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\begin{document}

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

\author{\IEEEauthorblockN{Kahina Ghidouche, Abderrahmane Sider }
  \IEEEauthorblockA{Laboratoire LIMED\\
    Faculté des sciences exactes\\
    Université de Bejaia, 06000, Algeria\\
Email: \{kahina.ghidouche,ar.sider\}@univ-bejaia.dz}
\and
\IEEEauthorblockN{Lilia Ziane Khodja, Raphaël Couturier}
\IEEEauthorblockA{FEMTO-ST Institute\\
  University of   Bourgogne Franche-Comte, France\\
Email: zianekhodja.lilia@gmail.com\\ raphael.couturier@univ-fcomte.fr}}


\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.
\end{abstract}

% no keywords
\LZK{Faut pas mettre des keywords?\KG{Oui d'après ça: "no keywords" qui se trouve dans leur fichier source!!, mais c'est Bizzard!!!}}


\IEEEpeerreviewmaketitle


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\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:
\begin{equation}
p(x) = \displaystyle\sum^n_{i=0}{a_ix^i},a_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 :
\begin{equation}
 p(x)=a_n\displaystyle\prod_{i=1}^n(x-z_i), a_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 and 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. The authors showed an interesting
speedup that is 20 times as fast as 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}
\KG{Je viens de supprimer les détails, la référence existe déja, a reverifier}
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.}
 \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. 
 \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.
\AS{La résolution du problème pour des polynomes pleins de degré 6M est une contribution aussi à mon avis}}

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. 


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\section{Parallel programming models}
\label{sec2}
Our objective consists in implementing a root-finding algorithm of polynomials on multiple GPUs. To this end, it is primordial to know how to manage CUDA contexts of different GPUs. A direct method for controlling the various GPUs is to use as many threads or processes as GPU devices. We investigate two parallel paradigms: OpenMP and MPI. In this case, the GPU indices are defined according to the identifiers of the OpenMP threads or the ranks of the MPI processes. In this section we present the parallel programming models: OpenMP, MPI and CUDA.
 
\subsection{OpenMP}


OpenMP (Open Multi-processing) is an application programming interface for parallel programming~\cite{openmp13}. It is a portable approach based on the multithreading designed for shared memory computers, where a master thread forks a number of slave threads which execute blocks of code in parallel. An OpenMP program alternates sequential regions and parallel regions of code, where the sequential regions are executed by the master thread and the parallel ones may be executed by multiple threads. During the execution of an OpenMP program the threads communicate their data (read and modified) in the shared memory. One advantage of OpenMP is the global view of the memory address space of an application. This allows relatively a fast development of parallel applications with easier maintenance. However, it is often difficult to get high rates of performances in large scale-applications. 

\subsection{MPI} 


MPI (Message Passing Interface) is a portable message passing style of the parallel programming designed especially for the distributed memory architectures~\cite{Peter96}. In most MPI implementations, a computation contains a fixed set of processes created at the initialization of the program in such way one process is created per processor. The processes synchronize their computations and communicate by sending/receiving messages to/from other processes. In this case, the data are explicitly exchanged by message passing while the data exchanges are implicit in a multithread programming model like OpenMP and Pthreads. However in the MPI programming model, the processes may either execute different programs referred to as multiple program multiple data (MPMD) or every process executes the same program (SPMD). The MPI approach is one of most used HPC programming model to solve large scale and complex applications.
 
\subsection{CUDA}


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.


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\section{The Ehrlich-Aberth algorithm on a GPU}
\label{sec3}

\subsection{The Ehrlich-Aberth method}

The Ehrlich-Aberth method is a simultaneous method~\cite{Aberth73} using the following iteration
\begin{equation}
\label{Eq:EA1}
z^{k+1}_{i}=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}
{1-\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}\sum_{j=1,j\neq i}^{j=n}{\frac{1}{(z_{i}^{k}-z_{j}^{k})}}}, i=1,\ldots,n
\end{equation}

This method contains 4 steps. The first step consists in the
initializing the polynomial. 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 $\epsilon$.

\begin{equation}
\label{eq:Aberth-Conv-Cond}
\forall i\in[1,n],~\vert\frac{z_i^k-z_i^{k-1}}{z_i^k}\vert<\epsilon
\end{equation}

\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.
 

In order to solve this problem, we propose to modify the iterative
function by using the logarithm and the exponential of a complex and
we propose a new version of the Ehrlich-Aberth method.  This method
allows us to exceed the computation of the polynomials of degree
100,000 and to reach a degree up to more than 1,000,000. The reformulation of the iteration~(\ref{Eq:EA1}) of the Ehrlich-Aberth method with exponential and logarithm is defined as follows, for $i=1,\dots,n$:

\begin{equation}
\label{Log_H2}
z^{k+1}_i = z_i^k - \exp(\ln(p(z_i^k)) - \ln(p'(z^k_i)) - \ln(1-Q(z^k_i))),
\end{equation}

where:

\begin{equation}
\label{Log_H1}
Q(z^k_i) = \exp(\ln(p(z^k_i)) - \ln(p'(z^k_i)) + \ln(\sum_{i\neq j}^n\frac{1}{z^k_i-z^k_j})).
\end{equation}



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}. 


\subsection{The Ehrlich-Aberth parallel implementation on CUDA}




The code is organized as kernels which are parts of code 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 transfered
from the CPU to the GPU (after allocation of the required memory on
the GPU) (line 3). Then at each iteration, if the error is greater
than a threshold, the following operations are performed. The previous
roots are saved using a kernel (line 5). Then the new root with the
new iterations are computed using the EA method with a Gauss Seidel
iteration modes in order to use the lastest roots updated (line
6). This improves the convergence. This kernel is, in pratice, 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{ $\epsilon$ (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{$error > \epsilon$}{
  $Z^{prev}$ = KernelSave($Z$)\;
  $Z$ = KernelUpdate($P,P',Z$)\;
  $error$ = KernelComputeError($Z,Z^{prev}$)\;
}
Copy $Z$ from GPU to CPU\;
\end{algorithm}


The development of this code is a rather long task, 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
section the GPU parallel implementation of Ehrlich-Aberth method with
OpenMP and MPI is presented.




 
\section{The EA algorithm on multiple GPUs}
\label{sec4}
\subsection{an OpenMP-CUDA approach}
Our OpenMP-CUDA implementation of EA algorithm is based on the hybrid
OpenMP and CUDA programming model. This algorithm is presented in
Algorithm~\ref{alg2-cuda-openmp}.  All the data are shared with OpenMP
amoung all the OpenMP threads. The shared data are the solution vector
$Z$, the polynomial to solve $P$, 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 to update its owns 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 (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 max of the local errors is computed on all OpenMP threads (lien 11). At
the end of an iteration, the updated roots are copied from the GPU to
the CPU (line 12) by direcly updating 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{ $\epsilon$ (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 ($Z$, $error$, $P$, $P'$ are shared variables)\;
Determine the local part of the OpenMP thread\;
Copy $P$, $P'$ from CPU to GPU\;
\While{$error > \epsilon$}{
  Copy $Z$ from CPU to GPU\;
  $Z^{prev}_{loc}$ = KernelSave($Z_{loc}$)\;
  $Z_{loc}$ = KernelUpdate($P,P',Z$)\;
  $error_{loc}$ = KernelComputeError($Z_{loc},Z^{prev}_{loc}$)\;
  $error = max(error_{loc})$\;
  Copy $Z_{loc}$ from GPU to $Z$ in CPU\;
}
\end{algorithm}





\subsection{a MPI-CUDA approach}

Our parallel implementation of EA to find roots of polynomials using a
CUDA-MPI approach follows a similar approach to the one used in
CUDA-OpenMP. Each process is responsible to compute 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 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. 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 transfering 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
(ligne 10) and the global error using $MPI\_Reduce$ (line 11). Then
the local roots are transfered from the GPU memory to the CPU memory
(line 12) before being exchanged between all processors (lige 13) in
order to give to all processors the last version of the roots (with
the MPI\_AlltoAll routine). If the convergence is not statisfied, an
new iteration is executed.



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

\KwIn{  $\epsilon$ (tolerance threshold)}

\KwOut {$Z$ (solution vector of roots)}

\BlankLine
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 $MPI\_AlltoAll$\;
Copy $P$, $P'$ from CPU to GPU\;
\While {$error > \epsilon$}{
  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\;
  $Z=MPI\_AlltoAll(Z_{loc})$\;
}
\end{algorithm}


\section{Experiments}
\label{sec5}
We study two categories of polynomials: sparse polynomials and full polynomials.\\
{\it A sparse polynomial} is a polynomial for which only some coefficients are not null. In this paper, we consider sparse polynomials for which the roots are distributed on 2 distinct circles:
\begin{equation}
        \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 test, 4 cards GPU tesla Kepler K40 are used.  In order to
evaluate both the GPU and Multi-GPU approaches, we performed a set of
experiments on a single GPU and multiple GPUs using OpenMP or MPI with
the EA algorithm, for both sparse and full polynomials of different
sizes.  All experimental results obtained are perfomed with double
precision float data and the convergence threshold of the EA method is
set to $10^{-7}$.  The initialization values of the vector solution of
the methods are given by Guggenheimer method~\cite{Gugg86}.


\subsection{Evaluation of the CUDA-OpenMP approach}

Here we report some experiments witt full and sparse polynomials of
different degrees with multiple GPUs.
\subsubsection{Execution times of the EA method to solve sparse polynomials on multiple GPUs}
 
In this experiments we report the execution time of the EA algorithm, on single GPU and multi-GPUs with (2,3,4) GPUs, for different sparse polynomial degrees ranging from 100,000 to 1,400,000.

\begin{figure}[htbp]
\centering
  \includegraphics[angle=-90,width=0.5\textwidth]{Sparse_omp}
\caption{Execution time 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.}
\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.}
\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}

\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.

\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.

\subsubsection{Solving sparse polynomials}
In this experiment three sparse polynomials of size 200K, 800K and 1,4M are investigated.
\begin{figure}[htbp]
\centering
 \includegraphics[angle=-90,width=0.5\textwidth]{Sparse}
\caption{Execution times  to solvs sparse polynomials of three
  distinct sizes on multiple GPUs using MPI and OpenMP with the
  Ehrlich-Aberth method}
\label{fig:05}
\end{figure}
In Figure~\ref{fig:05} there is one curve for CUDA-MPI and another one
for CUDA-OpenMP. We can see that the results are quite similar between
OpenMP and MPI for the polynomials size of 200K. For the size of 800K,
the MPI version is a little bit slower than the OpenMP approach but for
the 1,4 millions size, there is a slight advantage for the MPI
version.

\subsubsection{Solving full polynomials}
\begin{figure}[htbp]
\centering
 \includegraphics[angle=-90,width=0.5\textwidth]{Full}
\caption{Execution time for solving full polynomials of three distinct sizes on multiple GPUs using MPI and OpenMP approaches using Ehrlich-Aberth}
\label{fig:06}
\end{figure}
In Figure~\ref{fig:06}, we can see that when it comes to full polynomials, both approaches are almost equivalent.

\subsubsection{Solving sparse and full polynomials of the same size with CUDA-MPI}

In this experiment we compare the execution time of the EA algorithm
according to the number of GPUs to solve sparse and full
polynomials on multiples GPUs using MPI. We chose three sparse and full
polynomials of size 200K, 800K and 1,4M.
\begin{figure}[htbp]
\centering
 \includegraphics[angle=-90,width=0.5\textwidth]{MPI}
\caption{Execution times to solve sparse and full polynomials of three distinct sizes on multiple GPUs using MPI.}
\label{fig:07}
\end{figure}
In Figure~\ref{fig:07} we can see that CUDA-MPI can solve sparse and
full polynomials of high degrees, the execution times with sparse
polynomial are very low compared to full polynomials. With sparse
polynomials the number of monomials is reduced, consequently the number
of operations is reduced and the execution time decreases.

\subsubsection{Solving sparse and full polynomials of the same size with CUDA-OpenMP}

\begin{figure}[htbp]
\centering
 \includegraphics[angle=-90,width=0.5\textwidth]{OMP}
\caption{Execution time for solving sparse and full polynomials of three distinct sizes on multiple GPUs using OpenMP}
\label{fig:08}
\end{figure}

Figure ~\ref{fig:08} shows the impact of sparsity on the effectiveness of the CUDA-OpenMP approach. We can see that the impact follows the same pattern, a difference in execution time in favor of the sparse polynomials. 

\subsection{Scalability of the EA method on multiple GPUs to solve very high degree polynomials}
These experiments report the execution times of the EA method for
sparse and full polynomials ranging from 1,000,000 to 5,000,000.
\begin{figure}[htbp]
\centering
 \includegraphics[angle=-90,width=0.5\textwidth]{big}
 \caption{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials of high degree on 4 GPUs for sizes ranging from 1M to 5M}
\label{fig:09}
\end{figure}
In Figure~\ref{fig:09} we can see that both approaches are scalable
and can solve very high degree polynomials. In addition, with full polynomial as well as sparse ones, both
approaches give very similar results.

%SIDER JE viens de virer \c ca For sparse polynomials here are a noticeable difference in favour of MPI when the degree is
%above 4 millions. Between 1  and 3 millions, OpenMP is more effecient.
%Under 1 million, OpenMPI and MPI are almost equivalent.

%SIDER : il faut une explication sur les différences ici aussi.
 

\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. 


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).


\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.



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