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\newcommand{\LZK}[2][inline]{%
  \todo[color=red!10,#1]{\sffamily\textbf{LZK:} #2}\xspace}





\begin{document}
%
% paper title
% Titles are generally capitalized except for words such as a, an, and, as,
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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
% use a multiple column layout for up to three different
% affiliations
\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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%\author{\IEEEauthorblockN{Michael Shell\IEEEauthorrefmark{1},
%Homer Simpson\IEEEauthorrefmark{2},
%James Kirk\IEEEauthorrefmark{3}, 
%Montgomery Scott\IEEEauthorrefmark{3} and
%Eldon Tyrell\IEEEauthorrefmark{4}}
%\IEEEauthorblockA{\IEEEauthorrefmark{1}School of Electrical and Computer Engineering\\
%Georgia Institute of Technology,
%Atlanta, Georgia 30332--0250\\ Email: see http://www.michaelshell.org/contact.html}
%\IEEEauthorblockA{\IEEEauthorrefmark{2}Twentieth Century Fox, Springfield, USA\\
%Email: homer@thesimpsons.com}
%\IEEEauthorblockA{\IEEEauthorrefmark{3}Starfleet Academy, San Francisco, California 96678-2391\\
%Telephone: (800) 555--1212, Fax: (888) 555--1212}
%\IEEEauthorblockA{\IEEEauthorrefmark{4}Tyrell Inc., 123 Replicant Street, Los Angeles, California 90210--4321}}




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% make the title area
\maketitle

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

% no keywords




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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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}{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.

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_0\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}
\LZK{Pourquoi $a_0a_n\neq 0$ ?: $a_0$ pour la premiere equation et $a_n$ pour la deuxieme equation }

%The problem of finding the roots of polynomials can be encountered in numerous applications. \LZK{A mon avis on peut supprimer cette phrase}
Most of the numerical methods that deal with the polynomial root-finding problem are simultaneous methods, \textit{i.e.} the iterative methods to find simultaneous approximations of the $n$ polynomial roots. These methods start from the initial approximations of all $n$ polynomial roots and give a sequence of approximations that converge to the roots of the polynomial. Two examples of well-known simultaneous methods for root-finding problem of polynomials are  Durand-Kerner method~\cite{Durand60,Kerner66} and Ehrlich-Aberth method~\cite{Ehrlich67,Aberth73}.
\LZK{Pouvez-vous donner des références pour les deux méthodes?, c'est fait}

%The first method of this group is Durand-Kerner method:
%\begin{equation}
%\label{DK}
% DK: z_i^{k+1}=z_{i}^{k}-\frac{P(z_i^{k})}{\prod_{i\neq j}(z_i^{k}-z_j^{k})},   i = 1, \ldots, n,
%\end{equation}
%where $z_i^k$ is the $i^{th}$ root of the polynomial $p$ at the iteration $k$. Another method discovered by Borsch-Supan~\cite{ Borch-Supan63} and also described by Ehrlich~\cite{Ehrlich67} and Aberth~\cite{Aberth73} uses a different iteration form as follows:
%%\begin{center}
%\begin{equation}
%\label{Eq:EA}
 %EA: z_i^{k+1}=z_i^{k}-\frac{1}{{\frac {P'(z_i^{k})} {P(z_i^{k})}}-{\sum_{i\neq j}\frac{1}{(z_i^{k}-z_j^{k})}}}, i = 1, \ldots, n,
%\end{equation}
%%\end{center}
%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 Durand-Kerner method, Ehrlich-Aberth method and another method of the fourth order of convergence proposed by Farmer and Loizou~\cite{Loizou83} on a 8-processor linear chain, for polynomials of degree up-to 8. The method of Farmer and Loizou~\cite{Loizou83} often diverges, but the first two methods (Durand-Kerner and Ehrlich-Aberth methods) have a speed-up equals to 5.5. Later, Freeman and Bane~\cite{Freemanall90} considered asynchronous algorithms in which each processor continues to update its approximations even though the latest values of other approximations $z^{k}_{i}$ have not been received from the other processors, in contrast with synchronous algorithms where it would wait those values before making a new iteration. Couturier et al.~\cite{Raphaelall01} proposed two methods of parallelization for a shared memory architecture with OpenMP and for a distributed memory one with MPI. They are able to compute the roots of sparse polynomials of degree 10,000 in 116 seconds with OpenMP and 135 seconds with MPI only by using 8 personal computers and 2 communications per iteration. The authors showed an interesting speedup comparing to the sequential implementation which takes up-to 3,300 seconds to obtain same results.
\LZK{``only by using 8 personal computers and 2 communications per iteration''. Pour MPI? et Pour OpenMP: Rep: c'est MPI seulement}

Very few work had been performed since then until the appearing of the Compute Unified Device Architecture (CUDA)~\cite{CUDA15}, a parallel computing platform and a programming model invented by NVIDIA. The computing power of GPUs (Graphics Processing Units) has exceeded that of traditional processors CPUs. However, CUDA adopts a totally new computing architecture to use the hardware resources provided by the GPU in order to offer a stronger computing ability to the massive data computing. Ghidouche et al.~\cite{Kahinall14} proposed an implementation of the Durand-Kerner method on a single GPU. Their main results showed that a parallel CUDA implementation is about 10 times faster than the sequential implementation on a single CPU for sparse polynomials of degree 48,000.

%Finding polynomial roots rapidly and accurately is the main objective of our work. In this paper we propose the parallelization of Ehrlich-Aberth method using two parallel programming paradigms OpenMP and MPI on multi-GPU platforms. We consider two architectures: shared memory and distributed memory computers. The first parallel algorithm is implemented on shared memory computers by using OpenMP API. It is based on threads created from the same system process, such that each thread is attached to one GPU. In this case the communications between GPUs are done by OpenMP threads through shared memory. The second parallel algorithm uses the MPI API, such that each GPU is attached and managed by a MPI process. The GPUs exchange their data by message-passing communications. This latter approach is more used on distributed memory clusters to solve very complex problems that are too large for traditional supercomputers, which are very expensive to build and run.
%\LZK{Cette partie est réécrite. \\ Sinon qu'est ce qui a été fait pour l'accuracy dans ce papier (Finding polynomial roots rapidly and accurately is the main objective of our work.)?}
%\LZK{Les contributions ne sont pas définies !!}

In this paper we propose the parallelization of Ehrlich-Aberth method using two parallel programming paradigms OpenMP and MPI on CUDA multi-GPU platforms. Our CUDA/MPI and CUDA/OpenMP codes are the first implementations of Ehrlich-Aberth method with multiple GPUs for finding roots of polynomials. Our major contributions include:
\LZK{Pourquoi la méthode Ehrlich-Aberth et pas autres? the Ehrlich-Aberth have very good convergence  and it is suitable to be implemented in parallel computers.}
 \begin{itemize}
 \item An improvements for the Ehrlich-Aberth method using the exponential logarithm in order to be able to solve sparse and full polynomial of degree up to 1, 000, 000.
 \item A parallel implementation of Ehrlich-Aberth method on single GPU with CUDA.
\item The parallel implementation of Ehrlich-Aberth algorithm on a multi-GPU platform with a shared memory using OpenMP API. It is based on threads created from the same system process, such that each thread is attached to one GPU. In this case the communications between GPUs are done by OpenMP threads through shared memory.
\item The parallel implementation of Ehrlich-Aberth algorithm on a multi-GPU platform with a distributed memory using MPI API, such that each GPU is attached and managed by a MPI process. The GPUs exchange their data by message-passing communications. This latter approach is more used on clusters to solve very complex problems that are too large for traditional supercomputers, which are very expensive to build and run.
 \end{itemize}
\LZK{Pas d'autres contributions possibles?: j'ai rajouté 2}

%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 implementation of the Ehrlich-Aberth algorithm on a single GPU. In Section~\ref{sec4} we present the parallel implementations of the Ehrlich-Aberth algorithm on Multi-GPU using the OpenMP and MPI approaches. In section\ref{sec5} we present our experiments and discus it. Finally, Section~\ref{sec6} concludes this paper and gives some hints for future research directions in this topic. 
\LZK{A revoir toute cette organization: je viens de la revoir}

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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}
%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].

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

\subsection{MPI} 
%The MPI (Message Passing Interface) library allows to create computer programs that run on a distributed memory architecture. The various processes have their own environment of execution and execute their code in a asynchronous way, according to the MIMD model  (Multiple Instruction streams, Multiple Data streams); they communicate and synchronize by exchanging messages~\cite{Peter96}. MPI messages are explicitly sent, while the exchanges are implicit within the framework of a multi-thread programming environment like OpenMP or Pthreads.

MPI (Message Passing Interface) is a portable message passing style of the parallel programming designed especially for the distributed memory architectures~\cite{Peter96}. In most MPI implementations, a computation contains a fixed set of processes created at the initialization of the program in such way one process is created per processor. The processes synchronize their computations and communicate by sending/receiving messages to/from other processes. In this case, the data are explicitly exchanged by message passing while the data exchanges are implicit in a multithread programming model like OpenMP and Pthreads. However in the MPI programming model, the processes may either execute different programs referred to as multiple program multiple data (MPMD) or every process executes the same program (SPMD). The MPI approach is one of most used HPC programming model to solve large scale and complex applications.
 
\subsection{CUDA}
%CUDA (is an acronym of the Compute Unified Device Architecture) is a parallel computing architecture developed by NVIDIA~\cite{CUDA10}.The unit of execution in CUDA is called a thread. Each thread executes a kernel by the streaming processors in parallel. In CUDA, a group of threads that are executed together is called a thread block, and the computational grid consists of a grid of thread blocks. Additionally, a thread block can use the shared memory on a single multiprocessor while the grid executes a single CUDA program logically in parallel. Thus in CUDA programming, it is necessary to design carefully the arrangement of the thread blocks in order to ensure low latency and a proper usage of shared memory, since it can be shared only in a thread block scope. The effective bandwidth of each memory space depends on the memory access pattern. Since the global memory has lower bandwidth than the shared memory, the global memory accesses should be minimized.

CUDA (Compute Unified Device Architecture) is a parallel computing architecture developed by NVIDIA~\cite{CUDA15} for GPUs. It provides a high level GPGPU-based programming model to program GPUs for general purpose computations and non-graphic applications. The GPU is viewed as an accelerator such that data-parallel operations of a CUDA program running on a CPU are off-loaded onto GPU and executed by this later. The data-parallel operations executed by GPUs are called kernels. The same kernel is executed in parallel by a large number of threads organized in grids of thread blocks, such that each GPU multiprocessor executes one or more thread blocks in SIMD fashion (Single Instruction, Multiple Data) and in turn each core of the multiprocessor executes one or more threads within a block. Threads within a block can cooperate by sharing data through a fast shared memory and coordinate their execution through synchronization points. In contrast, within a grid of thread blocks, there is no synchronization at all between blocks. The GPU only works on data filled in the global memory and the final results of the kernel executions must be transferred out of the GPU. In the GPU, the global memory has lower bandwidth than the shared memory associated to each multiprocessor. Thus in the CUDA programming, it is necessary to design carefully the arrangement of the thread blocks in order to ensure low latency and a proper usage of the shared memory, and the global memory accesses should be minimized.

%We introduced three paradigms of parallel programming. Our objective consists in implementing a root finding polynomial algorithm on multiple GPUs. To this end, it is primordial to know how to manage CUDA contexts of different GPUs. A direct method for controlling the various GPUs is to use as many threads or processes as GPU devices. We can choose the GPU index based on the identifier of OpenMP thread or the rank of the MPI process. Both approaches will be investigated.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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

\subsection{The Ehrlich-Aberth method}
%A cubically convergent iteration method to find zeros of
%polynomials was proposed by O. Aberth~\cite{Aberth73}. The
%Ehrlich-Aberth (EA is short) method contains 4 main steps, presented in what
%follows.

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

%\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~\cite{,}, witch will make it possible to converge to the roots solution, provided that all the root are different.

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

contain 4 steps, start from the initial approximations of all the roots of the polynomial,the second step initialize the solution vector $Z$ using the Guggenheimer~\cite{Gugg86} method to assure the distinction of the initial vector roots,

 than in step 3 we apply the the iterative function based on the Newton's method~\cite{newt70} and Weiestrass operator~\cite{Weierstrass03}, wich will make it possible to converge to the roots solution, provided that all the root are different.


 At the end of each application of the iterative function, a stop condition is verified consists in stopping the iterative process when the whole of the modules of the roots are lower than a fixed value $\xi$. 

\begin{equation}
\label{eq:Aberth-Conv-Cond}
\forall i \in [1,n];\vert\frac{z_{i}^{k}-z_{i}^{k-1}}{z_{i}^{k}}\vert<\xi
\end{equation}
\subsection{Improving Ehrlich-Aberth method}
......
\subsection{Ehrlich-Aberth parallel implementation on CUDA}
We introduced three paradigms of parallel programming. Our objective consists in implementing a root finding polynomial algorithm on multiple GPUs. To this end, it is primordial to know how to manage CUDA contexts of different GPUs. A direct method for controlling the various GPUs is to use as many threads or processes as GPU devices. We can choose the GPU index based on the identifier of OpenMP thread or the rank of the MPI process. Both approaches will be investigated.




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 Multiple GPUs}
\label{sec4}
\subsection{M-GPU : an OpenMP-CUDA approach}
Our OpenMP-CUDA implementation of EA algorithm is based on the hybrid OpenMP and CUDA programming model. It works as follows.
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}
\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 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 M-GPU and Multi-GPU approaches, we performed a set of experiments on a single GPU and multiple GPUs using OpenMP or MPI by EA algorithm, for both sparse and full polynomials of different sizes.
All experimental results obtained are made in double precision data whereas the convergence threshold of the EA method is set to $10^{-7}$.
%Since we were more interested in the comparison of the
%performance behaviors of Ehrlich-Aberth and Durand-Kerner methods on
%CPUs versus on GPUs.
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 the M-GPU approach for full and sparse polynomials of different degrees, and we compare it with a Single GPU execution.
\subsubsection{Execution time of the EA method for solving sparse polynomials on multiple GPUs using the M-GPU approach}
 
In this experiments we report the execution time of the EA algorithm, on single GPU and Multi-GPU with (2,3,4) GPUs, for different sparse polynomial degrees ranging from 100,000 to 1,400,000.

\begin{figure}[htbp]
\centering
  \includegraphics[angle=-90,width=0.5\textwidth]{Sparse_omp}
\caption{Execution time in seconds of the Ehrlich-Aberth method for solving sparse polynomials on multiple GPUs using the M-GPU approach}
\label{fig:01}
\end{figure}

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

\subsubsection{Execution time in seconds of the Ehrlich-Aberth method for solving full polynomials on multiple GPUs using the M-GPU approach}

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

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

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

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

\subsubsection{Execution time of the Ehrlich-Aberth method for solving sparse polynomials on multiple GPUs using the Multi-GPU approach}

\begin{figure}[htbp]
\centering
  \includegraphics[angle=-90,width=0.5\textwidth]{Sparse_mpi}
\caption{Execution time in seconds of the Ehrlich-Aberth method for solving sparse polynomials on multiple GPUs using the Multi-GPU approach}
\label{fig:02}
\end{figure}
~\\
Figure~\ref{fig:02} shows execution time of EA algorithm, for a single GPU, and multiple GPUs (2, 3, 4) on respectively 2, 3 and four MPI nodes. We can clearly see that the curve for a single GPU is above the other curves, which shows overtime in execution time compared to the Multi-GPU approach. We can see also that the CUDA-MPI approach reduces the execution time by a factor of 10 for polynomials of degree more than 1,000,000. For example, at degree 1000000, the execution time with a single GPU amounted to 10 thousand seconds, while with 4 GPUs, it is lowered to about just one thousand seconds which makes it for a tenfold speedup.
%%SIDER : Je n'ai pas reformuler car je n'ai pas compris la phrase, merci de l'ecrire ici en fran\cais.
\\cette figure montre 4 courbes de temps d'exécution pour l'algorithme EA, une courbe avec un seul GPU, 3 courbes pour multiple GPUs(2, 3, 4), on peut constaté clairement que la courbe à un seul GPU est au-dessus des autres courbes, vue sa consomation en temps d'exècution. On peut voir aussi qu'avec l'approche Multi-GPU (CUDA-MPI) reduit le temps d'exècution jusqu'à l'echelle 100 pour le polynômes qui dépasse 1,000,000 tandis que Single GPU est de l'echelle 1000.

\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 GPUs using the Multi-GPU}
\label{fig:04}
\end{figure}


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

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



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

In the previuos section we saw that both approches are very effective in reducing execution time for sparse as well as full polynomials. At this stage, the interesting question is which approach is better. In the fellowing, we present appropriate experiments comparing the two Multi-GPU approaches to answer the question.

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

\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 for solving sparse and full polynomials on Multi-GPU using MPI. We chose three sparse and full polynomials of size 200K, 800K  and 1,4M. 
\begin{figure}[htbp]
\centering
 \includegraphics[angle=-90,width=0.5\textwidth]{MPI}
\caption{Execution time for solving sparse and full polynomials of three distinct sizes on multiple GPUs using MPI}
\label{fig:07}
\end{figure}
in figure ~\ref{fig:07} we can see that CUDA-MPI can solve sparse and full polynomials of high degrees, the execution time with sparse polynomial are very low comparing to full polynomials. with sparse polynomials the number of monomial are reduce, consequently the number of operation are reduce than the execution time decrease. 

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

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

Figure ~\ref{fig:08} shows the impact of sparsity on the effectiveness of the CUDA-OpenMP approach. We can see that the impact fellows the same pattern, a difference in execution time in favor of the sparse polynomials. 
%SIDER : il faut une explication ici. je ne vois pas de prime abords, qu'est-ce qui engendre cette différence, car quelques soient les coefficients nulls ou non nulls, c'est toutes les racines qui sont calculées qu'elles soient similaires ou non (degrés de multiplicité).
\subsection{Scalability of the EA method on Multi-GPU to solve very high degree polynomials}
These experiments report the execution time according to the degrees of polynomials ranging from 1,000,000 to 5,000,000 for both approaches with sparse and full  polynomials. 
\begin{figure}[htbp]
\centering
 \includegraphics[angle=-90,width=0.5\textwidth]{big}
 \caption{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials of high degree on 4 GPUs for sizes ranging from 1M to 5M}
\label{fig:09}
\end{figure}
In figure ~\ref{fig:09} we can see that both approaches are scalable and can solve very high degree polynomials. With full polynomial both approaches give interestingly very similar results. For the sparse case however, there are a noticeable difference in favour of MPI when the degree is above 4M. Between 1M and 3M, the OMP approach is more effective and under 1M degree, OMP and MPI approaches are almost equivalent.

%SIDER : il faut une explication sur les différences ici aussi.
 
%for sparse and full polynomials
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\section{Conclusion}
\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 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. 




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\section*{Acknowledgment}


The authors would like to thank...





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