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% paper title
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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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\IEEEauthorblockN{Homer Simpson}
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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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\section{Introduction}
Polynomials are mathematical algebraic structures used in science and engineering to capture physical phenomena and to express any outcome in the form of 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 $n$ values of the variable $x$ for which \textit{p(x)} is nullified. Such values are called zeros of $p$. If zeros are $\alpha_{i},\textit{i=1,...,n}$ the $p(x)$ 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 is encountered in different applications. Most of the numerical methods that deal with this problem are simultaneous ones. 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 implementation of simultaneous methods in parallel. Freeman [10] implemented and compared DK, EA and another method of the fourth order proposed by Farmer
and Loizou [9], 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 [11] 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. [12] 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 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 works had been performed since this last work until the appearing of the Compute Unified Device Architecture (CUDA) [13], 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 [14] 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 a parallel programming paradigms (OpenMP, MPI) on GPUs. We consider two architectures: Shared memory with OpenMP API based on threads from the same system process, which each thread is attached to one GPU and after the various memory allocation, each thread throws its part of calculation ( to do this you must first load on the GPU required data and after Suddenly repatriate the result on the host). Distributed memory with MPI: The MPI library is often used for parallel programming [11] in
cluster systems because it is a message-passing programming language. Each GPU are attached to one process MPI, 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 MGPU 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}%L'article en anglais Multi-GPU and multi-CPU accelerated FDTD scheme for vibroacoustic applications 
Open Multi-Processing (OpenMP) is a shared memory architecture API that provides multi thread capacity [22]. 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 allo-
cated 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, in OpenMP a usage of threads ids and managing data explicitly as done in an MPI
code can be considered, it defeats 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} %L'article en Français Programmation multiGPU – OpenMP versus MPI
 The library MPI allows to use a distributed memory architecture. The various processes have their own environment of execution and execute their codes in a asynchronous way, according to the model MIMD (Multiple Instruction streams, Multiple Dated streams); they communicate and synchronize by exchanges of messages [17]. MPI messages are explicitly sent, while the exchanges are implicit within the framework of a programming multi-thread (OpenMP/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 [28]. The
unit of execution in CUDA is called a thread. Each thread executes the kernel by the streaming processors in parallel. In CUDA,
a group of threads that are executed together is called thread blocks, and the computational grid consists of a grid of thread
blocks. Additionally, a thread block can use the shared memory on a single multiprocessor as while as 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 manage CUDA context of different GPUs. A direct method for controlling the various GPU is to use as many threads or processes that GPU. We can choose the GPU index based on the identifier of OpenMP thread or the rank of the MPI process. Both approaches will be created.

\section{The EA algorithm on single GPU}
\subsection{the EA method}
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{alg2-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}
\subsection{MGPU (MPI-CUDA)approach}

\section{experiments}

\begin{figure}[htbp]
\centering
  \includegraphics[angle=-90,width=0.5\textwidth]{Sparse_openmp}
\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}

\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}

\begin{figure}[htbp]
\centering
  \includegraphics[angle=-90,width=0.5\textwidth]{Full_openmp}
\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}

\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}

\begin{figure}[htbp]
\centering
  \includegraphics[angle=-90,width=0.5\textwidth]{Sparse_mpivsomp}
\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_mpivsomp}
\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_mpivsomp}
\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_mpivsomp}
\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}

% An example of a floating figure using the graphicx package.
% Note that \label must occur AFTER (or within) \caption.
% For figures, \caption should occur after the \includegraphics.
% Note that IEEEtran v1.7 and later has special internal code that
% is designed to preserve the operation of \label within \caption
% even when the captionsoff option is in effect. However, because
% of issues like this, it may be the safest practice to put all your
% \label just after \caption rather than within \caption{}.
%
% Reminder: the "draftcls" or "draftclsnofoot", not "draft", class
% option should be used if it is desired that the figures are to be
% displayed while in draft mode.
%
%\begin{figure}[!t]
%\centering
%\includegraphics[width=2.5in]{myfigure}
% where an .eps filename suffix will be assumed under latex, 
% and a .pdf suffix will be assumed for pdflatex; or what has been declared
% via \DeclareGraphicsExtensions.
%\caption{Simulation results for the network.}
%\label{fig_sim}
%\end{figure}

% Note that the IEEE typically puts floats only at the top, even when this
% results in a large percentage of a column being occupied by floats.


% An example of a double column floating figure using two subfigures.
% (The subfig.sty package must be loaded for this to work.)
% The subfigure \label commands are set within each subfloat command,
% and the \label for the overall figure must come after \caption.
% \hfil is used as a separator to get equal spacing.
% Watch out that the combined width of all the subfigures on a 
% line do not exceed the text width or a line break will occur.
%
%\begin{figure*}[!t]
%\centering
%\subfloat[Case I]{\includegraphics[width=2.5in]{box}%
%\label{fig_first_case}}
%\hfil
%\subfloat[Case II]{\includegraphics[width=2.5in]{box}%
%\label{fig_second_case}}
%\caption{Simulation results for the network.}
%\label{fig_sim}
%\end{figure*}
%
% Note that often IEEE papers with subfigures do not employ subfigure
% captions (using the optional argument to \subfloat[]), but instead will
% reference/describe all of them (a), (b), etc., within the main caption.
% Be aware that for subfig.sty to generate the (a), (b), etc., subfigure
% labels, the optional argument to \subfloat must be present. If a
% subcaption is not desired, just leave its contents blank,
% e.g., \subfloat[].


% An example of a floating table. Note that, for IEEE style tables, the
% \caption command should come BEFORE the table and, given that table
% captions serve much like titles, are usually capitalized except for words
% such as a, an, and, as, at, but, by, for, in, nor, of, on, or, the, to
% and up, which are usually not capitalized unless they are the first or
% last word of the caption. Table text will default to \footnotesize as
% the IEEE normally uses this smaller font for tables.
% The \label must come after \caption as always.
%
%\begin{table}[!t]
%% increase table row spacing, adjust to taste
%\renewcommand{\arraystretch}{1.3}
% if using array.sty, it might be a good idea to tweak the value of
% \extrarowheight as needed to properly center the text within the cells
%\caption{An Example of a Table}
%\label{table_example}
%\centering
%% Some packages, such as MDW tools, offer better commands for making tables
%% than the plain LaTeX2e tabular which is used here.
%\begin{tabular}{|c||c|}
%\hline
%One & Two\\
%\hline
%Three & Four\\
%\hline
%\end{tabular}
%\end{table}


% Note that the IEEE does not put floats in the very first column
% - or typically anywhere on the first page for that matter. Also,
% in-text middle ("here") positioning is typically not used, but it
% is allowed and encouraged for Computer Society conferences (but
% not Computer Society journals). Most IEEE journals/conferences use
% top floats exclusively. 
% Note that, LaTeX2e, unlike IEEE journals/conferences, places
% footnotes above bottom floats. This can be corrected via the
% \fnbelowfloat command of the stfloats package.




\section{Conclusion}
The conclusion goes here.




% conference papers do not normally have an appendix


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


The authors would like to thank...





% trigger a \newpage just before the given reference
% number - used to balance the columns on the last page
% adjust value as needed - may need to be readjusted if
% the document is modified later
%\IEEEtriggeratref{8}
% The "triggered" command can be changed if desired:
%\IEEEtriggercmd{\enlargethispage{-5in}}

% references section

% can use a bibliography generated by BibTeX as a .bbl file
% BibTeX documentation can be easily obtained at:
% http://mirror.ctan.org/biblio/bibtex/contrib/doc/
% The IEEEtran BibTeX style support page is at:
% http://www.michaelshell.org/tex/ieeetran/bibtex/
%\bibliographystyle{IEEEtran}
% argument is your BibTeX string definitions and bibliography database(s)
%\bibliography{IEEEabrv,../bib/paper}
%
% <OR> manually copy in the resultant .bbl file
% set second argument of \begin to the number of references
% (used to reserve space for the reference number labels box)
\begin{thebibliography}{1}

\bibitem{IEEEhowto:kopka}
H.~Kopka and P.~W. Daly, \emph{A Guide to \LaTeX}, 3rd~ed.\hskip 1em plus
  0.5em minus 0.4em\relax Harlow, England: Addison-Wesley, 1999.

\end{thebibliography}




% that's all folks
\end{document}