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\begin{document}
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We introduced three paradigms of parallel programming. Our objective consist to implement an algorithm of root finding polynomial on multiple GPUs. It primordial to know how manage CUDA context of different GPUs. A direct method for controlling the various GPU is to use as many threads or processes that GPU. We can choose the GPU index based on the identifier of OpenMP thread or the rank of the MPI process. Both approaches will be created.
\section{The EA algorithm on single GPU}
+\subsection{the EA method}
+the Ehrlich-Aberth method is an iterative method , contain 4 steps, start from the initial approximations of all the
+roots of the polynomial,the second step initialize the solution vector $Z$ using the Guggenheimer method to assure the distinction of the initial vector roots, than in step 3 we apply the the iterative function based on the Newton's method and Weiestrass operator[...,...], wich will make it possible to converge to the roots solution, provided that all the root are different. At the end of each application of the iterative function, a stop condition is verified consists in stopping the iterative process when the whole of the modules of the roots
+are lower than a fixed value $ε$
+\subsection{EA parallel implementation on CUDA}
+Like any parallel code, a GPU parallel implementation first
+requires to determine the sequential tasks and the
+parallelizable parts of the sequential version of the
+program/algorithm. In our case, all the operations that are easy
+to execute in parallel must be made by the GPU to accelerate
+the execution of the application, like the step 3 and step 4. On the other hand, all the
+sequential operations and the operations that have data
+dependencies between threads or recursive computations must
+be executed by only one CUDA or CPU thread (step 1 and step 2). Initially we specifies the organization of threads in parallel, need to specify the dimension of the grid Dimgrid: the number of block per grid and block by DimBlock: the number of threads per block required to process a certain task.
+
+we create the kernel, for step 3 we have two kernels, the
+first named \textit{save} is used to save vector $Z^{K-1}$ and the kernel
+\textit{update} is used to update the $Z^{K}$ vector. In step 4 a kernel is
+created to test the convergence of the method. In order to
+compute function H, we have two possibilities: either to use
+the Jacobi method, or the Gauss-Seidel method which uses the
+most recent computed roots. It is well known that the Gauss-
+Seidel mode converges more quickly. So, we used the Gauss-Seidel mode of iteration. To
+parallelize the code, we created kernels and many functions to
+be executed on the GPU for all the operations dealing with the
+computation on complex numbers and the evaluation of the
+polynomials. As said previously, we managed both functions
+of evaluation of a polynomial: the normal method, based on
+the method of Horner and the method based on the logarithm
+of the polynomial. All these methods were rather long to
+implement, as the development of corresponding kernels with
+CUDA is longer than on a CPU host. This comes in particular
+from the fact that it is very difficult to debug CUDA running
+threads like threads on a CPU host. In the following paragraph
+Algorithm 1 shows the GPU parallel implementation of Ehrlich-Aberth method.
+
+Algorithm~\ref{alg2-cuda} shows a sketch of the Ehrlich-Aberth method using CUDA.
+
+\begin{enumerate}
+\begin{algorithm}[htpb]
+\label{alg2-cuda}
+%\LinesNumbered
+\caption{CUDA Algorithm to find roots with the Ehrlich-Aberth method}
+
+\KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (Error tolerance
+ threshold), P (Polynomial to solve), Pu (Derivative of P), $n$ (Polynomial degrees), $\Delta z_{max}$ (Maximum value of stop condition)}
+
+\KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}
+
+\BlankLine
+
+\item Initialization of the of P\;
+\item Initialization of the of Pu\;
+\item Initialization of the solution vector $Z^{0}$\;
+\item Allocate and copy initial data to the GPU global memory\;
+\item k=0\;
+\While {$\Delta z_{max} > \epsilon$}{
+\item Let $\Delta z_{max}=0$\;
+\item $ kernel\_save(ZPrec,Z)$\;
+\item k=k+1\;
+\item $ kernel\_update(Z,P,Pu)$\;
+\item $kernel\_testConverge(\Delta z_{max},Z,ZPrec)$\;
+
+}
+\item Copy results from GPU memory to CPU memory\;
+\end{algorithm}
+\end{enumerate}
+~\\
+
\section{The EA algorithm on Multi-GPU}
-
\subsection{MGPU (OpenMP-CUDA)approach}
\subsection{MGPU (MPI-CUDA)approach}
+
\section{experiments}
+
\begin{figure}[htbp]
\centering
- \includegraphics[angle=-90,width=0.8\textwidth]{GPU_openmp}
-\caption{Execution times in seconds of the Ehrlich-Aberth method on GPUs using shared memory paradigm with OpenMP}
+ \includegraphics[angle=-90,width=0.5\textwidth]{Sparse_openmp}
+\caption{Execution times in seconds of the Ehrlich-Aberth method for solving sparse polynomials on GPUs using shared memory paradigm with OpenMP}
\label{fig:01}
\end{figure}
\begin{figure}[htbp]
\centering
- \includegraphics[angle=-90,width=0.8\textwidth]{GPU_mpi}
-\caption{Execution times in seconds of the Ehrlich-Aberth method on GPUs using distributed memory paradigm with MPI}
+ \includegraphics[angle=-90,width=0.5\textwidth]{Sparse_mpi}
+\caption{Execution times in seconds of the Ehrlich-Aberth method for solving sparse polynomials on GPUs using distributed memory paradigm with MPI}
\label{fig:02}
\end{figure}
+
+\begin{figure}[htbp]
+\centering
+ \includegraphics[angle=-90,width=0.5\textwidth]{Full_openmp}
+\caption{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials on GPUs using shared memory paradigm with OpenMP}
+\label{fig:03}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+ \includegraphics[angle=-90,width=0.5\textwidth]{Full_mpi}
+\caption{Execution times in seconds of the Ehrlich-Aberth method for full polynomials on GPUs using distributed memory paradigm with MPI}
+\label{fig:04}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+ \includegraphics[angle=-90,width=0.5\textwidth]{Sparse_mpivsomp}
+\caption{Comparaison between MPI and OpenMP versions of the Ehrlich-Aberth method for solving sparse plynomials on GPUs}
+\label{fig:05}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+ \includegraphics[angle=-90,width=0.5\textwidth]{Full_mpivsomp}
+\caption{Comparaison between MPI and OpenMP versions of the Ehrlich-Aberth method for solving full polynomials on GPUs}
+\label{fig:06}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+ \includegraphics[angle=-90,width=0.5\textwidth]{MPI_mpivsomp}
+\caption{Comparaison of execution times of the Ehrlich-Aberth method for solving sparse and full polynomials on GPUs with distributed memory paradigm using MPI}
+\label{fig:07}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+ \includegraphics[angle=-90,width=0.5\textwidth]{OMP_mpivsomp}
+\caption{Comparaison of execution times of the Ehrlich-Aberth method for solving sparse and full polynomials on GPUs with shared memory paradigm using OpenMP}
+\label{fig:08}
+\end{figure}
+
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