[kahina_paper2.git] / paper.tex

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\title{A parallel implementation of Ehrlich-Aberth algorithm  for root finding of polynomials
on Multi-GPU with OpenMP/MPI}


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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}
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Springfield, USA\\
Email: homer@thesimpsons.com}
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San Francisco, California 96678--2391\\
Telephone: (800) 555--1212\\
Fax: (888) 555--1212}}

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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 by expressing any outcome as a function of some unknown variables. Formally speaking,  a polynomial $p(x)$ of degree \textit{n} having $n$ coefficients in the complex plane \textit{C} is :
%%\begin{center}
\begin{equation}
     {\Large p(x)=\sum_{i=0}^{n}{a_{i}x^{i}}}.
\end{equation}
%%\end{center}

The root finding problem consists in finding the values of all the $n$ different values of the variable $x$ for which \textit{p(x)} is null. Such values are called zeros of $p$. If zeros are $\alpha_{i},\textit{i=1,...,n}$ then $p(x)$ can be written as :
\begin{equation}
     {\Large 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. Most of the numerical methods that deal with this problem are simultaneous ones, i.e that find concurrently all of $n$ zeroes. These methods start from the initial approximations of all the roots of the polynomial 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, . . . , n,
\end{equation}
%%\end{center}
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 and brought
in the following form by Ehrlich~\cite{Ehrlich67} and
Aberth~\cite{Aberth73} uses a different iteration formula given as:
%%\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, . . . , 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 is increased with the increasing of the degree of the polynomial. 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}, on a 8-processor linear chain, for polynomials of degree up to 8.
The third method often diverges, but the first two methods have speed-up equal 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 and al.~\cite{Raphaelall01} proposed two methods of parallelization for a shared memory architecture with \textit{OpenMP} and for distributed memory one with \textit{MPI}. They were 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. Comparing to the sequential implementation where it takes up to 3,300 seconds to obtain the same results, the authors show an interesting speedup.

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 GPU in order to offer a stronger computing ability to the massive data computing. Ghidouche and al~\cite{Kahinall14} proposed an implementation of the Durand-Kerner method on GPU. Their main result 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 parallel programming paradigms (OpenMP, MPI) on GPUs. We consider two architectures: shared memory with OpenMP API and distributed memory 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. 
 
This paper is organized as follows, in section 2 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.
 
 
\section{Parallel Programmings Model}
 
\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} %L'article en Français Programmation multiGPU – OpenMP versus MPI
%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 synchronise 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}%L'article en anglais Multi-GPU and multi-CPU accelerated FDTD scheme for vibroacoustic applications
CUDA (an acronym for Compute Unified Device Architecture) is a parallel computing architecture developed by NVIDIA~\cite{NVIDIA12}. 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 consist to implement an algorithm of root finding polynomial on multiple GPUs. It primordial to know how to manage CUDA contexts of different GPUs. A direct method for controlling the various GPU 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 single GPU}
\subsection{the EA method}

A cubically convergent iteration method to find zeros of
polynomials was proposed by O. Aberth~\cite{Aberth73}. The
Ehrlich-Aberth 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-spaced 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 is corresponding to the
following equation~\ref{Eq:EA1} which will enable the convergence towards
polynomial solutions, 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 specifies the organization of threads in parallel, need to specify the dimension of the grid Dimgrid: the number of block per grid and block by DimBlock: the number of threads per block required to process a certain task. 

we create the kernel, for step 3 we have two kernels, the
first named \textit{save} is used to save vector $Z^{K-1}$ and the kernel
\textit{update} is used to update the $Z^{K}$ vector. In step 4 a kernel is
created to test the convergence of the method. In order to
compute function H, we have two possibilities: either to use
the Jacobi method, or the Gauss-Seidel method 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 1 shows the GPU parallel implementation of Ehrlich-Aberth method.

Algorithm~\ref{alg2-cuda} shows a sketch of the Ehrlich-Aberth method using CUDA.

\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}

\subsection{MGPU (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 $P$. vector of error of stop condition $\Delta z$. Let(T\_omp) number of OpenMP threads is equal to the number of GPUs, each threads OpenMP checks one GPU,  and control a part of the shared memory, that is a part of the vector Z  like: $(n/num\_gpu)$ roots, n: the polynomial's degrees, $num\_gpu$ the number of GPUs. Each OpenMP thread copies its data from host memory to GPU’s device memory.Than every GPU will have a grid of computation organized with its performances and the size of data of which it checks and compute 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:

$num\_gpus$ thread OpenMP are created using \verb=omp_set_num_threads();=function (line,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}), than each OpenMP threads allocate and copy initial data from CPU memory to the GPU global memories, execute the kernels on GPU, and compute 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, OpenMP threads synchronize using \verb=#pragma omp barrier;= to recuperate all values of vector $\Delta z$, to compute the maximum stop condition in vector $\Delta z$(line 12, Algorithm \ref{alg2-cuda-openmp}).Finally,they copy the results from GPU memories to CPU memory. The OpenMP threads execute kernels until the roots converge sufficiently.  
\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 degrees), $\Delta z$ ( Vector of errors of stop condition), $num_gpus$ (number of OpenMP threads/ number of GPUs), $Size$ (number of roots)}

\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 $Z^{0}$\;
\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 (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 the Ehrlich-Aberth method to find root polynomial using (CUDA-MPI) approach, splits input data of the polynomial to solve between MPI processes. From Algorithm 3, the 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 size of the polynomial to be solved. The algorithm performs a simple data partitioning by creating $p$ portions, of at most $⌈n/p⌉$ roots to find per MPI process, for each element mentioned above. Consequently, each MPI process $k$ will have its own solution vector $Z_{k}$,polynomial to be solved $p_{k}$, the error of stop condition $\Delta z_{k}$, Than each MPI processes compute only $⌈n/p⌉$ roots.

Since a GPU works only on data of its memory, all local input data, $Z_{k}, p_{k}$ and $\Delta z_{k}$, must be transferred from CPU memories to the corresponding GPU memories. Afterward, the same EA algorithm (Algorithm 1) is run by all processes but on different sub-polynomial root $ p(x)_{k}=\sum_{i=k(\frac{n}{p})}^{k+1(\frac{n}{p})} a_{i}x^{i}, k=1,...,p$.  Each processes MPI execute the  loop \verb=(While(...)...do)= contain the kernels. Than each process MPI  compute only his portion of roots indicated with variable \textit{index} initialized in (line 5, Algorithm \ref{alg2-cuda-mpi}), used as input data in the $kernel\_update$ (line 10, Algorithm \ref{alg2-cuda-mpi}). After each iteration, MPI processes synchronize using \verb=MPI_Allreduce= function, in order to compute the maximum error stops condition $\Delta z_{k}$ computed by each process MPI line (line, Algorithm\ref{alg2-cuda-mpi}), and  copy the values of new roots computed from GPU memories to CPU memories, than communicate  her results to the neighboring processes,using \verb=MPI_Alltoallv=. If maximum stop condition $error > \epsilon$ the processes stay to execute the loop \verb= while(...)...do= until all the roots converge sufficiently.

\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 the P\;
\item Initialization of the Pu\;
\item Initialization of the solution vector $Z^{0}$\;
\item Allocate and copy initial data from CPU memories 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 $kernel\_testConverge(\Delta z,Z,ZPrec)$\;
\item ComputeMaxError($\Delta z$,error)\;
\item Copy results from GPU memories to CPU memories\;
\item Send $Z[id]$ to all neighboring processes\;
\item Receive $Z[j]$ from neighboring 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. 

We performed a set of experiments on single GPU and Multi-GPU using (OpenMP/MPI) to find roots polynomials with EA algorithm, for both sparse and full polynomials of different sizes. We took into account the execution times and the  polynomial size performed by sum or each experiment.
All experimental results obtained from the simulations 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{Test with Multi-GPU (CUDA OpenMP) approach}

In this part we performed  a set of experiments on Multi-GPU (CUDA OpenMP) approach for full and sparse polynomials of different degrees, compare it with Single GPU (CUDA).
 \subsubsection{Execution times in seconds of the Ehrlich-Aberth 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 (CUDA OpenMP) Multi-GPU approach reduce the execution time up to the scale 100 whereas single GPU is of scale 1000 for polynomial who exceed 1,000,000. It shows the advantage to use OpenMP parallel paradigm  to connect the performances of several GPUs and solve a  polynomial of high degrees.   

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

This experiments shows the execution time of the EA algorithm, on single GPU (CUDA) and Multi-GPU (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 GPUs using shared memory paradigm with OpenMP}
\label{fig:03}
\end{figure}

The second test with full polynomial shows a very important saving of time, for a polynomial of degrees 1,4M (CUDA OpenMP) approach with 4 GPUs compute and solve it 4 times as fast as single GPU. We notice that curves are positioned one below the other one, more the number of used GPUs increases more the execution time decreases.

\subsection{Test with Multi-GPU (CUDA MPI) approach}
In this part we perform a set of experiment 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 Multi-GPUs (2, 3, 4) GPUs. We see clearly 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 the approach Multi-GPU (CUDA MPI) reduces the execution time up to the scale 100 for polynomial of degrees more than 1,000,000 whereas single GPU is of the scale 1000.
\\
\subsubsection{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials on GPUs using distributed memory paradigm with MPI}

\begin{figure}[htbp]
\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}


this figure shows the execution time of the algorithm EA, on single GPU and Multi-GPUS with (2, 3, 4) GPUs for full polynomials. With (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 single GPU is well above the other curves.
This is due to the use of parallelization MPI paradigm that divides the polynomial into sub polynomials assigned to each GPU. unlike the single GPU which solves all the polynomial on a single GPU, consequently it engenders more execution time.

%\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}
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 Multi-GPUs using two parallel paradigm, shared memory with OpenMP, distributed memory with MPI.(CUDA-OpenMP) approach and (CUDA-MPI) approach, 
We have performed many experiments with the Ehrlich-Aberth method in single GPU, Multi-GPU with (CUDA-OpenMP) approach, Multi-GPU with (CUDA-MPI) approach for sparse and full polynomials. the experiments show that, using parallel programming model like (OpenMP, MPI) can effectively manage multiple graphics cards to work together to solve the same problem and accelerate parallel applications, like (CUDA MPI) approach with 4 GPUs can solve a polynomial of 1,000,000 4 speed up than on single GPU.


In future, we will evaluate our parallel implementation of Ehrlich-Aberth algorithm on other parallel programming model 


%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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