%roots of sparse polynomials of degree 10,000. The authors showed an interesting
%speedup that is 20 times as fast as the sequential implementation.
-With the recent advent of the Compute Unified Device Architecture
-(CUDA)~\cite{CUDA15}, a parallel computing platform and a programming
-model invented by NVIDIA had revived parallel programming interest for
+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 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.
-
-In this paper we propose the parallelization of the Ehrlich-Aberth (EA) method which has a cubic convergence rate which is much better than the quadratic rate of the Durand-Kerner method which 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:
+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.
\end{algorithm}
-The development of this code is a rather long task, 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
-section the GPU parallel implementation of the Ehrlich-Aberth method with
-OpenMP and MPI is presented.
+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.
\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 resource's.
+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}
