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\begin{document}

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

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


\maketitle

\begin{abstract}
Finding the roots of polynomials is a very important part of solving
real-life problems but the higher the degree of the polynomials is,
the less easy it becomes. 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 use 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 the roots of polynomials of degree up-to
1.4 million. Moreover, other experiments show it is possible to find the
roots of polynomials of degree up-to 5 million.
\end{abstract}

\begin{IEEEkeywords}
  root finding method, Ehrlich-Aberth method, GPU, MPI, OpenMP
\end{IEEEkeywords}

\IEEEpeerreviewmaketitle


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\section{Introduction}


Finding the roots of polynomials of very high degrees arises in many complex problems of 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}{\alpha_ix^i},\alpha_n\neq 0, 
\end{equation}
where $\{\alpha_i\}_{0\leq i\leq n}$ are complex coefficients and $n$ is a high integer number. If $\alpha_n\neq0$ 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)=\alpha_n\displaystyle\prod_{i=1}^n(x-z_i), \alpha_n\neq 0.
\end{equation}

Most of the numerical methods that deal with the polynomial
root-finding problems are simultaneous methods, \textit{i.e.} the
iterative methods to find simultaneous approximations of the $n$
polynomial roots. These methods start from the initial approximation
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
the Durand-Kerner method~\cite{Durand60,Kerner66} and the Ehrlich-Aberth method~\cite{Ehrlich67,Aberth73}.


The convergence time of simultaneous methods drastically increases
with the increasing of the polynomial's degree. The great challenge
with simultaneous methods is to parallelize them and to improve their
convergence. Many authors have proposed parallel simultaneous
methods~\cite{Freeman89,Loizou83,Freemanall90,bini96,cs01:nj,Couturier02},
using several paradigms of parallelization (synchronous or
asynchronous computations, mechanism of shared or distributed memory,
etc). However, so far until now, only polynomials not exceeding
degrees of less than 100,000 have been solved.

%The main problem of the simultaneous methods is that the necessary
%time needed for the convergence increases with the increasing of the
%polynomial's degree. Many authors have treated the problem of
%implementing  simultaneous methods in
%parallel. Freeman~\cite{Freeman89} implemented and compared
%Durand-Kerner method, Ehrlich-Aberth method and another method of the
%fourth order of convergence proposed by Farmer and
%Loizou~\cite{Loizou83} on a 8-processor linear chain, for polynomials
%of degree up-to 8. The method of Farmer and Loizou~\cite{Loizou83}
%often diverges, but the first two methods (Durand-Kerner and
%Ehrlich-Aberth methods) have a speed-up equals to 5.5. Later, Freeman
%and Bane~\cite{Freemanall90} considered asynchronous algorithms in
%which each processor continues to update its approximations even
%though the latest values of other approximations $z^{k}_{i}$ have not
%been received from the other processors, in contrast with synchronous
%algorithms where it would wait those values before making a new
%iteration. Couturier and al.~\cite{cs01:nj} proposed two methods
%of parallelization for a shared memory architecture with OpenMP and
%for a distributed memory one with MPI. They are able to compute the
%roots of sparse polynomials of degree 10,000. The authors showed an interesting
%speedup that is 20 times as fast as the sequential implementation. 

The recent advent of the Compute Unified Device Architecture
(CUDA)~\cite{CUDA15}, a programming
model and a parallel computing architecture developed by NVIDIA, has revived parallel programming interest in
this problem. Indeed, the computing power of GPUs (Graphics Processing
Units) has exceeded that of traditional  CPUs processors, which makes
it very appealing to the research community to investigate new
parallel implementations for a whole set of scientific problems in the
reasonable hope to solve bigger instances of well known
computationally demanding issues such as the one beforehand. However,
CUDA provides an efficient massive data computing model which is
suited to GPU architectures. Ghidouche et al.~\cite{Kahinall14} proposed an implementation of the Durand-Kerner method on a single GPU. Their main results showed that a parallel CUDA implementation is about 10 times faster than the sequential implementation on a single CPU for sparse polynomials of degree 48,000. 

In this paper we propose the parallelization of the Ehrlich-Aberth
(EA) method which has a much better cubic convergence rate than the
quadratic rate of the Durand-Kerner method that has already been investigated in \cite{Kahinall14}. In the other hand,  EA is suitable to be implemented in parallel computers according to the data-parallel paradigm. In this model, computing elements carry computations on the data they are assigned and communicate with other computing elements in order to get fresh data or to synchronize. Classically, two parallel programming paradigms OpenMP and MPI are used to code such solutions. But in our case, computing elements are CUDA multi-GPU platforms. This architectural setting poses new programming challenges but offers also new opportunities to efficiently solve huge problems, otherwise considered intractable until recently. To the best of our knowledge, our CUDA-MPI and CUDA-OpenMP codes are the first implementations of EA method with multiple GPUs for finding roots of polynomials. Our major contributions include:
 \begin{itemize}

\item The parallel implementation of EA 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 EA 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 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.
\item
  Our method is efficient to compute the roots of sparse and full
  polynomials of degree up to 5 million.
 \end{itemize}


The paper is organized as follows. In Section~\ref{sec2} we present three different parallel programming models OpenMP, MPI and CUDA. In Section~\ref{sec3} we present the implementation of the Ehrlich-Aberth algorithm on a single GPU. In Section~\ref{sec4} we present the parallel implementations of the Ehrlich-Aberth algorithm on multiple GPUs using the OpenMP and MPI approaches. In section~\ref{sec5} we present our experiments and discuss them. Finally, Section~\ref{sec6} concludes this paper and gives some hints for future research directions in this topic. 


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\section{Parallel programming models}
\label{sec2}
Our objective consists in implementing a root-finding algorithm of
polynomials on multiple GPUs. To this end, it is essential to know how
to manage the CUDA contexts of different GPUs. A direct method to control the various GPUs is to use as many threads or processes as GPU devices. We investigate two parallel paradigms: OpenMP and MPI. In this case, the GPU indices are defined according to the identifiers of the OpenMP threads or the ranks of the MPI processes. In this section we present the parallel programming models: OpenMP, MPI and CUDA.
 
\subsection{OpenMP}
OpenMP (Open Multi-processing) is an application programming interface
for parallel programming~\cite{openmp13}. It is a portable approach
based on the multithreading designed for shared memory computers,
where a master thread forks a number of slave threads which execute
blocks of code in parallel. An OpenMP program alternates sequential
regions and parallel regions of code, where the sequential regions are
executed by the master thread and the parallel ones may be executed by
multiple threads. During the execution of an OpenMP program the
threads communicate their data (read and modified) in the shared
memory. One advantage of OpenMP is the global view of the memory
address space of an application. This allows  a relatively fast development of parallel applications with easier maintenance. However, it is often difficult to get high rates of performances in large scale-applications. 

\subsection{MPI} 
MPI (Message Passing Interface) is a portable message passing style of
the parallel programming designed specifically for 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 a way that 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 the most used HPC programming model to solve large scale and complex applications.
 
\subsection{CUDA}
CUDA (Compute Unified Device Architecture) is a parallel computing
architecture developed by NVIDIA~\cite{CUDA15} for GPUs. It provides a
high level GPGPU-based programming model to program GPUs for general
purpose computations. 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 latter. 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 with CUDA programming,
it is necessary to design carefully the arrangement of the thread
blocks in order to ensure a low latency and a proper use of the shared
memory. As for the global memory accesses, it should also be minimized.


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

\subsection{The Ehrlich-Aberth method}

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

This method contains 4 steps. The first step consists in the
initializing the polynomial. The second step initializes the solution
vector $Z$ using the Guggenheimer method~\cite{Gugg86} to ensure that
initial roots are all distinct from each other. In step 3, the
iterative function based on the Newton's method~\cite{newt70} and
Weiestrass operator~\cite{Weierstrass03} is applied. In our case, the
Ehrlich-Aberth is applied as in~(\ref{Eq:EA1}). Iterations of the
EA method will converge to the roots of the considered
polynomial. In order to stop the iterative function, a stop condition
is applied, this is the 4th step. This condition checks that all the
root modules are lower than a fixed value $\epsilon$.

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

\subsection{Improving Ehrlich-Aberth method}
With high degree polynomials, the EA method suffers from floating point overflows due to the mantissa of floating points representations. This induces errors in the computation of $p(z)$ when $z$ is large.
 
In order to solve this problem, we propose to modify the iterative
function by using the logarithm and the exponential of a complex and
we propose a new version of the EA method.  This method
allows us to exceed the computation of the polynomials of degree
100,000 and to reach a degree up to more than 1,000,000. The reformulation of the iteration~(\ref{Eq:EA1}) of the EA method with exponential and logarithm operators is defined as follows, for $i=1,\dots,n$:

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

where:

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

Using the logarithm  and the exponential operators, we can replace any
multiplications and divisions with additions and
subtractions. Consequently, computations manipulate lower values in
absolute values~\cite{Karimall98}. In practice, the  exponential and
logarithm mode is used when a root is outside the circle unit represented by the radius $R$ evaluated in C language with:
\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 and $n$ is the degree of the polynomial.


\subsection{The Ehrlich-Aberth parallel implementation on CUDA}
%\KG{
  The algorithm ~\ref{alg1-cuda} shows sketch of the Ehrlich-Aberth method using CUDA.
The first steps consist in the initialization of the input data like, the polynomial P,derivative of P and the vector solution Z. Then, all data of the root finding problem
must be copied from the CPU memory to the GPU global memory,because
the GPUs only work on the data filled in their memories.
Next, all the data-parallel arithmetic operations inside the main loop
\verb=(while(...))= are executed as kernels by the GPU. The
first kernel named \textit{Kernelsave} in line 5 of Algorithm~\ref{alg1-cuda} consists in saving the vector of polynomial roots found at the previous time-step in GPU memory, in
order to check the convergence of the roots after each iteration (line
7, Algorithm~\ref{alg1-cuda}). Then the new roots with the
new iterations are computed using the EA method with a Gauss-Seidel
iteration mode in order to use the latest updated roots (line
6). This improves the convergence. This kernel is, in practice, very
long since it performs all the operations with complex numbers with
the normal mode of the EA method as in Eq.~\ref{Eq:EA1} but also with the logarithm-exponential one as in Eq.(~\ref{Log_H1},~\ref{Log_H2}). The last kernel checks the convergence of the roots after each update of $Z^{k}$, according to formula Eq.~\ref{eq:Aberth-Conv-Cond} line (7). We used the functions of the CUBLAS Library (CUDA Basic Linear Algebra Subroutines) to implement this kernel. 

The algorithm terminates its computations when all the roots have
converged.
%}


 %The code is organized as kernels which are parts of codes that are run
%on GPU devices. Algorithm~\ref{alg1-cuda} describes the CUDA
%implementation of the Ehrlich-Aberth on a GPU.  This algorithms starts
%by initializing the polynomial and its derivative (line 1). The
%initialization of the roots is performed (line 2). Data are transferred
%%from the CPU to the GPU (after the allocation of the required memory on
%the GPU) (line 3). Then at each iteration, if the error is greater
%%than the threshold, the following operations are performed. The previous
%roots are saved using a kernel (line 5). Then the new roots with the
%new iterations are computed using the EA method with a Gauss-Seidel
%iteration mode in order to use the latest updated roots (line
%6). This improves the convergence. This kernel is, in practice, very
%long since it performs all the operations with complex numbers with
%the normal mode of the EA method but also with the
%logarithm-exponential one. Then the error is computed with a final
%kernel (line 7). Finally when the EA method has converged, the roots
%are transferred from the GPU to the CPU.
\begin{algorithm}[htpb]
\label{alg1-cuda}
\LinesNumbered
\SetAlgoNoLine
\caption{Finding roots of polynomials with the Ehrlich-Aberth method on a GPU}
\KwIn{ $\epsilon$ (tolerance threshold)}
\KwOut{$Z$ (solution vector of roots)}
Initialize the polynomial $P$ and its derivative $P'$\;
Set the initial values of vector $Z$\;
Copy $P$, $P'$ and $Z$ from CPU to GPU\;
\While{$error > \epsilon$}{
  $Z^{prev}$ = KernelSave($Z$)\;
  $Z$ = KernelUpdate($P,P',Z$)\;
  $error$ = KernelComputeError($Z,Z^{prev}$)\;
}
Copy $Z$ from GPU to CPU\;
\end{algorithm}
\ \\
This figure shows the second kernel code
\begin{figure}[htbp]
\centering
\includegraphics[angle=+0,width=0.4\textwidth]{code1}
\caption{The Kernel Update code}
\label{fig:00}
\end{figure}

%We noticed that the code is executed by a large number of GPU threads organized as grid of to dimension (Number of block per grid (NbBlock), number of threads per block(Nbthread)), the Nbthread is fixed initially, the NbBlock is computed as fallow: 
%$ NbBlocks= \frac{N+Nbthreads-1}{Nbthreads} where N: the number of root$
%the  such that each thread in grid is in charge of the computation of one root.
 
The development of this code is a rather long task due to the
development of all the kernels that compute the parts ported on the
GPU.  This comes in particular from the fact that it is very difficult
to debug CUDA running threads like threads on a CPU host. In the
following section the GPU parallel implementation of the
Ehrlich-Aberth method with OpenMP and MPI is presented.

\section{The Ehrlich-Aberth algorithm on multiple GPUs}
\label{sec4}
\KG{we remind that to manage the CUDA contexts of different GPUs, We investigate two parallel paradigms: OpenMP and MPI. In this section we present the both \textit{OpenMP-CUDA} approach and the \textit{MPI-CUDA} approach} used to implement the Ehrlich-Aberth algorithm on multiple GPUs.
\subsection{An OpenMP-CUDA approach}
Our OpenMP-CUDA implementation of EA algorithm is based on the hybrid
OpenMP and CUDA programming model. This algorithm is presented in
Algorithm~\ref{alg2-cuda-openmp}.  All the data are shared with OpenMP
among all the OpenMP threads. The shared data are the solution vector
$Z$, the polynomial to solve $P$, its derivative $P'$, and the error
vector $error$. The number of OpenMP threads is equal to the number of
GPUs, each OpenMP thread binds to one GPU, and it controls a part of
the shared memory. More precisely each OpenMP thread will be
responsible for updating its own part of the vector $Z$. This part is
called $Z_{loc}$ in the following. Then all GPUs will have a grid of
computation organized according to the device performance and the size
of data on which it runs the computation kernels.

To compute one iteration of the EA method each GPU performs the
followings steps. First, roots are shared with OpenMP and the
computation of the local size for each GPU is performed (line 4). Each
thread starts by copying all the previous roots inside its GPU (line
5). At each iteration, the following operations are performed. First
the vector $Z$ is transferred from the CPU to the GPU (line 7).  Each
GPU copies the previous roots (line 8) and it computes an iteration of
the EA method on its own roots (line 9).  For that all the other roots
are used. The local error is computed on the new roots (line 10) and
the maximum of the local errors is computed on all OpenMP threads (line 11). At
the end of an iteration, the updated roots are copied from the GPU to
the CPU (line 12) and each CPU directly updates its own roots in the shared
memory arrays containing all the roots.



\begin{algorithm}[htpb]
\label{alg2-cuda-openmp}
\LinesNumbered
\SetAlgoNoLine
\caption{Finding roots of polynomials with the Ehrlich-Aberth method on multiple GPUs using OpenMP}
\KwIn{ $\epsilon$ (tolerance threshold)}
\KwOut{$Z$ (solution vector of roots)}
Initialize the polynomial $P$ and its derivative $P'$\;
Set the initial values of vector $Z$\;
Start of a parallel part with OpenMP ($Z$, $error$, $P$, $P'$ are shared variables)\;
Determine the local part of the OpenMP thread\;
Copy $P$, $P'$ from CPU to GPU\;
\While{$error > \epsilon$}{
  Copy $Z$ from CPU to GPU\;
  $Z^{prev}_{loc}$ = KernelSave($Z_{loc}$)\;
  $Z_{loc}$ = KernelUpdate($P,P',Z$)\;
  $error_{loc}$ = KernelComputeError($Z_{loc},Z^{prev}_{loc}$)\;
  $error = max(error_{loc})$\;
  Copy $Z_{loc}$ from GPU to $Z$ in CPU\;
}
\end{algorithm}





\subsection{A MPI-CUDA approach}
Our parallel implementation of EA to find the roots of polynomials using a
CUDA-MPI approach follows a similar approach to the one used in
CUDA-OpenMP. Each processor is responsible for computing its own part of
roots using all the roots computed by other processors at the previous
iteration. The difference between both approaches lies in the way
processors communicate and exchange data. With MPI, processors need to
send and receive data explicitly. So in
Algorithm~\ref{alg2-cuda-mpi}, after the initialization phase all the
processors have the same $Z$ vector. Then they need to compute the
parameters used by the $MPI\_AlltoAll$ routines (line 4). In practice,
each processor needs to compute its offset and its local
size. Processors need to allocate memory on their GPU and need to copy
their data on the GPU (line 5). At the beginning of each iteration, a
processor starts by transferring the whole vector $Z$ from the CPU to the
GPU (line 7). Only the local part of $Z^{prev}$ is saved (line
8). After that, a processor is able to compute an updated version of
its own roots (line 9) with the EA method. The local error is computed
(line 10) and the global error is also computed using $MPI\_Reduce$ (line 11). Then
the local roots are transferred from the GPU memory to the CPU memory
(line 12) before being exchanged between all processors (line 13) in
order to give to all processors the last version of the roots (with
the $MPI\_AlltoAll$ routine). If the convergence is not satisfied, a
new iteration is executed.

\begin{algorithm}[htpb]
\label{alg2-cuda-mpi}
\LinesNumbered
\SetAlgoNoLine
\caption{Finding roots of polynomials with the Ehrlich-Aberth method on multiple GPUs using MPI}

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

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

\BlankLine
Initialize the polynomial $P$ and its derivative $P'$\;
Set the initial values of vector $Z$\;
Determine the local part of the MPI process\;
Computation of the parameters for the $MPI\_AlltoAll$\;
Copy $P$, $P'$ from CPU to GPU\;
\While {$error > \epsilon$}{
  Copy $Z$ from CPU to GPU\;
  $Z^{prev}_{loc}$ = KernelSave($Z_{loc}$)\;
  $Z_{loc}$ = KernelUpdate($P,P',Z$)\;
  $error_{loc}$ = KernelComputeError($Z_{loc},Z^{prev}_{loc}$)\;
  $error=MPI\_Reduce(error_{loc})$\;
  Copy $Z_{loc}$ from GPU to CPU\;
  $Z=MPI\_AlltoAll(Z_{loc})$\;
}
\end{algorithm}


\section{Experiments}
\label{sec5}
We study two categories of polynomials: sparse polynomials and full polynomials.\\
{\it A sparse polynomial} is a polynomial for which only some coefficients are not null. In this paper, we consider sparse polynomials for which the roots are distributed on 2 distinct circles:
\begin{equation}
        \forall \alpha_{1} \alpha_{2} \in \mathbb{C},\forall n_{1},n_{2} \in \mathbb{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}
     {\Large \forall \alpha_{i} \in \mathbb{C}, i\in \mathbb{N};  p(x)=\sum^{n}_{i=0} \alpha_{i}.x^{i}} 
\end{equation}

\KG{For our tests, a CPU Intel(R) Xeon(R) CPU
X5650@2.40GHz and 4 GPUs cards Tesla Kepler K40,are used with CUDA version 7.5}.
 
 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
degrees.  All experimental results obtained are performed with double
precision floating-point 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 the Guggenheimer method~\cite{Gugg86}.

\subsection{Evaluation of the multi-GPUs approaches}
In this part, we evaluate the performances of the CUDA-OpenMP and
CUDA-MPI approaches of the EA algorithm on different GPU platforms
composed each of 1, 2, 3 or 4 GPUs. In this experiments we report the
experimental results of the EA algorithms to find the roots of different sparse and full polynomials of high degrees ranging from 100,000 to 1,400,000. Figures~\ref{fig:01} and~\ref{fig:02} show the execution times to solve, respectively, sparse and full polynomials with the CUDA-OpenMP algorithm, and Figures~\ref{fig:03} and~\ref{fig:04} show those to solve, respectively, sparse and full polynomials with the CUDA-MPI algorithm.

All these figures show that the CUDA-OpenMP and the CUDA-MPI approaches of the EA algorithm, compared to the single GPU version, are efficient and scale well with multiple GPUs. Both approaches allow us to solve sparse and full polynomials of very high degrees. Using 4 GPUs allows us to achieve a quasi-linear speedup.

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

\begin{figure}[htbp]
\centering
\includegraphics[angle=-90,width=0.5\textwidth]{Full_omp}
\caption{Execution times in seconds of the Ehrlich-Aberth method to solve full polynomials on multiple GPUs with CUDA-OpenMP.}
\label{fig:02}
\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 to solve sparse polynomials on multiple GPUs with CUDA-MPI.}
\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  multiple GPUs with CUDA-MPI.}
 \label{fig:04}
 \end{figure}


\subsection{Comparison between the CUDA-OpenMP and the CUDA-MPI approaches}
In the previous section we saw that both approaches are very efficient to reduce the execution times to solve sparse and full polynomials. In this section we try to compare these two approaches. In this experiment three sparse polynomials and three full polynomials of degrees 200,000, 800,000 and 1,400,000 are investigated. Figures~\ref{fig:05} and~\ref{fig:06} show the comparison between CUDA-OpenMP and CUDA-MPI algorithms of the EA method to solve sparse and full polynomials, respectively. 

\begin{figure}[htbp]
\centering
 \includegraphics[angle=-90,width=0.5\textwidth]{Sparse}
\caption{Execution times  to solve sparse polynomials of three distinct degrees on multiple GPUs using OpenMP and MPI with the Ehrlich-Aberth method}
\label{fig:05}
\end{figure}

\begin{figure}[htbp]
\centering
 \includegraphics[angle=-90,width=0.5\textwidth]{Full}
\caption{Execution times  to solve full polynomials of three distinct degrees on multiple GPUs using OpenMP and MPI with the Ehrlich-Aberth method}
\label{fig:06}
\end{figure}

In Figure~\ref{fig:05} there is one curve for CUDA-OpenMP and another one for CUDA-MPI for each polynomial investigated. We can see that the results are quite similar between OpenMP and MPI for the polynomial degree of 200K. For the degree of 800K, the MPI version is a little bit slower than the OpenMP version but for the degree of 1,4 million, there is a slight advantage for the MPI version. In Figure~\ref{fig:06}, we can see that when it comes to full polynomials, both approaches are almost equivalent.


\subsection{Solving sparse and full polynomials of the same degree on multiple GPUs}
In this experiment we compare the execution times of the EA algorithm
according to the number of GPUs to solve sparse and full polynomials
on multiple GPUs using OpenMP or MPI approaches. We chose three sparse
and three full polynomials of degrees 200,000, 800,000 and
1,400,000. Figures~\ref{fig:07} and~\ref{fig:08} show the execution
times to solve sparse and full polynomials of the same degrees with
the CUDA-OpenMP version and the CUDA-MPI version, respectively.

\begin{figure}[htbp]
\centering
 \includegraphics[angle=-90,width=0.5\textwidth]{OMP}
\caption{Execution times to solve sparse and full polynomials of three distinct degrees on multiple GPUs using OpenMP.}
\label{fig:07}
\end{figure}

\begin{figure}[htbp]
\centering
 \includegraphics[angle=-90,width=0.5\textwidth]{MPI}
\caption{Execution times to solve sparse and full polynomials of three distinct degrees on multiple GPUs using MPI.}
\label{fig:08}
\end{figure}

In Figure~\ref{fig:07} the execution times of the CUDA-OpenMP version to solve sparse polynomials are very low compared to those to solve full polynomials. With sparse polynomials the number of monomials is reduced, consequently the number of operations is reduced and the execution time decreases. Figure~\ref{fig:08} shows the impact of sparsity on the efficiency of the CUDA-MPI approach. We can see that the impact follows the same pattern, a difference in execution times 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 of high degrees ranging from 1,000,000 to 5,000,000. In Figure~\ref{fig:09} we can see that both approaches (CUDA-OpenMP and CUDA-MPI) 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.

\begin{figure}[htbp]
\centering
 \includegraphics[angle=-90,width=0.5\textwidth]{big}
 \caption{Execution times in seconds of the Ehrlich-Aberth method to solve sparse and full polynomials of high degree on 4 GPUs for degrees ranging from 1M to 5M}
\label{fig:09}
\end{figure}
 

\section{Conclusion}
\label{sec6}
In this paper, we have presented parallel implementations of the Ehrlich-Aberth algorithm to solve full and sparse polynomials, on a 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 or 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 a single GPU. 

Our next objective is to extend the model presented here to clusters of GPU nodes, with a three-level scheme: inter-node communications via MPI processes (distributed memory), management of multi-GPU nodes by OpenMP threads (shared memory). Actual platforms may probably also contain purely multi-core nodes without any GPU. This heterogeneous setting may lead to the integration of load balancing algorithms so as to allow an optimal use of hardware resources. 


\section*{Acknowledgment}
This  paper  is   partially  funded  by  the  Labex   ACTION  program  (contract
ANR-11-LABX-01-01). 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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