relecture de l'introduction

[kahina_paper2.git] / paper.tex

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\begin{document}
%
% paper title
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\title{Two parallel implementations of Ehrlich-Aberth algorithm for root finding of polynomials
on  multiple GPUs with OpenMP and MPI}


% author names and affiliations
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\author{\IEEEauthorblockN{Michael Shell}
\IEEEauthorblockA{School of Electrical and\\Computer Engineering\\
Georgia Institute of Technology\\
Atlanta, Georgia 30332--0250\\
Email: http://www.michaelshell.org/contact.html}
\and
\IEEEauthorblockN{Homer Simpson}
\IEEEauthorblockA{Twentieth Century Fox\\
Springfield, USA\\
Email: homer@thesimpsons.com}
\and
\IEEEauthorblockN{James Kirk\\ and Montgomery Scott}
\IEEEauthorblockA{Starfleet Academy\\
San Francisco, California 96678--2391\\
Telephone: (800) 555--1212\\
Fax: (888) 555--1212}}

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%Eldon Tyrell\IEEEauthorrefmark{4}}
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%Georgia Institute of Technology,
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% in the abstract
\begin{abstract}
\LZK{J'ai un peu modifié l'abstract. Sinon à revoir pour le degré max des polynômes testés après les tests de raph.}
Finding roots of polynomials is a very important part of solving real-life problems but it is not so easy for polynomials of high degrees. In this paper, we present two different parallel algorithms of the Ehrlich-Aberth method to find roots of sparse and fully defined polynomials of high degrees. Both algorithms are based on CUDA technology to be implemented on multi-GPU computing platforms but each using different parallel paradigms: OpenMP or MPI. The experiments show a quasi-linear speedup by using up-to 4 GPU devices to find roots of polynomials of degree up-to 1.4 billion. To our knowledge, this is the first paper to present this technology mix to solve such a highly demanding problem in parallel programming. \LZK{Je n'ai pas bien saisi la dernière phrase.}
\end{abstract}

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\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-1}{a_ix^i}.\end{equation}
\LZK{Dans ce cas le polynôme est de degré $n-1$!}

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 of $p$ (\textit{i.e.} roots). If zeros are $\alpha_{i}$, $i=1,\ldots,n$, then $p(x)$ can be written as :
\begin{equation}
 p(x)=a_n\prod_{i=1}^n(x-\alpha_i), a_0 a_n\neq 0.
\end{equation}

The 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}
 DK: z_i^{k+1}=z_{i}^{k}-\frac{P(z_i^{k})}{\prod_{i\neq j}(z_i^{k}-z_j^{k})},   i = 1, \ldots, n,
\end{equation}
where $z_i^k$ is the $i^{th}$ root of the polynomial $p$ at the iteration $k$. Another method discovered by Borsch-Supan~\cite{ Borch-Supan63} and also described by Ehrlich~\cite{Ehrlich67} and Aberth~\cite{Aberth73} uses a different iteration form as follows:
%%\begin{center}
\begin{equation}
\label{Eq:EA}
 EA: z_i^{k+1}=z_i^{k}-\frac{1}{{\frac {P'(z_i^{k})} {P(z_i^{k})}}-{\sum_{i\neq j}\frac{1}{(z_i^{k}-z_j^{k})}}}, i = 1, \ldots, n,
\end{equation}
%%\end{center}
where $p'(z)$ is the polynomial derivative of $p$ evaluated in the
point $z$.

%Aberth, Ehrlich and Farmer-Loizou~\cite{Loizou83} have proved that
%the Ehrlich-Aberth method (EA) has a cubic order of convergence for simple roots whereas the Durand-Kerner has a quadratic order of %convergence.

The main problem of the simultaneous methods is that the necessary time needed for the convergence increases with the increasing of the polynomial's degree. Many authors have treated the problem of implementing  simultaneous methods in parallel. Freeman~\cite{Freeman89} implemented and compared DK, EA and another method of the fourth order proposed by Farmer and Loizou~\cite{Loizou83} \LZK{of the fourth order ?? \\ Sinon peut on donner et citer le nom de la 3ième méthode?} on a 8-processor linear chain, for polynomials of degree up-to 8.
The third method often diverges, \LZK{C'est mieux de donner le nom de cette 3ième méthode} but the first two 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 $z^{k}_{i}$ 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 Unit) 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 et al~\cite{Kahinall14} proposed an implementation of the Durand-Kerner method on a single GPU. Their main results showed that a parallel CUDA implementation is about 10 times faster than the sequential implementation on a single CPU for sparse polynomials of degree 48,000.

Finding polynomial roots rapidly and accurately is the main objective of our work. In this paper we propose the parallelization of Ehrlich-Aberth method using two parallel programming paradigms OpenMP and MPI on multi-GPU platforms. {\color{red}{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.}}
\LZK{Trop détaillé et mal expliqué. \\ 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.}
 
{\color{red}{This paper is organized as follows. In Section~\ref{sec2} we recall the Ehrlich-Aberth method. In section 3 we present EA algorithm on single GPU. In section 4 we propose the EA algorithm implementation on Multi-GPU for (OpenMP-CUDA) approach and (MPI-CUDA) approach. In section 5 we present our experiments and discus it. Finally, Section~\ref{sec6} concludes this paper and gives some hints for future research directions in this topic.}}\LZK{A revoir toute cette organization}
 
 
\section{Parallel Programmings Model}
\label{sec2}
 
\subsection{OpenMP}
Open Multi-Processing (OpenMP) is a shared memory architecture API that provides multi thread capacity~\cite{openmp13}. OpenMP is
a portable approach for parallel programming on shared memory systems based on compiler directives, that can be included in order
to parallelize a loop. In this way, a set of loops can be distributed along the different threads that will access to different data allocated in local shared memory. One of the advantages of OpenMP is its global view of application memory address space that allows relatively fast development of parallel applications with easier maintenance. However, it is often difficult to get high rates of performance in large scale applications. Although usage of OpenMP  threads and managed data explicitly done with MPI can be considered, this approcache undermines the advantages of OpenMP.

%\subsection{OpenMP} 
%OpenMP is a shared memory programming API based on threads from
%the same system process. Designed for multiprocessor shared memory UMA or
%NUMA [10], it relies on the execution model SPMD ( Single Program, Multiple Data Stream )
%where the thread "master" and threads "slaves" asynchronously execute their codes
%communicate / synchronize via shared memory [7]. It also helps to build
%the loop parallelism and is very suitable for an incremental code parallelization
%Sequential natively. Threads share some or all of the available memory and can
%have private memory areas [6].

\subsection{MPI} 
The MPI (Message Passing Interface) library allows to create computer programs that run on a distributed memory architecture. The various processes have their own environment of execution and execute their code in a asynchronous way, according to the MIMD model  (Multiple Instruction streams, Multiple Data streams); they communicate and synchronize by exchanging messages~\cite{Peter96}. MPI messages are explicitly sent, while the exchanges are implicit within the framework of a multi-thread programming environment like OpenMP or Pthreads.
 
\subsection{CUDA}
CUDA (an acronym for Compute Unified Device Architecture) is a parallel computing architecture developed by NVIDIA~\cite{CUDA10}. The
unit of execution in CUDA is called a thread. Each thread executes a kernel by the streaming processors in parallel. In CUDA,
a group of threads that are executed together is called a thread block, and the computational grid consists of a grid of thread
blocks. Additionally, a thread block can use the shared memory on a single multiprocessor while the grid executes a single
CUDA program logically in parallel. Thus in CUDA programming, it is necessary to design carefully the arrangement of the thread
blocks in order to ensure low latency and a proper usage of shared memory, since it can be shared only in a thread block
scope. The effective bandwidth of each memory space depends on the memory access pattern. Since the global memory has lower
bandwidth than the shared memory, the global memory accesses should be minimized.


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}
\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.

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

%\begin{equation}
%w_{i}(z)=\prod_{j=1,j \neq i}^{n} (z-x_{j})
%\end{equation}

%And let a rational function $R_{i}(z)$ be the correction term of the
%Weistrass method~\cite{Weierstrass03}

%\begin{equation}
%R_{i}(z)=\frac{p(z)}{w_{i}(z)} , i=1,2,...,n.
%\end{equation}

%Differentiating the rational function $R_{i}(z)$ and applying the
%Newton method, we have:

%\begin{equation}
%\frac{R_{i}(z)}{R_{i}^{'}(z)}= \frac{p(z)}{p^{'}(z)-p(z)\frac{w_{i}(z)}{w_{i}^{'}(z)}}= \frac{p(z)}{p^{'}(z)-p(z) \sum _{j=1,j \neq i}^{n}\frac{1}{z-x_{j}}}, i=1,2,...,n
%\end{equation}
%where R_{i}^{'}(z)is the rational function derivative of F evaluated in the point z 
%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 $a_{i}$:

\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}.

\begin{equation}
\label{eq:radiusR}
%%\begin{align}
\sigma_{0}=\frac{u+v}{2};u=\frac{\sum_{i=1}^{n}u_{i}}{n.max_{i=1}^{n}u_{i}};
v=\frac{\sum_{i=0}^{n-1}v_{i}}{n.min_{i=0}^{n-1}v_{i}};\\
%%\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}

\subsubsection{Iterative Function}
The operator used by the Aberth method  corresponds to the
equation~\ref{Eq:EA1}, it enables the convergence towards
the polynomials zeros, provided all the roots are distinct.

%Here we give a second form of the iterative function used by the Ehrlich-Aberth method: 

\begin{equation}
\label{Eq:EA1}
EA: z^{k+1}_{i}=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}
{1-\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}\sum_{j=1,j\neq i}^{j=n}{\frac{1}{(z_{i}^{k}-z_{j}^{k})}}}, i=1,. . . .,n
\end{equation}

\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{figure}[htbp]
%\centering
 % \includegraphics[angle=-90,width=0.5\textwidth]{EA-Algorithm}
%\caption{The Ehrlich-Aberth algorithm on single GPU}
%\label{fig:03}
%\end{figure}

%the Ehrlich-Aberth method is an iterative  method, contain 4 steps, start from the initial approximations of all the
%roots of the polynomial,the second step initialize the solution vector $Z$ using the Guggenheimer method to assure the distinction of the initial vector roots, than in step 3 we apply the the iterative function based on the Newton's method and Weiestrass operator[...,...], wich will make it possible to converge to the roots solution, provided that all the root are different. At the end of each application of the iterative function, a stop condition is verified consists in stopping the iterative process when the whole of the modules of the roots
%are lower than a fixed value $ε$ 


\subsection{EA parallel implementation on CUDA}
Like any parallel code, a GPU parallel implementation first
requires to determine the sequential tasks and the
parallelizable parts of the sequential version of the
program/algorithm. In our case, all the operations that are easy
to execute in parallel must be made by the GPU to accelerate
the execution of the application, like the step 3 and step 4. On the other hand, all the
sequential operations and the operations that have data
dependencies between threads or recursive computations must
be executed by only one CUDA or CPU thread (step 1 and step 2). Initially, we specify the organization of parallel threads, by specifying the dimension of the grid Dimgrid, the number of blocks per grid DimBlock and the number of threads per block. 

The code is organzed  by what is named kernels, portions o code that are run on GPU devices. For step 3, there are two kernels, the
first named \textit{save} is used to save vector $Z^{K-1}$ and the seconde one is named
\textit{update} and is used to update the $Z^{K}$ vector. For step 4, a kernel 
tests the convergence of the method. In order to
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 used the Gauss-Seidel mode of iteration. To
parallelize the code, we created kernels and many functions to
be executed on the GPU for all the operations dealing with the
computation on complex numbers and the evaluation of the
polynomials. As said previously, we managed both functions
of evaluation of a polynomial: the normal method, based on
the method of Horner and the method based on the logarithm
of the polynomial. All these methods were rather long to
implement, as the development of corresponding kernels with
CUDA is longer than on a CPU host. This comes in particular
from the fact that it is very difficult to debug CUDA running
threads like threads on a CPU host. In the following paragraph
Algorithm~\ref{alg1-cuda} shows the GPU parallel implementation of Ehrlich-Aberth method.

\begin{enumerate}
\begin{algorithm}[htpb]
\label{alg1-cuda}
%\LinesNumbered
\caption{CUDA Algorithm to find roots with the Ehrlich-Aberth method}

\KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (Error tolerance
  threshold), P (Polynomial to solve), Pu (Derivative of P), $n$ (Polynomial degrees), $\Delta z_{max}$ (Maximum value of stop condition)}

\KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}

%\BlankLine

\item Initialization of the of P\;
\item Initialization of the of Pu\;
\item Initialization of the solution vector $Z^{0}$\;
\item Allocate and copy initial data to the GPU global memory\;
\item k=0\;
\While {$\Delta z_{max} > \epsilon$}{
\item Let $\Delta z_{max}=0$\;
\item $ kernel\_save(ZPrec,Z)$\;
\item  k=k+1\;
\item $ kernel\_update(Z,P,Pu)$\;
\item $kernel\_testConverge(\Delta z_{max},Z,ZPrec)$\;

}
\item Copy results from GPU memory to CPU memory\;
\end{algorithm}
\end{enumerate}
~\\ 


 
\section{The EA algorithm on Multi-GPU}
\label{sec4}
\subsection{MGPU : an OpenMP-CUDA approach}
Our OpenMP-CUDA implementation of EA algorithm is based on the hybrid OpenMP and CUDA programming model. It works as follows.
Based on the metadata, a shared memory is used to make data evenly shared among OpenMP threads. The shared data are the solution vector $Z$, the polynomial to solve $P$, and  the error vector $\Delta z$. Let (T\_omp) the number of OpenMP threads be equal to the number of GPUs, each OpenMP thread binds to one GPU,  and controls a part of the shared memory, that is a part of the vector Z , that is $(n/num\_gpu)$ roots where $n$ is the polynomial's degree and $num\_gpu$ the total number of available GPUs. Each OpenMP thread copies its data from host memory to GPU’s device memory.Then every GPU will have a grid of computation organized according to the device performance and the size of data on which it runs the computation kernels. %In principle a grid is set by two parameter DimGrid, the number of block per grid, DimBloc: the number of threads per block. The following schema  shows the architecture of (CUDA,OpenMP).

%\begin{figure}[htbp]
%\centering
 % \includegraphics[angle=-90,width=0.5\textwidth]{OpenMP-CUDA}
%\caption{The OpenMP-CUDA architecture}
%\label{fig:03}
%\end{figure}
%Each thread OpenMP compute the kernels on GPUs,than after each iteration they copy out the data from GPU memory to CPU shared memory. The kernels are re-runs is up to the roots converge sufficiently. Here are below the corresponding algorithm:

$num\_gpus$ OpenMP threads  are created using \verb=omp_set_num_threads();=function (step $3$, Algorithm \ref{alg2-cuda-openmp}), the shared memory is created using \verb=#pragma omp parallel shared()= OpenMP function (line $5$, Algorithm\ref{alg2-cuda-openmp}), then each OpenMP thread allocates memory and copies initial data from CPU memory to GPU global memory, executes the kernels on GPU, but computes only his portion of roots indicated with variable \textit{index} initialized in (line 5, Algorithm \ref{alg2-cuda-openmp}), used as input data in the $kernel\_update$ (line 10, Algorithm \ref{alg2-cuda-openmp}). After each iteration, all OpenMP threads synchronize using \verb=#pragma omp barrier;= to gather all the correct values of $\Delta z$, thus allowing the computation the maximum stop condition on vector $\Delta z$ (line 12, Algorithm \ref{alg2-cuda-openmp}). Finally, threads copy the results from GPU memories to CPU memory. The OpenMP threads execute kernels until the roots sufficiently converge.  
\begin{enumerate}
\begin{algorithm}[htpb]
\label{alg2-cuda-openmp}
%\LinesNumbered
\caption{CUDA-OpenMP Algorithm to find roots with the Ehrlich-Aberth method}

\KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (Error tolerance
  threshold), P (Polynomial to solve), Pu (Derivative of P), $n$ (Polynomial degree), $\Delta z$ ( Vector of errors for stop condition), $num_gpus$ (number of OpenMP threads/ Number of GPUs), $Size$ (number of roots)}

\KwOut {$Z$ ( Root's vector), $ZPrec$ (Previous root's vector)}

\BlankLine

\item Initialization of P\;
\item Initialization of Pu\;
\item Initialization of the solution vector $Z^{0}$\;
\verb=omp_set_num_threads(num_gpus);=
\verb=#pragma omp parallel shared(Z,$\Delta$ z,P);=
\verb=cudaGetDevice(gpu_id);=
\item Allocate and copy initial data from CPU memory to the GPU global memories\;
\item index= $Size/num\_gpus$\;
\item k=0\;
\While {$error > \epsilon$}{
\item Let $\Delta z=0$\;
\item $ kernel\_save(ZPrec,Z)$\;
\item  k=k+1\;
\item $ kernel\_update(Z,P,Pu,index)$\;
\item $kernel\_testConverge(\Delta z[gpu\_id],Z,ZPrec)$\;
%\verb=#pragma omp barrier;=
\item error= Max($\Delta z$)\;
}

\item Copy results from GPU memories to CPU memory\;
\end{algorithm}
\end{enumerate}
~\\ 



\subsection{Multi-GPU : an MPI-CUDA approach}
%\begin{figure}[htbp]
%\centering
 % \includegraphics[angle=-90,width=0.2\textwidth]{MPI-CUDA}
%\caption{The MPI-CUDA architecture }
%\label{fig:03}
%\end{figure}
Our parallel implementation of EA to find root of polynomials using a CUDA-MPI approach is a data parallel approach. It splits input data of the polynomial to solve among MPI processes. In Algorithm \ref{alg2-cuda-mpi}, input data are the polynomial to solve $P$, the solution vector $Z$, the previous solution vector $ZPrev$, and the value of errors of stop condition $\Delta z$. Let $p$ denote the number of MPI processes on and $n$ the degree of the polynomial to be solved. The algorithm performs a simple data partitioning by creating $p$ portions, of at most $\lceil n/p \rceil$ roots to find per MPI process, for each $Z$ and $ZPrec$. Consequently, each MPI process of rank $k$ will have its own solution vector $Z_{k}$ and $ZPrec$, the error related to the stop condition $\Delta z_{k}$, enabling each MPI process to compute $\lceil n/p \rceil$ roots.

Since a GPU works only on data already allocated in its memory, all local input data, $Z_{k}$, $ZPrec$ and $\Delta z_{k}$, must be transferred from CPU memories to the corresponding GPU memories. Afterwards, the same EA algorithm (Algorithm \ref{alg1-cuda}) is run by all processes but on different polynomial subset of roots $ p(x)_{k}=\sum_{i=1}^{n} a_{i}x^{i}, k=1,...,p$.  Each MPI process executes the  loop \verb=(While(...)...do)= containing the CUDA kernels but each MPI process  computes only its own portion of the roots according to the rule ``''owner computes``''. The local range of roots is indicated with the \textit{index} variable initialized at (line 5, Algorithm \ref{alg2-cuda-mpi}), and passed as an input variable to $kernel\_update$ (line 10, Algorithm \ref{alg2-cuda-mpi}). After each iteration, MPI processes synchronize  (\verb=MPI_Allreduce= function) by a reduction on $\Delta z_{k}$ in order to compute the maximum error related to the stop condition.   Finally, processes copy the values of new computed roots  from GPU memories to CPU memories, then communicate their results to other processes with \verb=MPI_Alltoall= broadcast. If the stop condition is not verified ($error > \epsilon$) then processes stay withing the loop \verb= while(...)...do= until all the roots sufficiently converge.

\begin{enumerate}
\begin{algorithm}[htpb]
\label{alg2-cuda-mpi}
%\LinesNumbered
\caption{CUDA-MPI Algorithm to find roots with the Ehrlich-Aberth method}

\KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (Error tolerance
  threshold), P (Polynomial to solve), Pu (Derivative of P), $n$ (Polynomial degrees), $\Delta z$ ( error of stop condition), $num_gpus$ (number of MPI processes/ number of GPUs), Size (number of roots)}

\KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}

\BlankLine
\item Initialization of P\;
\item Initialization of Pu\;
\item Initialization of the solution vector $Z^{0}$\;
\item Allocate and copy initial data from CPU memories to GPU global memories\;
\item $index= Size/num_gpus$\;
\item k=0\;
\While {$error > \epsilon$}{
\item Let $\Delta z=0$\;
\item $kernel\_save(ZPrec,Z)$\;
\item  k=k+1\;
\item $kernel\_update(Z,P,Pu,index)$\;
\item $kernel\_testConverge(\Delta z,Z,ZPrec)$\;
\item ComputeMaxError($\Delta z$,error)\;
\item Copy results from GPU memories to CPU memories\;
\item Send $Z[id]$ to all processes\;
\item Receive $Z[j]$ from every other process j\;
}
\end{algorithm}
\end{enumerate}
~\\ 

\section{Experiments}
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})
\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}} 
\end{equation}
For our tests, a CPU Intel(R) Xeon(R) CPU E5620@2.40GHz and a GPU K40 (with 6 Go of ram) are used. 
%SIDER : Une meilleure présentation de l'architecture est à faire ici.

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

\subsection{Evaluating the M-GPU (CUDA-OpenMP) approach}

We report here  the results of the set of experiments with M-GPU approach for full and sparse polynomials of different degrees, and we compare it with a Single GPU execution.
\subsubsection{Execution times in seconds of the EA method for solving sparse polynomials on GPUs using shared memory paradigm with OpenMP}
 
In this experiments we report the execution time of the EA algorithm, on single GPU and Multi-GPU with (2,3,4) GPUs, for different sparse polynomial degrees ranging from 100,000 to 1,400,000.

\begin{figure}[htbp]
\centering
  \includegraphics[angle=-90,width=0.5\textwidth]{Sparse_omp}
\caption{Execution times in seconds of the Ehrlich-Aberth method for solving sparse polynomials on GPUs using shared memory paradigm with OpenMP}
\label{fig:01}
\end{figure}

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

\subsubsection{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials on GPUs using shared memory paradigm with OpenMP}

The experiments shows the execution time of the EA algorithm, on a single GPU and on multiple GPUs using the CUDA OpenMP approach for full polynomials of degrees ranging from 100,000 to 1,400,000.

\begin{figure}[htbp]
\centering
  \includegraphics[angle=-90,width=0.5\textwidth]{Full_omp}
\caption{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials on multiple GPUs using shared memory paradigm with OpenMP}
\label{fig:03}
\end{figure}

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

\subsection{Evaluting the Multi-GPU (CUDA-MPI) approach}
In this part we perform a set of experiments to compare Multi-GPU (CUDA MPI) approach with single GPU, for solving full and sparse polynomials of degrees ranging from 100,000 to 1,400,000.

\subsubsection{Execution times in seconds of the Ehrlich-Aberth method for solving sparse polynomials on GPUs using distributed memory paradigm with MPI}

\begin{figure}[htbp]
\centering
  \includegraphics[angle=-90,width=0.5\textwidth]{Sparse_mpi}
\caption{Execution times in seconds of the Ehrlich-Aberth method for solving sparse polynomials on GPUs using distributed memory paradigm with MPI}
\label{fig:02}
\end{figure}
~\\
This figure shows 4 curves of execution time of EA algorithm, a curve with single GPU, 3 curves with multiple GPUs (2, 3, 4). We can clearly see that the curve with single GPU is above the other curves, which shows consumption in execution time compared to the Multi-GPU. We can see also that the CUDA-MPI approach reduces the execution time by a factor of 100 for polynomials of degree more than 1,000,000 whereas a single GPU is of the scale 1000.
%%SIDER : Je n'ai pas reformuler car je n'ai pas compris la phrase, merci de l'ecrire ici en fran\cais.
\\
\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]
\centering
  \includegraphics[angle=-90,width=0.5\textwidth]{Full_mpi}
\caption{Execution times in seconds of the Ehrlich-Aberth method for full polynomials on GPUs using distributed memory paradigm with MPI}
\label{fig:04}
\end{figure}
%SIDER : Corriger le point de la courbe 3-GPUs qui correpsond à un degré de 600000

Figure \ref{fig:04} shows the execution time of the algorithm on single GPU and on multipe GPUs with (2, 3, 4) GPUs for full polynomials. With the CUDA-MPI approach, we notice that the three curves are distinct from each other, more we use GPUs more the execution time decreases. On the other hand the curve with a single GPU is well above the other curves.

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




%\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}

%\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}

%\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}

%\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}

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\section{Conclusion}
\label{sec5}
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 future, we will evaluate our parallel implementation of Ehrlich-Aberth algorithm on other parallel programming model 

Our next objective is to extend the model presented here at clusters of nodes featuring  multiple GPUs, with a three-level scheme: inter-node communication via MPI processes (distributed memory), management of multi-GPU node by OpenMP threads (shared memory).

%present a communication approach between multiple GPUs. The comparison between MPI and OpenMP as GPUs controllers shows that these
%solutions can effectively manage multiple graphics cards to work together
%to solve the same problem


 %than we have presented two communication approach between multiple GPUs.(CUDA-OpenMP) approach and (CUDA-MPI) approach, in the objective to manage multiple graphics cards to work together and solve the same problem. in the objective to manage multiple graphics cards to work together and solve the same problem. 




% conference papers do not normally have an appendix


% use section* for acknowledgment
\section*{Acknowledgment}


The authors would like to thank...





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