method for solving sparse and full polynomials on GPUs with MPI and with OpenMP.

 
 

-%\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}
+\subsubsection{Comparison between MPI and OpenMP versions of the Ehrlich-Aberth method for solving sparse polynomials on GPUs}
+In this experiment we chose three polynomials of different size like (200K, 800K, 1,4M). We compare their execution time according to the number of the GPUs.
+\begin{figure}[htbp]
+\centering
+ \includegraphics[angle=-90,width=0.5\textwidth]{Sparse}
+\caption{Comparison between MPI and OpenMP versions of the Ehrlich-Aberth method for solving sparse polynomials on GPUs.}
+\label{fig:05}
+\end{figure}
+in figure ~\ref{fig:05} we have two curves: MPI curve and OpenMP curve for each polynomials size. We can see that the results are similar between OpenMP curves and MPI curves for the polynomials size (200K, 1,4M), but there is a slight different between MPI curve and OpenMP curve for the polynomial of size 800K. ...

 
 

-%\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{Comparison between MPI and OpenMP versions of the Ehrlich-Aberth method for solving full polynomials on GPUs}
+\begin{figure}[htbp]
+\centering
+ \includegraphics[angle=-90,width=0.5\textwidth]{Full}
+\caption{Comparison between MPI and OpenMP versions of the Ehrlich-Aberth method for solving full polynomials on GPUs.}
+\label{fig:06}
+\end{figure}
+in figure ~\ref{fig:06}, we can see that the two paradigm MPI and OpenMP give the same result for solving full polynomials with EA algorithm.
+% size (200k,800K, 1,4M) are very similar for solving full polynomials with the EA algorithm.

 
 

-%\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{Comparison of execution times of the Ehrlich-Aberth method for solving sparse and full polynomials on GPUs with distributed memory paradigm using MPI}
+in this experiment we compare the execution time of EA algorithm according to the number of the GPU for solving sparse and full polynomials on Multi-GPU using MPI. We chose three sparse and full polynomials of different size like (200K, 800K, 1,4M). 
+\begin{figure}[htbp]
+\centering
+ \includegraphics[angle=-90,width=0.5\textwidth]{MPI}
+\caption{Comparison 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}
+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. 

 
 

-%\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{Comparison of execution times of the Ehrlich-Aberth method for solving sparse and full polynomials on GPUs with shared memory paradigm using OpenMP}
+
+\begin{figure}[htbp]
+\centering
+ \includegraphics[angle=-90,width=0.5\textwidth]{OMP}
+\caption{Comparison of execution times of the Ehrlich-Aberth method for solving sparse and full polynomials on GPUs with shared memory paradigm using OpenMP.}
+\label{fig:08}
+\end{figure}

 
 

+in figure ~\ref{fig:08}
+\subsection{Scalability of the EA method on Multi-GPU to solve very high polynomials degrees}
+ This experiment we report the execution time according to the degrees polynomials ranging from 1,000,000 to 5,000,000 for both approaches (cuda-OpenMP) and (CUDA-MPI) 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 degrees on 4 GPUs.}
+\label{fig:09}
+\end{figure}
+in figure ~\ref{fig:09} we can see that both (cuda-OpenMP) and (CUDA-MPI) approaches are scalable can solve very high polynomials degrees. with full polynomial the both approaches give very interesting ans similar results for polynomials of 5,000,000 degrees we not reach 30,000 s 
+%for sparse and full polynomials

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@@ -986,7 +1038,7 @@ 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.