{M}itwirkung einer von der k{\"o}niglich
preu\ss{}ischen Akademie der {W}issenschaften
eingesetzten {C}ommission)",
- year = "1872",
+ year = "1872","in French"
volume = "2",
series = "Mathematische Werke",
}
@InCollection{newt70,
author = "Isaac Newton",
- year = "1670--71?",
+ year = "1670--1671",
title = "Tractatus de Methodis Serierum et Fluxionum",
booktitle = "The Mathematical Papers of Isaac Newton, III",
editor = "D. T. Whiteside",
\subsection{The Ehrlich-Aberth parallel implementation on CUDA}
-The algorithm ~\ref{alg1-cuda} shows sketch of the Ehrlich-Aberth method using 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.
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.
+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
\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
{\Large \forall \alpha_{i} \in \mathbb{C}, i\in \mathbb{N}; p(x)=\sum^{n}_{i=0} \alpha_{i}.x^{i}}
\end{equation}
-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
+\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
