-\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}
-%\begin{figure}[htbp]
-%\centering
- % \includegraphics[angle=-90,width=0.5\textwidth]{EA-Algorithm}
-%\caption{The Ehrlich-Aberth algorithm on single GPU}
-%\label{fig:03}
-%\end{figure}
-
-the Ehrlich-Aberth method is an iterative method, contain 4 steps, start from the initial approximations of all the
-roots of the polynomial,the second step initialize the solution vector $Z$ using the Guggenheimer method to assure the distinction of the initial vector roots, than in step 3 we apply the the iterative function based on the Newton's method and Weiestrass operator[...,...], wich will make it possible to converge to the roots solution, provided that all the root are different. At the end of each application of the iterative function, a stop condition is verified consists in stopping the iterative process when the whole of the modules of the roots
-are lower than a fixed value $ε$
-\subsection{EA parallel implementation on CUDA}
-Like any parallel code, a GPU parallel implementation first
-requires to determine the sequential tasks and the
-parallelizable parts of the sequential version of the
-program/algorithm. In our case, all the operations that are easy
-to execute in parallel must be made by the GPU to accelerate
-the execution of the application, like the step 3 and step 4. On the other hand, all the
-sequential operations and the operations that have data
-dependencies between threads or recursive computations must
-be executed by only one CUDA or CPU thread (step 1 and step 2). Initially we specifies the organization of threads in parallel, need to specify the dimension of the grid Dimgrid: the number of block per grid and block by DimBlock: the number of threads per block required to process a certain task.
-
-we create the kernel, for step 3 we have two kernels, the
-first named \textit{save} is used to save vector $Z^{K-1}$ and the kernel
-\textit{update} is used to update the $Z^{K}$ vector. In step 4 a kernel is
-created to test the convergence of the method. In order to
-compute function H, we have two possibilities: either to use
-the Jacobi method, or the Gauss-Seidel method which uses the
-most recent computed roots. It is well known that the Gauss-
-Seidel mode converges more quickly. So, we used the Gauss-Seidel mode of iteration. To
-parallelize the code, we created kernels and many functions to
-be executed on the GPU for all the operations dealing with the
-computation on complex numbers and the evaluation of the
-polynomials. As said previously, we managed both functions
-of evaluation of a polynomial: the normal method, based on
-the method of Horner and the method based on the logarithm
-of the polynomial. All these methods were rather long to
-implement, as the development of corresponding kernels with
-CUDA is longer than on a CPU host. This comes in particular
-from the fact that it is very difficult to debug CUDA running
-threads like threads on a CPU host. In the following paragraph
-Algorithm 1 shows the GPU parallel implementation of Ehrlich-Aberth method.
-
-Algorithm~\ref{alg2-cuda} shows a sketch of the Ehrlich-Aberth method using CUDA.
-
-\begin{enumerate}
-\begin{algorithm}[htpb]
-\label{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 $kernel\_testConverge(\Delta z_{max},Z,ZPrec)$\;
+\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.
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\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}
