\usepackage[textsize=footnotesize]{todonotes}
\newcommand{\LZK}[2][inline]{%
\todo[color=red!10,#1]{\sffamily\textbf{LZK:} #2}\xspace}
+\newcommand{\RC}[2][inline]{%
+ \todo[color=blue!10,#1]{\sffamily\textbf{RC:} #2}\xspace}
+\newcommand{\KG}[2][inline]{%
+ \todo[color=green!10,#1]{\sffamily\textbf{KG:} #2}\xspace}
+\newcommand{\AS}[2][inline]{%
+ \todo[color=orange!10,#1]{\sffamily\textbf{AS:} #2}\xspace}
% As a general rule, do not put math, special symbols or citations
% in the abstract
\begin{abstract}
-\LZK{J'ai un peu modifié l'abstract. Sinon à revoir pour le degré max des polynômes après les tests de raph.}
-Finding roots of polynomials is a very important part of solving real-life problems but it is not so easy for polynomials of high degrees. 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 using different parallel paradigms: OpenMP or MPI. The experiments show a quasi-linear speedup by using up-to 4 GPU devices to find roots of polynomials of degree up-to 1.4 billion. To our knowledge, this is the first paper to present this technology mix to solve such a highly demanding problem in parallel programming.
-\LZK{Je n'ai pas bien saisi la dernière phrase.}
+Finding roots of polynomials is a very important part of solving
+real-life problems but it is not so easy for polynomials of high
+degrees. 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
+using 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 roots of polynomials of degree up-to 1.4
+million. Moreover, other experiments show it is possible to find roots
+of polynomials of degree up to 5 millions.
\end{abstract}
% no keywords
Finding roots of polynomials of very high degrees arises in many complex problems in 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}{a_ix^i},a_0\neq 0.
+p(x) = \displaystyle\sum^n_{i=0}{a_ix^i},a_n\neq 0.
\end{equation}
where $\{a_i\}_{0\leq i\leq n}$ are complex coefficients and $n$ is a high integer number. If $a_n\neq0$ 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)=a_n\displaystyle\prod_{i=1}^n(x-z_i), a_n\neq 0.
\end{equation}
-\LZK{Pourquoi $a_0a_n\neq 0$ ?: $a_0$ pour la premiere equation et $a_n$ pour la deuxieme equation }
+%\LZK{Pourquoi $a_0a_n\neq 0$ ?: $a_0$ pour la premiere equation et $a_n$ pour la deuxieme equation }
%The problem of finding the roots of polynomials can be encountered in numerous applications. \LZK{A mon avis on peut supprimer cette phrase}
Most of the numerical methods that deal with the polynomial root-finding problem are simultaneous methods, \textit{i.e.} the iterative methods to find simultaneous approximations of the $n$ polynomial roots. These methods start from the initial approximations 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 Durand-Kerner method~\cite{Durand60,Kerner66} and Ehrlich-Aberth method~\cite{Ehrlich67,Aberth73}.
-\LZK{Pouvez-vous donner des références pour les deux méthodes?, c'est fait}
+%\LZK{Pouvez-vous donner des références pour les deux méthodes?, c'est fait}
%The first method of this group is Durand-Kerner method:
%\begin{equation}
%Aberth, Ehrlich and Farmer-Loizou~\cite{Loizou83} have proved that
%the Ehrlich-Aberth method (EA) has a cubic order of convergence for simple roots whereas the Durand-Kerner has a quadratic order of %convergence.
-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 et al.~\cite{Raphaelall01} 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 in 116 seconds with OpenMP and 135 seconds with MPI only by using 8 personal computers and 2 communications per iteration. The authors showed an interesting speedup comparing to the sequential implementation which takes up-to 3,300 seconds to obtain same results.
+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 et al.~\cite{Raphaelall01} 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 in 116 seconds with
+OpenMP and 135 seconds with MPI only by using 8 personal computers and
+2 communications per iteration. \RC{si on donne des temps faut donner
+ le proc, comme c'est vieux à mon avis faut supprimer ca, votre avis?} The authors showed an interesting
+speedup comparing to the sequential implementation which takes up-to
+3,300 seconds to obtain same results.
\LZK{``only by using 8 personal computers and 2 communications per iteration''. Pour MPI? et Pour OpenMP: Rep: c'est MPI seulement}
Very few work had been performed since then until the appearing of the Compute Unified Device Architecture (CUDA)~\cite{CUDA15}, a parallel computing platform and a programming model invented by NVIDIA. The computing power of GPUs (Graphics Processing Units) has exceeded that of traditional processors CPUs. 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 Ehrlich-Aberth method using two parallel programming paradigms OpenMP and MPI on CUDA multi-GPU platforms. Our CUDA/MPI and CUDA/OpenMP codes are the first implementations of Ehrlich-Aberth method with multiple GPUs for finding roots of polynomials. Our major contributions include:
\LZK{Pourquoi la méthode Ehrlich-Aberth et pas autres? the Ehrlich-Aberth have very good convergence and it is suitable to be implemented in parallel computers.}
\begin{itemize}
- \item An improvements for the Ehrlich-Aberth method using the exponential logarithm in order to be able to solve sparse and full polynomial of degree up to 1, 000, 000.
- \item A parallel implementation of Ehrlich-Aberth method on single GPU with CUDA.
+ \item An improvements for the Ehrlich-Aberth method using the exponential logarithm in order to be able to solve sparse and full polynomial of degree up to 1, 000, 000.\RC{j'ai envie de virer ca, car c'est pas la nouveauté dans ce papier}
+ \item A parallel implementation of Ehrlich-Aberth method on single GPU with CUDA.\RC{idem}
\item The parallel implementation of Ehrlich-Aberth 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 Ehrlich-Aberth 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 latter 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.
\end{itemize}
%This paper is organized as follows. In Section~\ref{sec2} we recall the Ehrlich-Aberth method. In section~\ref{sec3} we present EA algorithm on single GPU. In section~\ref{sec4} we propose the EA algorithm implementation on Multi-GPU for (OpenMP-CUDA) approach and (MPI-CUDA) approach. In sectioné\ref{sec5} we present our experiments and discus it. Finally, Section~\ref{sec6} concludes this paper and gives some hints for future research directions in this topic.}
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 Multi-GPU using the OpenMP and MPI approaches. In section\ref{sec5} we present our experiments and discus it. Finally, Section~\ref{sec6} concludes this paper and gives some hints for future research directions in this topic.
-\LZK{A revoir toute cette organization: je viens de la revoir}
+%\LZK{A revoir toute cette organization: je viens de la revoir}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Here we give a second form of the iterative function used by the Ehrlich-Aberth method:
%\begin{equation}
-%\label{Eq:EA1}
+%\label{Eq:EA-1}
%EA: 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,. . . .,n
%\end{equation}
%The convergence condition determines the termination of the algorithm. It consists in stopping the %iterative function when the roots are sufficiently stable. We consider that the method converges %sufficiently when:
%\begin{equation}
-%\label{eq:Aberth-Conv-Cond}
+%\label{eq:AAberth-Conv-Cond}
%\forall i \in [1,n];\vert\frac{z_{i}^{k}-z_{i}^{k-1}}{z_{i}^{k}}\vert<\xi
%\end{equation}
%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~\cite{,}, witch will make it possible to converge to the roots solution, provided that all the root are different.
-The Ehrlich-Aberth method is a simultaneous method~\cite{} using the following iteration
+The Ehrlich-Aberth method is a simultaneous method~\cite{Aberth73} using the following iteration
\begin{equation}
\label{Eq:EA1}
EA: 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,. . . .,n
\end{equation}
-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~\cite{Gugg86} method to assure the distinction of the initial vector roots,
+This methods contains 4 steps. The first step consists of the initial
+approximations of all the roots of the polynomial. The second step
+initializes the solution vector $Z$ using the Guggenheimer
+method~\cite{Gugg86} to ensure the distinction of the initial vector
+roots. In step 3, the iterative function based on the Newton's
+method~\cite{newt70} and Weiestrass operator~\cite{Weierstrass03} is
+applied. With this step the computation of roots will converge,
+provided that all roots are different.
- than in step 3 we apply the the iterative function based on the Newton's method~\cite{newt70} and Weiestrass operator~\cite{Weierstrass03}, 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 $\xi$.
+In order to stop the iterative function, a stop condition is
+applied. This condition checks that all the root modules are lower
+than a fixed value $\xi$.
\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<\xi
\end{equation}
\subsection{Improving Ehrlich-Aberth method}
-With high degree polynomial, the Ehrlich-Aberth method suffer from overflow because the limited number in the mantissa of floating points representations, which makes the computation of $p(z)$ wrong when z is large.
+With high degree polynomials, the Ehrlich-Aberth 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.
+
+%Experimentally, it is very difficult to solve polynomials with the Ehrlich-Aberth method and have roots which except the circle of unit, represented by the radius $r$ evaluated as:
+
+%\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.
-Experimentally, it is very difficult to solve polynomials with Ehrlich-Aberth method and have roots which except the circle of unit, represented by the radius $r$ evaluated as:
+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 Ehrlich-Aberth 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. This new
+version of the Ehrlich-Aberth method with exponential and logarithm is
+defined as follows:
\begin{equation}
-\label{R.EL}
-R = exp(log(DBL\_MAX)/(2*n) );
+\label{Log_H2}
+z^{k+1}_{i}=z_{i}^{k}-\exp \left(\ln \left(
+p(z_{i}^{k})\right)-\ln\left(p'(z^{k}_{i})\right)- \ln\left(1-Q(z^{k}_{i})\right)\right),
\end{equation}
+where:
-%\begin{equation}
+\begin{eqnarray}
+\label{Log_H1}
+Q(z^{k}_{i})=\exp\left( \ln (p(z^{k}_{i}))-\ln(p'(z^{k}_{i}))+\ln \left(
+\sum_{i\neq j}^{n}\frac{1}{z^{k}_{i}-z^{k}_{j}}\right)\right) \nonumber \\
+i=1,...,n
+\end{eqnarray}
-%R = \exp( \log(DBL\_MAX) / (2*n) )
-%\end{equation}
- where \verb=DBL_MAX= stands for the maximum representable \verb=double= value.
-This problem was discussed earlier in~\cite{Karimall98} for the Durand-Kerner method. The authors
-propose to use the logarithm and the exponential of a complex in order to compute the power at a high exponent. Using the logarithm and the exponential operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations manipulate lower absolute values and the roots for large polynomial degrees can be looked for successfully~\cite{Karimall98}.
+%We propose to use the logarithm and the exponential of a complex in order to compute the power at a high exponent.
+Using the logarithm and the exponential operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations manipulate lower absolute values~\cite{Karimall98}.
+
+%This problem was discussed earlier in~\cite{Karimall98} for the Durand-Kerner method. The authors
+%propose to use the logarithm and the exponential of a complex in order to compute the power at a high exponent. Using the logarithm and the exponential operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations manipulate lower absolute values and the roots for large polynomial degrees can be looked for successfully~\cite{Karimall98}.
\subsection{Ehrlich-Aberth parallel implementation on CUDA}
-We introduced three paradigms of parallel programming. Our objective consists in implementing a root finding polynomial algorithm on multiple GPUs. To this end, it is primordial to know how to manage CUDA contexts of different GPUs. A direct method for controlling the various GPUs is to use as many threads or processes as GPU devices. We can choose the GPU index based on the identifier of OpenMP thread or the rank of the MPI process. Both approaches will be investigated.
-
-
-
-
-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 specify the organization of parallel threads, by specifying the dimension of the grid Dimgrid, the number of blocks per grid DimBlock and the number of threads per block.
-
-The code is organized by what is named kernels, portions code that are run on GPU devices. For step 3, there are two kernels, the
-first named \textit{save} is used to save vector $Z^{K-1}$ and the seconde one is named
-\textit{update} and is used to update the $Z^{K}$ vector. For step 4, a kernel
-tests the convergence of the method. In order to
-compute the function H, we have two possibilities: either to use
-the Jacobi mode, or the Gauss-Seidel mode of iterating 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
+%We introduced three paradigms of parallel programming.
+
+Our objective consists in implementing a root finding polynomial
+algorithm on multiple GPUs. To this end, it is primordial to know how
+to manage CUDA contexts of different GPUs. A direct method for
+controlling the various GPUs is to use as many threads or processes as
+GPU devices. We can choose the GPU index based on the identifier of
+OpenMP thread or the rank of the MPI process. Both approaches will be
+investigated.
+
+
+
+
+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 specify the organization of parallel
+threads, by specifying the dimension of the grid Dimgrid, the number
+of blocks per grid DimBlock and the number of threads per block.
+
+The code is organized kernels which are part of code that are run on
+GPU devices. For step 3, there are two kernels, the first named
+\textit{save} is used to save vector $Z^{K-1}$ and the second one is
+named \textit{update} and is used to update the $Z^{K}$ vector. For
+step 4, a kernel tests the convergence of the method. In order to
+compute the function H, we have two possibilities: either to use the
+Jacobi mode, or the Gauss-Seidel mode of iterating which uses the most
+recent computed roots. It is well known that the Gauss-Seidel mode
+converges more quickly. So, we use Gauss-Seidel iterations. To
+parallelize the code, we create 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~\ref{alg1-cuda} shows the GPU parallel implementation of Ehrlich-Aberth method.
+polynomials. As said previously, we manage both functions of
+evaluation: 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~\ref{alg1-cuda} shows the GPU parallel
+implementation of Ehrlich-Aberth method.
\begin{enumerate}
\begin{algorithm}[htpb]
%\BlankLine
-\item Initialization of the of P\;
-\item Initialization of the of Pu\;
+\item Initialization of P\;
+\item Initialization of Pu\;
\item Initialization of the solution vector $Z^{0}$\;
\item Allocate and copy initial data to the GPU global memory\;
\item k=0\;
\end{enumerate}
~\\
-
+\RC{Au final, on laisse ce code, on l'explique, si c'est kahina qui
+ rajoute l'explication, il faut absolument ajouter \KG{dfsdfsd}, car
+ l'anglais sera à relire et je ne veux pas tout relire... }
\section{The EA algorithm on Multiple GPUs}
\label{sec4}
\subsection{M-GPU : an OpenMP-CUDA approach}
-Our OpenMP-CUDA implementation of EA algorithm is based on the hybrid OpenMP and CUDA programming model. It works as follows.
-Based on the metadata, a shared memory is used to make data evenly shared among OpenMP threads. The shared data are the solution vector $Z$, the polynomial to solve $P$, and the error vector $\Delta z$. Let (T\_omp) the number of OpenMP threads be equal to the number of GPUs, each OpenMP thread binds to one GPU, and controls a part of the shared memory, that is a part of the vector Z , that is $(n/num\_gpu)$ roots where $n$ is the polynomial's degree and $num\_gpu$ the total number of available GPUs. Each OpenMP thread copies its data from host memory to GPU’s device memory. Then every GPU will have a grid of computation organized according to the device performance and the size of data on which it runs the computation kernels. %In principle a grid is set by two parameter DimGrid, the number of block per grid, DimBloc: the number of threads per block. The following schema shows the architecture of (CUDA,OpenMP).
+Our OpenMP-CUDA implementation of EA algorithm is based on the hybrid
+OpenMP and CUDA programming model. All the data
+are shared with OpenMP amoung all the OpenMP threads. The shared data
+are the solution vector $Z$, the polynomial to solve $P$, and the
+error vector $\Delta z$. 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 owns of
+the vector Z, that is $(n/num\_gpu)$ roots where $n$ is the
+polynomial's degree and $num\_gpu$ the total number of available
+GPUs. 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. Each thread
+starts by copying all the previous roots inside its GPU. Then each GPU
+will compute an iteration of the EA method on its own roots. For that
+all the other roots are used. At the end of an iteration, the updated
+roots are copied from the GPU to the CPU. The convergence is checked
+on the new roots. Finally each CPU will update its own roots in the
+shared memory arrays containing all the roots.
+
+%In principle a grid is set by two parameter DimGrid, the number of block per grid, DimBloc: the number of threads per block. The following schema shows the architecture of (CUDA,OpenMP).
%\begin{figure}[htbp]
%\centering
%\end{figure}
%Each thread OpenMP compute the kernels on GPUs,than after each iteration they copy out the data from GPU memory to CPU shared memory. The kernels are re-runs is up to the roots converge sufficiently. Here are below the corresponding algorithm:
-$num\_gpus$ OpenMP threads are created using \verb=omp_set_num_threads();=function (step $3$, Algorithm \ref{alg2-cuda-openmp}), the shared memory is created using \verb=#pragma omp parallel shared()= OpenMP function (line $5$, Algorithm\ref{alg2-cuda-openmp}), then each OpenMP thread allocates memory and copies initial data from CPU memory to GPU global memory, executes the kernels on GPU, but computes only his portion of roots indicated with variable \textit{index} initialized in (line 5, Algorithm \ref{alg2-cuda-openmp}), used as input data in the $kernel\_update$ (line 10, Algorithm \ref{alg2-cuda-openmp}). After each iteration, all OpenMP threads synchronize using \verb=#pragma omp barrier;= to gather all the correct values of $\Delta z$, thus allowing the computation the maximum stop condition on vector $\Delta z$ (line 12, Algorithm \ref{alg2-cuda-openmp}). Finally, threads copy the results from GPU memories to CPU memory. The OpenMP threads execute kernels until the roots sufficiently converge.
-\begin{enumerate}
-\begin{algorithm}[htpb]
-\label{alg2-cuda-openmp}
-%\LinesNumbered
-\caption{CUDA-OpenMP 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 degree), $\Delta z$ ( Vector of errors for stop condition), $num_gpus$ (number of OpenMP threads/ Number of GPUs), $Size$ (number of roots)}
-
-\KwOut {$Z$ ( Root's vector), $ZPrec$ (Previous root's vector)}
-
-\BlankLine
-
-\item Initialization of P\;
-\item Initialization of Pu\;
-\item Initialization of the solution vector $Z^{0}$\;
-\verb=omp_set_num_threads(num_gpus);=
-\verb=#pragma omp parallel shared(Z,$\Delta$ z,P);=
-\verb=cudaGetDevice(gpu_id);=
-\item Allocate and copy initial data from CPU memory to the GPU global memories\;
-\item index= $Size/num\_gpus$\;
-\item k=0\;
-\While {$error > \epsilon$}{
-\item Let $\Delta z=0$\;
-\item $ kernel\_save(ZPrec,Z)$\;
-\item k=k+1\;
-\item $ kernel\_update(Z,P,Pu,index)$\;
-\item $kernel\_testConverge(\Delta z[gpu\_id],Z,ZPrec)$\;
-%\verb=#pragma omp barrier;=
-\item error= Max($\Delta z$)\;
-}
-
-\item Copy results from GPU memories to CPU memory\;
-\end{algorithm}
-\end{enumerate}
-~\\
-
+%% \RC{Surement à virer ou réécrire pour etre compris sans algo}
+%% $num\_gpus$ OpenMP threads are created using
+%% \verb=omp_set_num_threads();=function (step $3$, Algorithm
+%% \ref{alg2-cuda-openmp}), the shared memory is created using
+%% \verb=#pragma omp parallel shared()= OpenMP function (line $5$,
+%% Algorithm\ref{alg2-cuda-openmp}), then each OpenMP thread allocates
+%% memory and copies initial data from CPU memory to GPU global memory,
+%% executes the kernels on GPU, but computes only his portion of roots
+%% indicated with variable \textit{index} initialized in (line 5,
+%% Algorithm \ref{alg2-cuda-openmp}), used as input data in the
+%% $kernel\_update$ (line 10, Algorithm \ref{alg2-cuda-openmp}). After
+%% each iteration, all OpenMP threads synchronize using
+%% \verb=#pragma omp barrier;= to gather all the correct values of
+%% $\Delta z$, thus allowing the computation the maximum stop condition
+%% on vector $\Delta z$ (line 12, Algorithm
+%% \ref{alg2-cuda-openmp}). Finally, threads copy the results from GPU
+%% memories to CPU memory. The OpenMP threads execute kernels until the
+%% roots sufficiently converge.
+
+
+%% \begin{enumerate}
+%% \begin{algorithm}[htpb]
+%% \label{alg2-cuda-openmp}
+%% %\LinesNumbered
+%% \caption{CUDA-OpenMP 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 degree), $\Delta z$ ( Vector of errors for stop condition), $num_gpus$ (number of OpenMP threads/ Number of GPUs), $Size$ (number of roots)}
+
+%% \KwOut {$Z$ ( Root's vector), $ZPrec$ (Previous root's vector)}
+
+%% \BlankLine
+
+%% \item Initialization of P\;
+%% \item Initialization of Pu\;
+%% \item Initialization of the solution vector $Z^{0}$\;
+%% \verb=omp_set_num_threads(num_gpus);=
+%% \verb=#pragma omp parallel shared(Z,$\Delta$ z,P);=
+%% \verb=cudaGetDevice(gpu_id);=
+%% \item Allocate and copy initial data from CPU memory to the GPU global memories\;
+%% \item index= $Size/num\_gpus$\;
+%% \item k=0\;
+%% \While {$error > \epsilon$}{
+%% \item Let $\Delta z=0$\;
+%% \item $ kernel\_save(ZPrec,Z)$\;
+%% \item k=k+1\;
+%% \item $ kernel\_update(Z,P,Pu,index)$\;
+%% \item $kernel\_testConverge(\Delta z[gpu\_id],Z,ZPrec)$\;
+%% %\verb=#pragma omp barrier;=
+%% \item error= Max($\Delta z$)\;
+%% }
+
+%% \item Copy results from GPU memories to CPU memory\;
+%% \end{algorithm}
+%% \end{enumerate}
+%% ~\\
+%% \RC{C'est encore pire ici, on ne voit pas les comm CPU <-> GPU }
\subsection{Multi-GPU : an MPI-CUDA approach}
Since a GPU works only on data already allocated in its memory, all local input data, $Z_{k}$, $ZPrec$ and $\Delta z_{k}$, must be transferred from CPU memories to the corresponding GPU memories. Afterwards, the same EA algorithm (Algorithm \ref{alg1-cuda}) is run by all processes but on different polynomial subset of roots $ p(x)_{k}=\sum_{i=1}^{n} a_{i}x^{i}, k=1,...,p$. Each MPI process executes the loop \verb=(While(...)...do)= containing the CUDA kernels but each MPI process computes only its own portion of the roots according to the rule ``''owner computes``''. The local range of roots is indicated with the \textit{index} variable initialized at (line 5, Algorithm \ref{alg2-cuda-mpi}), and passed as an input variable to $kernel\_update$ (line 10, Algorithm \ref{alg2-cuda-mpi}). After each iteration, MPI processes synchronize (\verb=MPI_Allreduce= function) by a reduction on $\Delta z_{k}$ in order to compute the maximum error related to the stop condition. Finally, processes copy the values of new computed roots from GPU memories to CPU memories, then communicate their results to other processes with \verb=MPI_Alltoall= broadcast. If the stop condition is not verified ($error > \epsilon$) then processes stay withing the loop \verb= while(...)...do= until all the roots sufficiently converge.
-\begin{enumerate}
-\begin{algorithm}[htpb]
-\label{alg2-cuda-mpi}
-%\LinesNumbered
-\caption{CUDA-MPI 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), $\Delta z$ ( error of stop condition), $num_gpus$ (number of MPI processes/ number of GPUs), Size (number of roots)}
-
-\KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}
-
-\BlankLine
-\item Initialization of P\;
-\item Initialization of Pu\;
-\item Initialization of the solution vector $Z^{0}$\;
-\item Allocate and copy initial data from CPU memories to GPU global memories\;
-\item $index= Size/num_gpus$\;
-\item k=0\;
-\While {$error > \epsilon$}{
-\item Let $\Delta z=0$\;
-\item $kernel\_save(ZPrec,Z)$\;
-\item k=k+1\;
-\item $kernel\_update(Z,P,Pu,index)$\;
-\item $kernel\_testConverge(\Delta z,Z,ZPrec)$\;
-\item ComputeMaxError($\Delta z$,error)\;
-\item Copy results from GPU memories to CPU memories\;
-\item Send $Z[id]$ to all processes\;
-\item Receive $Z[j]$ from every other process j\;
-}
-\end{algorithm}
-\end{enumerate}
-~\\
+%% \begin{enumerate}
+%% \begin{algorithm}[htpb]
+%% \label{alg2-cuda-mpi}
+%% %\LinesNumbered
+%% \caption{CUDA-MPI 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), $\Delta z$ ( error of stop condition), $num_gpus$ (number of MPI processes/ number of GPUs), Size (number of roots)}
+
+%% \KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}
+
+%% \BlankLine
+%% \item Initialization of P\;
+%% \item Initialization of Pu\;
+%% \item Initialization of the solution vector $Z^{0}$\;
+%% \item Allocate and copy initial data from CPU memories to GPU global memories\;
+%% \item $index= Size/num_gpus$\;
+%% \item k=0\;
+%% \While {$error > \epsilon$}{
+%% \item Let $\Delta z=0$\;
+%% \item $kernel\_save(ZPrec,Z)$\;
+%% \item k=k+1\;
+%% \item $kernel\_update(Z,P,Pu,index)$\;
+%% \item $kernel\_testConverge(\Delta z,Z,ZPrec)$\;
+%% \item ComputeMaxError($\Delta z$,error)\;
+%% \item Copy results from GPU memories to CPU memories\;
+%% \item Send $Z[id]$ to all processes\;
+%% \item Receive $Z[j]$ from every other process j\;
+%% }
+%% \end{algorithm}
+%% \end{enumerate}
+%% ~\\
+
+%% \RC{ENCORE ENCORE PIRE}
\section{Experiments}
\label{sec5}
\begin{equation}
{\Large \forall a_{i} \in C, i\in N; p(x)=\sum^{n}_{i=0} a_{i}.x^{i}}
\end{equation}
-For our tests, a CPU Intel(R) Xeon(R) CPU E5620@2.40GHz and a GPU K40 (with 6 Go of ram) are used.
-%SIDER : Une meilleure présentation de l'architecture est à faire ici.
-In order to evaluate both the M-GPU and Multi-GPU approaches, we performed a set of experiments on a single GPU and multiple GPUs using OpenMP or MPI by EA algorithm, for both sparse and full polynomials of different sizes.
-All experimental results obtained are made in double precision data whereas the convergence threshold of the EA method is set to $10^{-7}$.
-%Since we were more interested in the comparison of the
-%performance behaviors of Ehrlich-Aberth and Durand-Kerner methods on
-%CPUs versus on GPUs.
-The initialization values of the vector solution
-of the methods are given in %Section~\ref{sec:vec_initialization}.
+For our test, 4 cards GPU tesla Kepler K40 are used. 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
+sizes. All experimental results obtained are perfomed with double
+precision float 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 Guggenheimer method~\cite{Gugg86}.
-\subsection{Evaluating the M-GPU (CUDA-OpenMP) approach}
-We report here the results of the set of experiments with the M-GPU approach for full and sparse polynomials of different degrees, and we compare it with a Single GPU execution.
-\subsubsection{Execution time of the EA method for solving sparse polynomials on multiple GPUs using the M-GPU approach}
+\subsection{Evaluation of the CUDA-OpenMP approach}
+
+Here we report some experiments witt full and sparse polynomials of
+different degrees with multiple GPUs.
+\subsubsection{Execution times of the EA method to solve sparse polynomials on multiple GPUs}
-In this experiments we report the execution time of the EA algorithm, on single GPU and Multi-GPU with (2,3,4) GPUs, for different sparse polynomial degrees ranging from 100,000 to 1,400,000.
+In this experiments we report the execution time of the EA algorithm, on single GPU and multi-GPUs with (2,3,4) GPUs, for different sparse polynomial degrees ranging from 100,000 to 1,400,000.
\begin{figure}[htbp]
\centering
\includegraphics[angle=-90,width=0.5\textwidth]{Sparse_omp}
-\caption{Execution time in seconds of the Ehrlich-Aberth method for solving sparse polynomials on multiple GPUs using the M-GPU approach}
+\caption{Execution time in seconds of the Ehrlich-Aberth method to solve sparse polynomials on multiple GPUs.}
\label{fig:01}
\end{figure}
-This figure~\ref{fig:01} shows that the (CUDA-OpenMP) M-GPU approach reduces the execution time by a factor up to 100 w.r.t the single GPU approach and a by a factor of 1000 for polynomials exceeding degree 1,000,000. It shows the advantage to use the OpenMP parallel paradigm to gather the capabilities of several GPUs and solve polynomials of very high degrees.
+Figure~\ref{fig:01} shows that the CUDA-OpenMP approach scales well
+with multiple GPUs. This version allows us to solve sparse polynomials
+of very high degrees.
-\subsubsection{Execution time in seconds of the Ehrlich-Aberth method for solving full polynomials on multiple GPUs using the M-GPU approach}
+\subsubsection{Execution times of the EA method to solve full polynomials on multiple GPUs}
-The experiments shows the execution time of the EA algorithm, on a single GPU and on multiple GPUs using the CUDA OpenMP approach for full polynomials of degrees ranging from 100,000 to 1,400,000.
+These experiments show the execution times of the EA algorithm, on a single GPU and on multiple GPUs using the CUDA OpenMP approach for full polynomials of degrees ranging from 100,000 to 1,400,000.
\begin{figure}[htbp]
\centering
\includegraphics[angle=-90,width=0.5\textwidth]{Full_omp}
-\caption{Execution time in seconds of the Ehrlich-Aberth method for solving full polynomials on multiple GPUs using the M-GPU appraoch}
-\label{fig:03}
+\caption{Execution time in seconds of the Ehrlich-Aberth method to solve full polynomials on multiple GPUs}
+\label{fig:02}
\end{figure}
-Results with full polynomials show very important savings in execution time. For a polynomial of degree 1,4 million, the CUDA-OpenMP approach with 4 GPUs solves it 4 times as fast as single GPU, thus achieving a quasi-linear speedup.
+In Figure~\ref{fig:02}, we can observe that with full polynomials the EA version with
+CUDA-OpenMP scales also well. Using 4 GPUs allows us to achieve a
+quasi-linear speedup.
-\subsection{Evaluating the Multi-GPU (CUDA-MPI) approach}
-In this part we perform a set of experiments to compare the Multi-GPU (CUDA MPI) approach with a single GPU, for solving full and sparse polynomials of degrees ranging from 100,000 to 1,400,000.
+\subsection{Evaluation of the CUDA-MPI approach}
+In this part we perform some experiments to evaluate the CUDA-MPI
+approach to solve full and sparse polynomials of degrees ranging from
+100,000 to 1,400,000.
-\subsubsection{Execution time of the Ehrlich-Aberth method for solving sparse polynomials on multiple GPUs using the Multi-GPU approach}
+\subsubsection{Execution times of the EA method to solve sparse polynomials on multiple GPUs}
\begin{figure}[htbp]
\centering
\includegraphics[angle=-90,width=0.5\textwidth]{Sparse_mpi}
-\caption{Execution time in seconds of the Ehrlich-Aberth method for solving sparse polynomials on multiple GPUs using the Multi-GPU approach}
-\label{fig:02}
+\caption{Execution time in seconds of the Ehrlich-Aberth method to solve sparse polynomials on multiple GPUs.}
+\label{fig:03}
\end{figure}
-~\\
-Figure~\ref{fig:02} shows execution time of EA algorithm, for a single GPU, and multiple GPUs (2, 3, 4) on respectively 2, 3 and four MPI nodes. We can clearly see that the curve for a single GPU is above the other curves, which shows overtime in execution time compared to the Multi-GPU approach. We can see also that the CUDA-MPI approach reduces the execution time by a factor of 10 for polynomials of degree more than 1,000,000. For example, at degree 1000000, the execution time with a single GPU amounted to 10 thousand seconds, while with 4 GPUs, it is lowered to about just one thousand seconds which makes it for a tenfold speedup.
-%%SIDER : Je n'ai pas reformuler car je n'ai pas compris la phrase, merci de l'ecrire ici en fran\cais.
-\\cette figure montre 4 courbes de temps d'exécution pour l'algorithme EA, une courbe avec un seul GPU, 3 courbes pour multiple GPUs(2, 3, 4), on peut constaté clairement que la courbe à un seul GPU est au-dessus des autres courbes, vue sa consomation en temps d'exècution. On peut voir aussi qu'avec l'approche Multi-GPU (CUDA-MPI) reduit le temps d'exècution jusqu'à l'echelle 100 pour le polynômes qui dépasse 1,000,000 tandis que Single GPU est de l'echelle 1000.
+Figure~\ref{fig:03} shows the execution times of te EA algorithm,
+for a single GPU, and multiple GPUs (2, 3, 4) with the CUDA-MPI approach.
\subsubsection{Execution time of the Ehrlich-Aberth method for solving full polynomials on multiple GPUs using the Multi-GPU appraoch}
\label{fig:04}
\end{figure}
-
- Figure \ref{fig:04} shows execution time for a single GPU, and multiple GPUs (2, 3, 4) on respectively 2, 3 and four MPI nodes. With the CUDA-MPI approach, we notice that the three curves are distinct from each other, more we use GPUs more the execution time decreases. On the other hand the curve for a single GPU is well above the other curves.
-
-This is due to the use of MPI parallel paradigm that divides the problem computations and assigns portions to each GPU. But unlike the single GPU which carries all the computations on a single GPU, data communications are introduced, consequently engendering more execution time. But experiments show that execution time is still highly reduced.
-
-
-
+In Figure~\ref{fig:04}, we can also observe that the CUDA-MPI approach
+is also efficient to solve full polynimails on multiple GPUs.
\subsection{Comparing the CUDA-OpenMP approach and the CUDA-MPI approach}
In the previuos section we saw that both approches are very effective in reducing execution time for sparse as well as full polynomials. At this stage, the interesting question is which approach is better. In the fellowing, we present appropriate experiments comparing the two Multi-GPU approaches to answer the question.
\caption{Execution time for solving sparse polynomials of three distinct sizes on multiple GPUs using MPI and OpenMP approaches using Ehrlich-Aberth}
\label{fig:05}
\end{figure}
-In Figure~\ref{fig:05} there two curves for each polynomial size : one for the MPI-CUDA and another for the OpenMP. We can see that the results are similar between OpenMP and MPI for the polynomials size of 200K. For the size of 800K, the MPI version is a little slower than the OpenMP approach but for for the 1,4M size, there is a slight advantage for the MPI version.
+In Figure~\ref{fig:05} there two curves for each polynomial size : one for the MPI-CUDA and another for the OpenMP. We can see that the results are similar between OpenMP and MPI for the polynomials size of 200K. For the size of 800K, the MPI version is a little slower than the OpenMP approach but for the 1,4 millions size, there is a slight advantage for the MPI version.
\subsubsection{Solving full polynomials}
\begin{figure}[htbp]
