\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}
% 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
%The issue of finding the roots of polynomials of very high degrees arises in many complex problems in various fields, such as algebra, biology, finance, physics or climatology [1]. In algebra for example, finding eigenvalues or eigenvectors of any real/complex matrix amounts to that of finding the roots of the so-called characteristic polynomial.
-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
+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},
+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_0 a_n\neq 0.
+ p(x)=a_n\displaystyle\prod_{i=1}^n(x-z_i), a_n\neq 0.
\end{equation}
-\LZK{Pourquoi $a_0a_n\neq 0$ ?}
+%\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{} and Ehrlich-Aberth method~\cite{}.
-\LZK{Pouvez-vous donner des références pour les deux méthodes?}
+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}
%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.
-\LZK{``only by using 8 personal computers and 2 communications per iteration''. Pour MPI? et Pour OpenMP}
+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.
%\LZK{Les contributions ne sont pas définies !!}
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?}
+\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 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}
-\LZK{Pas d'autres contributions possibles?}
+\LZK{Pas d'autres contributions possibles?: j'ai rajouté 2}
%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.
-\LZK{A revoir toute cette organization}
+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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Ehrlich-Aberth algorithm on a GPU}
\label{sec3}
-\subsection{The EA method}
+\subsection{The Ehrlich-Aberth method}
%A cubically convergent iteration method to find zeros of
%polynomials was proposed by O. Aberth~\cite{Aberth73}. The
%Ehrlich-Aberth (EA is short) method contains 4 main steps, presented in what
%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}
%\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~\cite{,}, wich will make it possible to converge to the roots solution, provided that all the root are different.
+%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
\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}
- 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 $\xi$
+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,
+
+ 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 $\xi$.
\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.
+
+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:
+
+\begin{equation}
+\label{R.EL}
+R = exp(log(DBL\_MAX)/(2*n) );
+\end{equation}
+
-\subsection{EA parallel implementation on CUDA}
+%\begin{equation}
+
+%R = \exp( \log(DBL\_MAX) / (2*n) )
+%\end{equation}
+ where \verb=DBL_MAX= stands for the maximum representable \verb=double= value.
+
+In order to hold into account the limit of size of floats, we propose to modifying the iterative function and compute the logarithm of:
+
+\begin{equation}
+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}
+
+This method allows, indeed, to exceed the computation of the polynomials of degree 100,000 and to reach a degree upper to 1,000,000. For that purpose, it is necessary to use the logarithm and the exponential of a complex. The iterative function of Ehrlich-Aberth method with exponential and logarithm is given as following:
+
+\begin{equation}
+\label{Log_H2}
+EA.EL: 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}
+\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)i=1,...,n,
+\end{equation}
+
+
+%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 and the roots for large polynomial degrees can be looked for successfully~\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.
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 organzed by what is named kernels, portions o code that are run on GPU devices. For step 3, there are two kernels, the
+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
\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)$\;
+\item \While {$\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 $kernel\_testConverge(\Delta z_{max},Z,ZPrec)$\;
}
