[kahina_paper2.git] / paper.tex

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\begin{document}
%
% paper title
% Titles are generally 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 title.
% Linebreaks \\ can be used within to get better formatting as desired.
% Do not put math or special symbols in the title.
\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{Kahina Guidouche, Abderrahmane Sider }
  \IEEEauthorblockA{Laboratoire LIMED\\
    Faculté des sciences exactes\\
    Université de Bejaia, 06000, Algeria\\
Email: \{kahina.ghidouche,ar.sider\}@univ-bejaia.dz}
\and
\IEEEauthorblockN{Lilia Ziane Khodja, Raphaël Couturier}
\IEEEauthorblockA{FEMTO-ST Institute\\
  University of   Bourgogne Franche-Comte, France\\
Email: zianekhodja.lilia@gmail.com\\ raphael.couturier@univ-fcomte.fr}}

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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}
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%\IEEEauthorblockA{\IEEEauthorrefmark{4}Tyrell Inc., 123 Replicant Street, Los Angeles, California 90210--4321}}




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%\IEEEspecialpapernotice{(Invited Paper)}




% make the title area
\maketitle

% As a general rule, do not put math, special symbols or citations
% in the abstract
\begin{abstract}
Finding roots of polynomials is a very important part of solving
real-life problems but it is not so easy for polynomials of high
degrees. In this paper, we present two different parallel algorithms
of the Ehrlich-Aberth method to find roots of sparse and fully defined
polynomials of high degrees. Both algorithms are based on CUDA
technology to be implemented on multi-GPU computing platforms but each
using different parallel paradigms: OpenMP or MPI. The experiments
show a quasi-linear speedup by using up-to 4 GPU devices compared to 1
GPU to find roots of polynomials of degree up-to 1.4
million. Moreover, other experiments show it is possible to find roots
of polynomials of degree up to 5 millions.
\end{abstract}

% no keywords




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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_n\neq 0. 
\end{equation}
where $\{a_i\}_{0\leq i\leq n}$ are complex coefficients and $n$ is a high integer number. If $a_n\neq0$ then $n$ is called the degree of the polynomial. The root-finding problem consists in finding the $n$ different values of the unknown variable $x$ for which $p(x)=0$. Such values are called roots of $p(x)$. Let $\{z_i\}_{1\leq i\leq n}$ be the roots of polynomial $p(x)$, then $p(x)$ can be written as :
\begin{equation}
 p(x)=a_n\displaystyle\prod_{i=1}^n(x-z_i), a_n\neq 0.
\end{equation}
%\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. \RC{si on donne des temps faut donner
  le proc, comme c'est vieux à mon avis faut supprimer ca, votre avis?} 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.\RC{j'ai envie de virer ca, car c'est pas la nouveauté dans ce papier}
 \item A parallel implementation of Ehrlich-Aberth method on single GPU with CUDA.\RC{idem}
\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}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
\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:EA-1}
%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:AAberth-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{Aberth73} 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}

This methods contains 4 steps. The first step consists of the initial
approximations of all the roots of the polynomial. The second step
initializes the solution vector $Z$ using the Guggenheimer
method~\cite{Gugg86} to ensure the distinction of the initial vector
roots. In step 3, the iterative function based on the Newton's
method~\cite{newt70} and Weiestrass operator~\cite{Weierstrass03} is
applied. With this step the computation of roots will converge,
provided that all roots are different.


In order to stop the iterative function, a stop condition is
applied. This condition checks that all the root modules 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}
With high degree polynomials, the Ehrlich-Aberth method suffers from
floating point overflows due to the mantissa of floating points
representations. This induces errors in the computation of $p(z)$ when
$z$ is large.
 
%Experimentally, it is very difficult to solve polynomials with the Ehrlich-Aberth method and have roots which except the circle of unit, represented by the radius $r$ evaluated as: 

%\begin{equation}
%\label{R.EL}
%R = exp(log(DBL\_MAX)/(2*n) );
%\end{equation}



% where \verb=DBL_MAX= stands for the maximum representable \verb=double= value.
 
In order to solve this problem, we propose to modify the iterative
function by using the logarithm and the exponential of a complex and
we propose a new version of the Ehrlich-Aberth method.  This method
allows us to exceed the computation of the polynomials of degree
100,000 and to reach a degree up to more than 1,000,000. This new
version of the Ehrlich-Aberth method with exponential and logarithm is
defined as follows:

\begin{equation}
\label{Log_H2}
z^{k+1}_{i}=z_{i}^{k}-\exp \left(\ln \left(
p(z_{i}^{k})\right)-\ln\left(p'(z^{k}_{i})\right)- \ln\left(1-Q(z^{k}_{i})\right)\right),
\end{equation}

where:

\begin{eqnarray}
\label{Log_H1}
Q(z^{k}_{i})=\exp\left( \ln (p(z^{k}_{i}))-\ln(p'(z^{k}_{i}))+\ln \left(
\sum_{i\neq j}^{n}\frac{1}{z^{k}_{i}-z^{k}_{j}}\right)\right) \nonumber  \\
i=1,...,n  
\end{eqnarray}


%We propose to use the logarithm and the exponential of a complex in order to compute the power at a high exponent. 
Using the logarithm  and the exponential operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations manipulate lower absolute values~\cite{Karimall98}.

%This problem was discussed earlier in~\cite{Karimall98} for the Durand-Kerner method. The authors
%propose to use the logarithm and the exponential of a complex in order to compute the power at a high exponent. Using the logarithm  and the exponential operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations manipulate lower absolute values and the roots for large polynomial degrees can be looked for successfully~\cite{Karimall98}.

\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 organized kernels which are part of 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 second 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 use Gauss-Seidel iterations. To
parallelize the code, we create 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 manage both functions of
evaluation: 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{algorithm}[htpb]
\label{alg1-cuda}
\LinesNumbered
\SetAlgoNoLine
\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

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

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

 
\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.  All the data
are shared with OpenMP amoung all the OpenMP threads. The shared data
are the solution vector $Z$, the polynomial to solve $P$, and the
error vector $\Delta z$. The number of OpenMP threads is equal to the
number of GPUs, each OpenMP thread binds to one GPU, and it controls a
part of the shared memory. More precisely each OpenMP thread owns 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. Then all GPUs will have a grid of computation organized
according to the device performance and the size of data on which it
runs the computation kernels.

To compute one iteration of the EA method each GPU performs the
followings steps. First roots are shared with OpenMP. Each thread
starts by copying all the previous roots inside its GPU. Then each GPU
will compute an iteration of the EA method on its own roots. For that
all the other roots are used. At the end of an iteration, the updated
roots are copied from the GPU to the CPU. The convergence is checked
on the new roots. Finally each CPU will update its own roots in the
shared memory arrays containing all the roots.

%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:

%% \RC{Surement à virer ou réécrire pour etre compris sans algo}
%% $num\_gpus$ 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{algorithm}[htpb]
\label{alg2-cuda-openmp}
\LinesNumbered
\SetAlgoNoLine
\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

Initialization of P\;
Initialization of Pu\;
Initialization of the solution vector $Z^{0}$\;
omp\_set\_num\_threads(num\_gpus)\;
\#pragma omp parallel shared(Z,$\Delta$ z,P)\;
\Indp
{
gpu\_id=cudaGetDevice()\;
Allocate memory on GPU\;
Compute local size and offet according to gpu\_id\;
\While {$error > \epsilon$}{
  copy Z from CPU to GPU\;
$ ZPrec_{loc}=kernel\_save(Z_{loc})$\;
$ Z_{loc}=kernel\_update(Z,P,Pu)$\;
$\Delta z[gpu\_id] = kernel\_testConv(Z_{loc},ZPrec_{loc})$\;
$  error= Max(\Delta z)$\;
  copy $Z_{loc}$ from GPU to Z in CPU
}
\Indm}
\end{algorithm}



\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}
%% ~\\ 

%% \RC{ENCORE ENCORE PIRE}

\section{Experiments}
\label{sec5}
We study two categories of polynomials: sparse polynomials and full polynomials.\\
{\it A sparse polynomial} is a polynomial for which only some coefficients are not null. In this paper, we consider sparse polynomials for which the roots are distributed on 2 distinct circles:
\begin{equation}
        \forall \alpha_{1} \alpha_{2} \in C,\forall n_{1},n_{2} \in N^{*}; P(z)= (z^{n_{1}}-\alpha_{1})(z^{n_{2}}-\alpha_{2})
\end{equation}\noindent
{\it A full polynomial} is, in contrast, a polynomial for which all the coefficients are not null. A full polynomial is defined by:
%%\begin{equation}
        %%\forall \alpha_{i} \in C,\forall n_{i}\in N^{*}; P(z)= \sum^{n}_{i=1}(z^{n^{i}}.a_{i})
%%\end{equation}

\begin{equation}
     {\Large \forall a_{i} \in C, i\in N;  p(x)=\sum^{n}_{i=0} a_{i}.x^{i}} 
\end{equation}

For our test, 4 cards GPU tesla Kepler K40 are used.  In order to
evaluate both the GPU and Multi-GPU approaches, we performed a set of
experiments on a single GPU and multiple GPUs using OpenMP or MPI with
the EA algorithm, for both sparse and full polynomials of different
sizes.  All experimental results obtained are perfomed with double
precision float data and the convergence threshold of the EA method is
set to $10^{-7}$.  The initialization values of the vector solution of
the methods are given by Guggenheimer method~\cite{Gugg86}.


\subsection{Evaluation of the CUDA-OpenMP approach}

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

\begin{figure}[htbp]
\centering
  \includegraphics[angle=-90,width=0.5\textwidth]{Sparse_omp}
\caption{Execution time in seconds of the Ehrlich-Aberth method to
  solve sparse polynomials on multiple GPUs with CUDA-OpenMP.}
\label{fig:01}
\end{figure}

Figure~\ref{fig:01} shows that the CUDA-OpenMP approach scales well
with multiple GPUs. This version allows us to solve sparse polynomials
of very high degrees.

\subsubsection{Execution times of the EA method to solve full polynomials on multiple GPUs}

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

\begin{figure}[htbp]
\centering
  \includegraphics[angle=-90,width=0.5\textwidth]{Full_omp}
\caption{Execution time in seconds of the Ehrlich-Aberth method to
  solve full polynomials on multiple GPUs with CUDA-OpenMP.}
\label{fig:02}
\end{figure}

In Figure~\ref{fig:02}, we can observe that with full polynomials the EA version with
CUDA-OpenMP scales also well. Using 4 GPUs allows us to achieve a
quasi-linear speedup.

\subsection{Evaluation of the CUDA-MPI approach}
In this part we perform some experiments to evaluate the CUDA-MPI
approach to solve full and sparse polynomials of degrees ranging from
100,000 to 1,400,000.

\subsubsection{Execution times of the EA method to solve sparse polynomials on multiple GPUs}

\begin{figure}[htbp]
\centering
  \includegraphics[angle=-90,width=0.5\textwidth]{Sparse_mpi}
\caption{Execution time in seconds of the Ehrlich-Aberth method to
  solve sparse polynomials on multiple GPUs with CUDA-MPI.}
\label{fig:03}
\end{figure}
Figure~\ref{fig:03} shows the execution times of te EA algorithm,
for a single GPU, and multiple GPUs (2, 3, 4) with the CUDA-MPI approach.

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

\begin{figure}[htbp]
\centering
 \includegraphics[angle=-90,width=0.5\textwidth]{Full_mpi}
\caption{Execution times in seconds of the Ehrlich-Aberth method for
  full polynomials on  multiple GPUs with CUDA-MPI.}
\label{fig:04}
\end{figure}

In Figure~\ref{fig:04}, we can also observe that the CUDA-MPI approach
is also efficient to solve full polynimails on multiple GPUs.

\subsection{Comparison of  the CUDA-OpenMP and the CUDA-MPI approaches}

In the previuos section we saw that both approches are very effecient
to  reduce the execution times the  sparse and full polynomials. In
this section we try to compare these two approaches.

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

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

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

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

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

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

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

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

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

%SIDER : il faut une explication sur les différences ici aussi.
 
%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 to solve full and sparse polynomials, on
single GPU with CUDA and on multiple GPUs using two parallel
paradigms: shared memory with OpenMP and distributed memory with
MPI. These architectures were addressed by a CUDA-OpenMP approach and
CUDA-MPI approach, respectively.  Experiments show that, using
parallel programming model like (OpenMP, MPI). We can efficiently
manage multiple graphics cards to solve the same
problem and accelerate the parallel execution with 4 GPUs and solve a
polynomial of degree up to 5,000,000, four times faster than on single
GPU. 


%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 with clusters
of GPU nodes, with a three-level scheme: inter-node communication via
MPI processes (distributed memory), management of multi-GPU node by
OpenMP threads (shared memory).

%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}

Computations have been performed on the supercomputer facilities of
the Mésocentre de calcul de Franche-Comté. We also would like to thank
Nvidia for hardware donation under CUDA Research Center 2014.






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