-
-%% bare_conf.tex
-%% V1.4b
-%% 2015/08/26
-%% by Michael Shell
-%% See:
-%% http://www.michaelshell.org/
-%% for current contact information.
-%%
-%% This is a skeleton file demonstrating the use of IEEEtran.cls
-%% (requires IEEEtran.cls version 1.8b or later) with an IEEE
-%% conference paper.
-%%
-%% Support sites:
-%% http://www.michaelshell.org/tex/ieeetran/
-%% http://www.ctan.org/pkg/ieeetran
-%% and
-%% http://www.ieee.org/
-
-%%*************************************************************************
-%% Legal Notice:
-%% This code is offered as-is without any warranty either expressed or
-%% implied; without even the implied warranty of MERCHANTABILITY or
-%% FITNESS FOR A PARTICULAR PURPOSE!
-%% User assumes all risk.
-%% In no event shall the IEEE or any contributor to this code be liable for
-%% any damages or losses, including, but not limited to, incidental,
-%% consequential, or any other damages, resulting from the use or misuse
-%% of any information contained here.
-%%
-%% All comments are the opinions of their respective authors and are not
-%% necessarily endorsed by the IEEE.
-%%
-%% This work is distributed under the LaTeX Project Public License (LPPL)
-%% ( http://www.latex-project.org/ ) version 1.3, and may be freely used,
-%% distributed and modified. A copy of the LPPL, version 1.3, is included
-%% in the base LaTeX documentation of all distributions of LaTeX released
-%% 2003/12/01 or later.
-%% Retain all contribution notices and credits.
-%% ** Modified files should be clearly indicated as such, including **
-%% ** renaming them and changing author support contact information. **
-%%*************************************************************************
-
-
-% *** Authors should verify (and, if needed, correct) their LaTeX system ***
-% *** with the testflow diagnostic prior to trusting their LaTeX platform ***
-% *** with production work. The IEEE's font choices and paper sizes can ***
-% *** trigger bugs that do not appear when using other class files. *** ***
-% The testflow support page is at:
-% http://www.michaelshell.org/tex/testflow/
-
-
-
\documentclass[conference]{IEEEtran}
-% Some Computer Society conferences also require the compsoc mode option,
-% but others use the standard conference format.
-%
-% If IEEEtran.cls has not been installed into the LaTeX system files,
-% manually specify the path to it like:
-% \documentclass[conference]{../sty/IEEEtran}
-
-
-
-
-
-% Some very useful LaTeX packages include:
-% (uncomment the ones you want to load)
-
-
-% *** MISC UTILITY PACKAGES ***
-%
-%\usepackage{ifpdf}
-% Heiko Oberdiek's ifpdf.sty is very useful if you need conditional
-% compilation based on whether the output is pdf or dvi.
-% usage:
-% \ifpdf
-% % pdf code
-% \else
-% % dvi code
-% \fi
-% The latest version of ifpdf.sty can be obtained from:
-% http://www.ctan.org/pkg/ifpdf
-% Also, note that IEEEtran.cls V1.7 and later provides a builtin
-% \ifCLASSINFOpdf conditional that works the same way.
-% When switching from latex to pdflatex and vice-versa, the compiler may
-% have to be run twice to clear warning/error messages.
-
-
-
-
-
-
-% *** CITATION PACKAGES ***
-%
-%\usepackage{cite}
-% cite.sty was written by Donald Arseneau
-% V1.6 and later of IEEEtran pre-defines the format of the cite.sty package
-% \cite{} output to follow that of the IEEE. Loading the cite package will
-% result in citation numbers being automatically sorted and properly
-% "compressed/ranged". e.g., [1], [9], [2], [7], [5], [6] without using
-% cite.sty will become [1], [2], [5]--[7], [9] using cite.sty. cite.sty's
-% \cite will automatically add leading space, if needed. Use cite.sty's
-% noadjust option (cite.sty V3.8 and later) if you want to turn this off
-% such as if a citation ever needs to be enclosed in parenthesis.
-% cite.sty is already installed on most LaTeX systems. Be sure and use
-% version 5.0 (2009-03-20) and later if using hyperref.sty.
-% The latest version can be obtained at:
-% http://www.ctan.org/pkg/cite
-% The documentation is contained in the cite.sty file itself.
-
-
-
-
-
-
-% *** GRAPHICS RELATED PACKAGES ***
-%
-\ifCLASSINFOpdf
- \usepackage[pdftex]{graphicx}
-
- % declare the path(s) where your graphic files are
- % \graphicspath{{../pdf/}{../jpeg/}}
- % and their extensions so you won't have to specify these with
- % every instance of \includegraphics
- % \DeclareGraphicsExtensions{.pdf,.jpeg,.png}
-\else
- % or other class option (dvipsone, dvipdf, if not using dvips). graphicx
- % will default to the driver specified in the system graphics.cfg if no
- % driver is specified.
- % \usepackage[dvips]{graphicx}
- % declare the path(s) where your graphic files are
- % \graphicspath{{../eps/}}
- % and their extensions so you won't have to specify these with
- % every instance of \includegraphics
- % \DeclareGraphicsExtensions{.eps}
-\fi
-% graphicx was written by David Carlisle and Sebastian Rahtz. It is
-% required if you want graphics, photos, etc. graphicx.sty is already
-% installed on most LaTeX systems. The latest version and documentation
-% can be obtained at:
-% http://www.ctan.org/pkg/graphicx
-% Another good source of documentation is "Using Imported Graphics in
-% LaTeX2e" by Keith Reckdahl which can be found at:
-% http://www.ctan.org/pkg/epslatex
-%
-% latex, and pdflatex in dvi mode, support graphics in encapsulated
-% postscript (.eps) format. pdflatex in pdf mode supports graphics
-% in .pdf, .jpeg, .png and .mps (metapost) formats. Users should ensure
-% that all non-photo figures use a vector format (.eps, .pdf, .mps) and
-% not a bitmapped formats (.jpeg, .png). The IEEE frowns on bitmapped formats
-% which can result in "jaggedy"/blurry rendering of lines and letters as
-% well as large increases in file sizes.
-%
-% You can find documentation about the pdfTeX application at:
-% http://www.tug.org/applications/pdftex
-
-
-
-
-
-% *** MATH PACKAGES ***
-%
-\usepackage{amsmath}
-% A popular package from the American Mathematical Society that provides
-% many useful and powerful commands for dealing with mathematics.
-%
-% Note that the amsmath package sets \interdisplaylinepenalty to 10000
-% thus preventing page breaks from occurring within multiline equations. Use:
-%\interdisplaylinepenalty=2500
-% after loading amsmath to restore such page breaks as IEEEtran.cls normally
-% does. amsmath.sty is already installed on most LaTeX systems. The latest
-% version and documentation can be obtained at:
-% http://www.ctan.org/pkg/amsmath
-
-
-
-
-
-% *** SPECIALIZED LIST PACKAGES ***
-%
-
-% algorithmic.sty was written by Peter Williams and Rogerio Brito.
-% This package provides an algorithmic environment fo describing algorithms.
-% You can use the algorithmic environment in-text or within a figure
-% environment to provide for a floating algorithm. Do NOT use the algorithm
-% floating environment provided by algorithm.sty (by the same authors) or
-% algorithm2e.sty (by Christophe Fiorio) as the IEEE does not use dedicated
-% algorithm float types and packages that provide these will not provide
-% correct IEEE style captions. The latest version and documentation of
-% algorithmic.sty can be obtained at:
-% http://www.ctan.org/pkg/algorithms
-% Also of interest may be the (relatively newer and more customizable)
-% algorithmicx.sty package by Szasz Janos:
-% http://www.ctan.org/pkg/algorithmicx
\usepackage[ruled,vlined]{algorithm2e}
-
-
-
-% *** ALIGNMENT PACKAGES ***
-%
-%\usepackage{array}
-% Frank Mittelbach's and David Carlisle's array.sty patches and improves
-% the standard LaTeX2e array and tabular environments to provide better
-% appearance and additional user controls. As the default LaTeX2e table
-% generation code is lacking to the point of almost being broken with
-% respect to the quality of the end results, all users are strongly
-% advised to use an enhanced (at the very least that provided by array.sty)
-% set of table tools. array.sty is already installed on most systems. The
-% latest version and documentation can be obtained at:
-% http://www.ctan.org/pkg/array
-
-
-% IEEEtran contains the IEEEeqnarray family of commands that can be used to
-% generate multiline equations as well as matrices, tables, etc., of high
-% quality.
-
-
-
-
-% *** SUBFIGURE PACKAGES ***
-%\ifCLASSOPTIONcompsoc
-% \usepackage[caption=false,font=normalsize,labelfont=sf,textfont=sf]{subfig}
-%\else
-% \usepackage[caption=false,font=footnotesize]{subfig}
-%\fi
-% subfig.sty, written by Steven Douglas Cochran, is the modern replacement
-% for subfigure.sty, the latter of which is no longer maintained and is
-% incompatible with some LaTeX packages including fixltx2e. However,
-% subfig.sty requires and automatically loads Axel Sommerfeldt's caption.sty
-% which will override IEEEtran.cls' handling of captions and this will result
-% in non-IEEE style figure/table captions. To prevent this problem, be sure
-% and invoke subfig.sty's "caption=false" package option (available since
-% subfig.sty version 1.3, 2005/06/28) as this is will preserve IEEEtran.cls
-% handling of captions.
-% Note that the Computer Society format requires a larger sans serif font
-% than the serif footnote size font used in traditional IEEE formatting
-% and thus the need to invoke different subfig.sty package options depending
-% on whether compsoc mode has been enabled.
-%
-% The latest version and documentation of subfig.sty can be obtained at:
-% http://www.ctan.org/pkg/subfig
-
-
-
-
-% *** FLOAT PACKAGES ***
-%
-%\usepackage{fixltx2e}
-% fixltx2e, the successor to the earlier fix2col.sty, was written by
-% Frank Mittelbach and David Carlisle. This package corrects a few problems
-% in the LaTeX2e kernel, the most notable of which is that in current
-% LaTeX2e releases, the ordering of single and double column floats is not
-% guaranteed to be preserved. Thus, an unpatched LaTeX2e can allow a
-% single column figure to be placed prior to an earlier double column
-% figure.
-% Be aware that LaTeX2e kernels dated 2015 and later have fixltx2e.sty's
-% corrections already built into the system in which case a warning will
-% be issued if an attempt is made to load fixltx2e.sty as it is no longer
-% needed.
-% The latest version and documentation can be found at:
-% http://www.ctan.org/pkg/fixltx2e
-
-
-%\usepackage{stfloats}
-% stfloats.sty was written by Sigitas Tolusis. This package gives LaTeX2e
-% the ability to do double column floats at the bottom of the page as well
-% as the top. (e.g., "\begin{figure*}[!b]" is not normally possible in
-% LaTeX2e). It also provides a command:
-%\fnbelowfloat
-% to enable the placement of footnotes below bottom floats (the standard
-% LaTeX2e kernel puts them above bottom floats). This is an invasive package
-% which rewrites many portions of the LaTeX2e float routines. It may not work
-% with other packages that modify the LaTeX2e float routines. The latest
-% version and documentation can be obtained at:
-% http://www.ctan.org/pkg/stfloats
-% Do not use the stfloats baselinefloat ability as the IEEE does not allow
-% \baselineskip to stretch. Authors submitting work to the IEEE should note
-% that the IEEE rarely uses double column equations and that authors should try
-% to avoid such use. Do not be tempted to use the cuted.sty or midfloat.sty
-% packages (also by Sigitas Tolusis) as the IEEE does not format its papers in
-% such ways.
-% Do not attempt to use stfloats with fixltx2e as they are incompatible.
-% Instead, use Morten Hogholm'a dblfloatfix which combines the features
-% of both fixltx2e and stfloats:
-%
-% \usepackage{dblfloatfix}
-% The latest version can be found at:
-% http://www.ctan.org/pkg/dblfloatfix
-
-
-
-
-% *** PDF, URL AND HYPERLINK PACKAGES ***
-%
-%\usepackage{url}
-% url.sty was written by Donald Arseneau. It provides better support for
-% handling and breaking URLs. url.sty is already installed on most LaTeX
-% systems. The latest version and documentation can be obtained at:
-% http://www.ctan.org/pkg/url
-% Basically, \url{my_url_here}.
-
-
-
-
-% *** Do not adjust lengths that control margins, column widths, etc. ***
-% *** Do not use packages that alter fonts (such as pslatex). ***
-% There should be no need to do such things with IEEEtran.cls V1.6 and later.
-% (Unless specifically asked to do so by the journal or conference you plan
-% to submit to, of course. )
-
-
-% correct bad hyphenation here
\hyphenation{op-tical net-works semi-conduc-tor}
-%\usepackage{graphicx}
\bibliographystyle{IEEEtran}
-% argument is your BibTeX string definitions and bibliography database(s)
-%\bibliography{IEEEabrv,../bib/paper}
-%\bibliographystyle{elsarticle-num}
-
-
-
\usepackage{amsfonts}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[textsize=footnotesize]{todonotes}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{float}
\newcommand{\LZK}[2][inline]{%
\todo[color=red!10,#1]{\sffamily\textbf{LZK:} #2}\xspace}
\newcommand{\RC}[2][inline]{%
\todo[color=orange!10,#1]{\sffamily\textbf{AS:} #2}\xspace}
-
-
-
\begin{document}
-%
-% paper title
-% Titles are generally capitalized except for words such as a, an, and, as,
-% at, but, by, for, in, nor, of, on, or, the, to and up, which are usually
-% not capitalized unless they are the first or last word of the title.
-% Linebreaks \\ can be used within to get better formatting as desired.
-% Do not put math or special symbols in the title.
-\title{Two parallel implementations of Ehrlich-Aberth algorithm for root-finding of polynomials on multiple GPUs with OpenMP and MPI}
+\title{Two parallel implementations of Ehrlich-Aberth algorithm for root-finding of polynomials on multiple GPUs with OpenMP and MPI}
-% author names and affiliations
-% use a multiple column layout for up to three different
-% affiliations
-\author{\IEEEauthorblockN{Kahina Guidouche, Abderrahmane Sider }
+\author{\IEEEauthorblockN{Kahina Ghidouche, Abderrahmane Sider }
\IEEEauthorblockA{Laboratoire LIMED\\
Faculté des sciences exactes\\
Université de Bejaia, 06000, Algeria\\
University of Bourgogne Franche-Comte, France\\
Email: zianekhodja.lilia@gmail.com\\ raphael.couturier@univ-fcomte.fr}}
-% conference papers do not typically use \thanks and this command
-% is locked out in conference mode. If really needed, such as for
-% the acknowledgment of grants, issue a \IEEEoverridecommandlockouts
-% after \documentclass
-
-% for over three affiliations, or if they all won't fit within the width
-% of the page, use this alternative format:
-%
-%\author{\IEEEauthorblockN{Michael Shell\IEEEauthorrefmark{1},
-%Homer Simpson\IEEEauthorrefmark{2},
-%James Kirk\IEEEauthorrefmark{3},
-%Montgomery Scott\IEEEauthorrefmark{3} and
-%Eldon Tyrell\IEEEauthorrefmark{4}}
-%\IEEEauthorblockA{\IEEEauthorrefmark{1}School of Electrical and Computer Engineering\\
-%Georgia Institute of Technology,
-%Atlanta, Georgia 30332--0250\\ Email: see http://www.michaelshell.org/contact.html}
-%\IEEEauthorblockA{\IEEEauthorrefmark{2}Twentieth Century Fox, Springfield, USA\\
-%Email: homer@thesimpsons.com}
-%\IEEEauthorblockA{\IEEEauthorrefmark{3}Starfleet Academy, San Francisco, California 96678-2391\\
-%Telephone: (800) 555--1212, Fax: (888) 555--1212}
-%\IEEEauthorblockA{\IEEEauthorrefmark{4}Tyrell Inc., 123 Replicant Street, Los Angeles, California 90210--4321}}
-
-
-
-
-% use for special paper notices
-%\IEEEspecialpapernotice{(Invited Paper)}
-
-
-
-% make the title area
\maketitle
-% As a general rule, do not put math, special symbols or citations
-% in the abstract
\begin{abstract}
-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.
+Finding the roots of polynomials is a very important part of solving
+real-life problems but the higher the degree of the polynomials is,
+the less easy it becomes. 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 use 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 the roots of polynomials of degree up-to
+1.4 million. Moreover, other experiments show it is possible to find the
+roots of polynomials of degree up-to 5 million.
\end{abstract}
-% no keywords
-\LZK{Faut pas mettre des keywords?}
-
-
+\begin{IEEEkeywords}
+ root finding method, Ehrlich-Aberth method, GPU, MPI, OpenMP
+\end{IEEEkeywords}
-
-% For peer review papers, you can put extra information on the cover
-% page as needed:
-% \ifCLASSOPTIONpeerreview
-% \begin{center} \bfseries EDICS Category: 3-BBND \end{center}
-% \fi
-%
-% For peerreview papers, this IEEEtran command inserts a page break and
-% creates the second title. It will be ignored for other modes.
\IEEEpeerreviewmaketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
-%Polynomials are mathematical algebraic structures that play an important role in science and engineering by capturing physical phenomena and expressing any outcome as a function of some unknown variables. Formally speaking, a polynomial $p(x)$ of degree $n$ having $n$ coefficients in the complex plane $\mathbb{C}$ is:
-%\begin{equation}
-%p(x)=\sum_{i=0}^{n}{a_ix^i}.
-%\end{equation}
-%\LZK{Dans ce cas le polynôme a $n+1$ coefficients et non pas $n$!}
-%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 the roots of polynomials of very high degrees arises in many complex problems of 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_n\neq 0,
+p(x) = \displaystyle\sum^n_{i=0}{\alpha_ix^i},\alpha_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 :
+where $\{\alpha_i\}_{0\leq i\leq n}$ are complex coefficients and $n$ is a high integer number. If $\alpha_n\neq0$ 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.
+ p(x)=\alpha_n\displaystyle\prod_{i=1}^n(x-z_i), \alpha_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 }
-
-%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}
-
-%The first method of this group is Durand-Kerner method:
-%\begin{equation}
-%\label{DK}
-% DK: z_i^{k+1}=z_{i}^{k}-\frac{P(z_i^{k})}{\prod_{i\neq j}(z_i^{k}-z_j^{k})}, i = 1, \ldots, n,
-%\end{equation}
-%where $z_i^k$ is the $i^{th}$ root of the polynomial $p$ at the iteration $k$. Another method discovered by Borsch-Supan~\cite{ Borch-Supan63} and also described by Ehrlich~\cite{Ehrlich67} and Aberth~\cite{Aberth73} uses a different iteration form as follows:
-%%\begin{center}
-%\begin{equation}
-%\label{Eq:EA}
- %EA: z_i^{k+1}=z_i^{k}-\frac{1}{{\frac {P'(z_i^{k})} {P(z_i^{k})}}-{\sum_{i\neq j}\frac{1}{(z_i^{k}-z_j^{k})}}}, i = 1, \ldots, n,
-%\end{equation}
-%%\end{center}
-%where $p'(z)$ is the polynomial derivative of $p$ evaluated in the point $z$.
-
-%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.
-\RC{si on donne des temps faut donner le proc, comme c'est vieux à mon avis faut supprimer ca, votre avis?}
-\LZK{Supprimons ces détails et mettons une référence s'il y en a une}
-
-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.
-
-%Finding polynomial roots rapidly and accurately is the main objective of our work. In this paper we propose the parallelization of Ehrlich-Aberth method using two parallel programming paradigms OpenMP and MPI on multi-GPU platforms. We consider two architectures: shared memory and distributed memory computers. The first parallel algorithm is implemented on shared memory computers by 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. The second parallel algorithm uses the 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 distributed memory clusters to solve very complex problems that are too large for traditional supercomputers, which are very expensive to build and run.
-%\LZK{Cette partie est réécrite. \\ Sinon qu'est ce qui a été fait pour l'accuracy dans ce papier (Finding polynomial roots rapidly and accurately is the main objective of our work.)?}
-%\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? the Ehrlich-Aberth have very good convergence and it is suitable to be implemented in parallel computers.}
-In this paper we propose the parallelization of Ehrlich-Aberth method which has a good convergence and it is suitable to be implemented in parallel computers. We use 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{J'ai ajouté une phrase pour justifier notre choix de la méthode Ehrlich-Aberth. A revérifier.}
+
+Most of the numerical methods that deal with the polynomial
+root-finding problems are simultaneous methods, \textit{i.e.} the
+iterative methods to find simultaneous approximations of the $n$
+polynomial roots. These methods start from the initial approximation
+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
+the Durand-Kerner method~\cite{Durand60,Kerner66} and the Ehrlich-Aberth method~\cite{Ehrlich67,Aberth73}.
+
+
+The convergence time of simultaneous methods drastically increases
+with the increasing of the polynomial's degree. The great challenge
+with simultaneous methods is to parallelize them and to improve their
+convergence. Many authors have proposed parallel simultaneous
+methods~\cite{Freeman89,Loizou83,Freemanall90,bini96,cs01:nj,Couturier02},
+using several paradigms of parallelization (synchronous or
+asynchronous computations, mechanism of shared or distributed memory,
+etc). However, so far until now, only polynomials not exceeding
+degrees of less than 100,000 have been solved.
+
+%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 and al.~\cite{cs01:nj} 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. The authors showed an interesting
+%speedup that is 20 times as fast as the sequential implementation.
+
+The recent advent of the Compute Unified Device Architecture
+(CUDA)~\cite{CUDA15}, a programming
+model and a parallel computing architecture developed by NVIDIA, has revived parallel programming interest in
+this problem. Indeed, the computing power of GPUs (Graphics Processing
+Units) has exceeded that of traditional CPUs processors, which makes
+it very appealing to the research community to investigate new
+parallel implementations for a whole set of scientific problems in the
+reasonable hope to solve bigger instances of well known
+computationally demanding issues such as the one beforehand. However,
+CUDA provides an efficient massive data computing model which is
+suited to GPU architectures. 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 the Ehrlich-Aberth
+(EA) method which has a much better cubic convergence rate than the
+quadratic rate of the Durand-Kerner method that has already been investigated in \cite{Kahinall14}. In the other hand, EA is suitable to be implemented in parallel computers according to the data-parallel paradigm. In this model, computing elements carry computations on the data they are assigned and communicate with other computing elements in order to get fresh data or to synchronize. Classically, two parallel programming paradigms OpenMP and MPI are used to code such solutions. But in our case, computing elements are CUDA multi-GPU platforms. This architectural setting poses new programming challenges but offers also new opportunities to efficiently solve huge problems, otherwise considered intractable until recently. To the best of our knowledge, our CUDA-MPI and CUDA-OpenMP codes are the first implementations of EA method with multiple GPUs for finding roots of polynomials. Our major contributions include:
\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.\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.
+
+\item The parallel implementation of EA 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 EA 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 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.
+\item
+ Our method is efficient to compute the roots of sparse and full
+ polynomials of degree up to 5 million.
\end{itemize}
-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.
-\LZK{Pas d'autres contributions possibles? J'ai supprimé les deux premiers points proposés précédemment.}
-%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 multiple GPUs using the OpenMP and MPI approaches. In section~\ref{sec5} we present our experiments and discuss them. 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{Parallel programming models}
\label{sec2}
-Our objective consists in implementing a root-finding algorithm of polynomials 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 investigate two parallel paradigms: OpenMP and MPI. In this case, the GPU indices are defined according to the identifiers of the OpenMP threads or the ranks of the MPI processes. In this section we present the parallel programming models: OpenMP, MPI and CUDA.
+Our objective consists in implementing a root-finding algorithm of
+polynomials on multiple GPUs. To this end, it is essential to know how
+to manage the CUDA contexts of different GPUs. A direct method to control the various GPUs is to use as many threads or processes as GPU devices. We investigate two parallel paradigms: OpenMP and MPI. In this case, the GPU indices are defined according to the identifiers of the OpenMP threads or the ranks of the MPI processes. In this section we present the parallel programming models: OpenMP, MPI and CUDA.
\subsection{OpenMP}
-%Open Multi-Processing (OpenMP) is a shared memory architecture API that provides multi thread capacity~\cite{openmp13}. 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 allocated 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 usage of OpenMP threads and managed data explicitly done with MPI can be considered, this approcache undermines the advantages of OpenMP.
-
-%\subsection{OpenMP}
-%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].
-
-OpenMP (Open Multi-processing) is an application programming interface for parallel programming~\cite{openmp13}. It is a portable approach based on the multithreading designed for shared memory computers, where a master thread forks a number of slave threads which execute blocks of code in parallel. An OpenMP program alternates sequential regions and parallel regions of code, where the sequential regions are executed by the master thread and the parallel ones may be executed by multiple threads. During the execution of an OpenMP program the threads communicate their data (read and modified) in the shared memory. One advantage of OpenMP is the global view of the memory address space of an application. This allows relatively a fast development of parallel applications with easier maintenance. However, it is often difficult to get high rates of performances in large scale-applications.
+OpenMP (Open Multi-processing) is an application programming interface
+for parallel programming~\cite{openmp13}. It is a portable approach
+based on the multithreading designed for shared memory computers,
+where a master thread forks a number of slave threads which execute
+blocks of code in parallel. An OpenMP program alternates sequential
+regions and parallel regions of code, where the sequential regions are
+executed by the master thread and the parallel ones may be executed by
+multiple threads. During the execution of an OpenMP program the
+threads communicate their data (read and modified) in the shared
+memory. One advantage of OpenMP is the global view of the memory
+address space of an application. This allows a relatively fast development of parallel applications with easier maintenance. However, it is often difficult to get high rates of performances in large scale-applications.
\subsection{MPI}
-%The MPI (Message Passing Interface) library allows to create computer programs that run on a distributed memory architecture. The various processes have their own environment of execution and execute their code in a asynchronous way, according to the MIMD model (Multiple Instruction streams, Multiple Data streams); they communicate and synchronize by exchanging messages~\cite{Peter96}. MPI messages are explicitly sent, while the exchanges are implicit within the framework of a multi-thread programming environment like OpenMP or Pthreads.
-
-MPI (Message Passing Interface) is a portable message passing style of the parallel programming designed especially for the distributed memory architectures~\cite{Peter96}. In most MPI implementations, a computation contains a fixed set of processes created at the initialization of the program in such way one process is created per processor. The processes synchronize their computations and communicate by sending/receiving messages to/from other processes. In this case, the data are explicitly exchanged by message passing while the data exchanges are implicit in a multithread programming model like OpenMP and Pthreads. However in the MPI programming model, the processes may either execute different programs referred to as multiple program multiple data (MPMD) or every process executes the same program (SPMD). The MPI approach is one of most used HPC programming model to solve large scale and complex applications.
+MPI (Message Passing Interface) is a portable message passing style of
+the parallel programming designed specifically for distributed memory
+architectures~\cite{Peter96}. In most MPI implementations, a
+computation contains a fixed set of processes created at the
+initialization of the program in such a way that one process is
+created per processor. The processes synchronize their computations
+and communicate by sending/receiving messages to/from other
+processes. In this case, the data are explicitly exchanged by message
+passing while the data exchanges are implicit in a multithread
+programming model like OpenMP and Pthreads. However in the MPI
+programming model, the processes may either execute different programs
+referred to as multiple program multiple data (MPMD) or every process
+executes the same program (SPMD). The MPI approach is one of the most used HPC programming model to solve large scale and complex applications.
\subsection{CUDA}
-%CUDA (is an acronym of the Compute Unified Device Architecture) is a parallel computing architecture developed by NVIDIA~\cite{CUDA10}.The unit of execution in CUDA is called a thread. Each thread executes a kernel by the streaming processors in parallel. In CUDA, a group of threads that are executed together is called a thread block, and the computational grid consists of a grid of thread blocks. Additionally, a thread block can use the shared memory on a single multiprocessor while 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.
+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.
-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 and non-graphic applications. 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 later. 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 in the 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 the shared memory, and the global memory accesses should be minimized.
-
-%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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec3}
\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
-%follows.
-
-%The Aberth method is a purely algebraic derivation.
-%To illustrate the derivation, we let $w_{i}(z)$ be the product of linear factors
-
-%\begin{equation}
-%w_{i}(z)=\prod_{j=1,j \neq i}^{n} (z-x_{j})
-%\end{equation}
-
-%And let a rational function $R_{i}(z)$ be the correction term of the
-%Weistrass method~\cite{Weierstrass03}
-
-%\begin{equation}
-%R_{i}(z)=\frac{p(z)}{w_{i}(z)} , i=1,2,...,n.
-%\end{equation}
-
-%Differentiating the rational function $R_{i}(z)$ and applying the
-%Newton method, we have:
-
-%\begin{equation}
-%\frac{R_{i}(z)}{R_{i}^{'}(z)}= \frac{p(z)}{p^{'}(z)-p(z)\frac{w_{i}(z)}{w_{i}^{'}(z)}}= \frac{p(z)}{p^{'}(z)-p(z) \sum _{j=1,j \neq i}^{n}\frac{1}{z-x_{j}}}, i=1,2,...,n
-%\end{equation}
-%where R_{i}^{'}(z)is the rational function derivative of F evaluated in the point z
-%Substituting $x_{j}$ for $z_{j}$ we obtain the Aberth iteration method.%
-
-
-%\subsubsection{Polynomials Initialization}
-%The initialization of a polynomial $p(z)$ is done by setting each of the $n$ complex coefficients %$a_{i}$:
-
-%\begin{equation}
-%\label{eq:SimplePolynome}
-% p(z)=\sum{a_{i}z^{n-i}} , a_{n} \neq 0,a_{0}=1, a_{i}\subset C
-%\end{equation}
-
-
-%\subsubsection{Vector $Z^{(0)}$ Initialization}
-%\label{sec:vec_initialization}
-%As for any iterative method, we need to choose $n$ initial guess points $z^{0}_{i}, i = 1, . . . , %n.$
-%The initial guess is very important since the number of steps needed by the iterative method to %reach
-%a given approximation strongly depends on it.
-%In~\cite{Aberth73} the Ehrlich-Aberth iteration is started by selecting $n$
-%equi-distant points on a circle of center 0 and radius r, where r is
-%an upper bound to the moduli of the zeros. Later, Bini and al.~\cite{Bini96}
-%performed this choice by selecting complex numbers along different
-%circles which relies on the result of~\cite{Ostrowski41}.
-
-%\begin{equation}
-%\label{eq:radiusR}
-%%\begin{align}
-%\sigma_{0}=\frac{u+v}{2};u=\frac{\sum_{i=1}^{n}u_{i}}{n.max_{i=1}^{n}u_{i}};
-%v=\frac{\sum_{i=0}^{n-1}v_{i}}{n.min_{i=0}^{n-1}v_{i}};\\
-%%\end{align}
-%\end{equation}
-%Where:
-%\begin{equation}
-%u_{i}=2.|a_{i}|^{\frac{1}{i}};
-%v_{i}=\frac{|\frac{a_{n}}{a_{i}}|^{\frac{1}{n-i}}}{2}.
-%\end{equation}
-
-%\subsubsection{Iterative Function}
-%The operator used by the Aberth method corresponds to the
-%equation~\ref{Eq:EA1}, it enables the convergence towards
-%the polynomials zeros, provided all the roots are distinct.
-
-%Here we give a second form of the iterative function used by the Ehrlich-Aberth method:
-
-%\begin{equation}
-%\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}
-
-%\subsubsection{Convergence Condition}
-%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:AAberth-Conv-Cond}
-%\forall i \in [1,n];\vert\frac{z_{i}^{k}-z_{i}^{k-1}}{z_{i}^{k}}\vert<\xi
-%\end{equation}
-
-
-%\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~\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{Aberth73} using the following iteration
\begin{equation}
\end{equation}
This method contains 4 steps. The first step consists in the
-initializing the polynomial.\LZK{Pas compris?? \RC{changé}}.
-The second step initializes the solution vector $Z$ using the
-Guggenheimer method~\cite{Gugg86} to ensure that initial roots are all
-distinct from each other. \LZK{Quelle est la différence entre la 1st
- step et la 2nd step? Que veut dire " to ensure the distinction of
- the initial vector roots"? \RC{reformulé}}
-In step 3, the iterative function based on the Newton's
-method~\cite{newt70} and Weiestrass operator~\cite{Weierstrass03} is
-applied. In our case, the Ehrlich-Aberth is applied as in (\ref{Eq:EA1}).
-Iterations of the EA method will converge to the roots of the
-considered polynomial.\LZK{On ne peut pas expliquer un peu plus
- comment? Donner des formules comment elle se base sur la méthode de
- Newton et de l'opérateur de Weiestrass? \RC{amélioré}}
-\LZK{Elle est où la 4th step??}
-\LZK{Conclusion: Méthode mal présentée et j'ai presque rien compris!
- \RC{après} }
-
-
-In order to stop the iterative function, a stop condition is applied,
-this is the 4th step. This condition checks that all the root modules
-are lower than a fixed value $\epsilon$.
+initializing the polynomial. The second step initializes the solution
+vector $Z$ using the Guggenheimer method~\cite{Gugg86} to ensure that
+initial roots are all distinct from each other. In step 3, the
+iterative function based on the Newton's method~\cite{newt70} and
+Weiestrass operator~\cite{Weierstrass03} is applied. In our case, the
+Ehrlich-Aberth is applied as in~(\ref{Eq:EA1}). Iterations of the
+EA method will converge to the roots of the considered
+polynomial. In order to stop the iterative function, a stop condition
+is applied, this is the 4th step. This condition checks that all the
+root modules are lower than a fixed value $\epsilon$.
\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<\epsilon
\end{equation}
-\LZK{On ne dit pas plutôt "the relative errors" à la place de "root
- modules"? Raph nous confirmera quelle critère d'arrêt a
- utilisé. \RC{normalement c'est bon, l'erreur est calculée avec le
- module de chaque racine}}
-
\subsection{Improving Ehrlich-Aberth method}
-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.
+With high degree polynomials, the EA 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.
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
+we propose a new version of the EA 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. The reformulation of the iteration~(\ref{Eq:EA1}) of the Ehrlich-Aberth method with exponential and logarithm is defined as follows, for $i=1,\dots,n$:
+100,000 and to reach a degree up to more than 1,000,000. The reformulation of the iteration~(\ref{Eq:EA1}) of the EA method with exponential and logarithm operators is defined as follows, for $i=1,\dots,n$:
\begin{equation}
\label{Log_H2}
Q(z^k_i) = \exp(\ln(p(z^k_i)) - \ln(p'(z^k_i)) + \ln(\sum_{i\neq j}^n\frac{1}{z^k_i-z^k_j})).
\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 values in absolute
-values~\cite{Karimall98}. \LZK{Je n'ai pas compris cette dernière
- phrase? \RC{changé : on veut dire on manipule des valeurs plus petites en valeur absolues}}
+subtractions. Consequently, computations manipulate lower values in
+absolute values~\cite{Karimall98}. In practice, the exponential and
+logarithm mode is used when a root is outside the circle unit represented by the radius $R$ evaluated in C language with:
+\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 and $n$ is the degree of the polynomial.
-%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{The 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. \LZK{Répétition! Le même texte est déjà écrit comme
- intro dans la section II. Sinon ici on parle seulement de
- l'implémentation cuda sans mpi et openmp! \RC{Je suis d'accord à
- revoir après, quand les 2 parties suivantes seront plus stables}}
-
-
-
-
-Like any parallel code, a GPU parallel implementation first requires to determine the sequential code and the data-parallel operations of a algorithm. In fact, all the operations that are easy to execute in parallel must be made by the GPU to accelerate the execution, like the steps 3 and 4. On the other hand, all the sequential operations and the operations that have data dependencies between CUDA threads or recursive computations must be executed by only one CUDA thread or a CPU thread (the steps 1 and 2).\LZK{La méthode est déjà mal présentée, dans ce cas c'est encore plus difficile de comprendre que représentent ces différentes étapes!} Initially, we specify the organization of parallel threads by specifying the dimension of the grid \verb+Dimgrid+, the number of blocks per grid \verb+DimBlock+ and the number of threads per block.
-
-The code is organized as kernels which are parts of code that are run on GPU devices. For step 3, there are two kernels, the first is 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 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.
-\LZK{Vaut mieux expliquer l'implémentation en faisant référence à l'algo séquentiel que de parler des différentes steps.}
-
+%\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.
+Next, all the data-parallel arithmetic operations inside the main loop
+\verb=(while(...))= are executed as kernels by the GPU. The
+first kernel named \textit{Kernelsave} in line 5 of Algorithm~\ref{alg1-cuda} consists in saving the vector of polynomial roots found at the previous time-step in GPU memory, in
+order to check the convergence of the roots after each iteration (line
+7, Algorithm~\ref{alg1-cuda}). Then the new roots with the
+new iterations are computed using the EA method with a Gauss-Seidel
+iteration mode in order to use the latest updated roots (line
+6). This improves the convergence. This kernel is, in practice, very
+long since it performs all the operations with complex numbers with
+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.
+%}
+
+
+ %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
+%by initializing the polynomial and its derivative (line 1). The
+%initialization of the roots is performed (line 2). Data are transferred
+%%from the CPU to the GPU (after the allocation of the required memory on
+%the GPU) (line 3). Then at each iteration, if the error is greater
+%%than the threshold, the following operations are performed. The previous
+%roots are saved using a kernel (line 5). Then the new roots with the
+%new iterations are computed using the EA method with a Gauss-Seidel
+%iteration mode in order to use the latest updated roots (line
+%6). This improves the convergence. This kernel is, in practice, very
+%long since it performs all the operations with complex numbers with
+%the normal mode of the EA method but also with the
+%logarithm-exponential one. Then the error is computed with a final
+%kernel (line 7). Finally when the EA method has converged, the roots
+%are transferred from the GPU to the CPU.
\begin{algorithm}[htpb]
\label{alg1-cuda}
\LinesNumbered
\SetAlgoNoLine
-\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
-
-Initialization of P\;
-Initialization of Pu\;
-Initialization of the solution vector $Z^{0}$\;
-Allocate and copy initial data to the GPU global memory\;
-\While {$\Delta z_{max} > \epsilon$}{
- $ ZPres=kernel\_save(Z)$\;
- $ Z=kernel\_update(Z,P,Pu)$\;
- $\Delta z_{max}=kernel\_testConv(Z,ZPrec)$\;
-
+\caption{Finding roots of polynomials with the Ehrlich-Aberth method on a GPU}
+\KwIn{ $\epsilon$ (tolerance threshold)}
+\KwOut{$Z$ (solution vector of roots)}
+Initialize the polynomial $P$ and its derivative $P'$\;
+Set the initial values of vector $Z$\;
+Copy $P$, $P'$ and $Z$ from CPU to GPU\;
+\While{$error > \epsilon$}{
+ $Z^{prev}$ = KernelSave($Z$)\;
+ $Z$ = KernelUpdate($P,P',Z$)\;
+ $error$ = KernelComputeError($Z,Z^{prev}$)\;
}
-Copy results from GPU memory to CPU memory\;
+Copy $Z$ from GPU to CPU\;
\end{algorithm}
-\RC{Si l'algo vous convient, il faudrait le détailler précisément}
+\ \\
+This figure shows the second kernel code
+\begin{figure}[htbp]
+\centering
+\includegraphics[angle=+0,width=0.5\textwidth]{code}
+\caption{The Kernel Update code}
+\label{fig:00}
+\end{figure}
+
+%We noticed that the code is executed by a large number of GPU threads organized as grid of to dimension (Number of block per grid (NbBlock), number of threads per block(Nbthread)), the Nbthread is fixed initially, the NbBlock is computed as fallow:
+%$ NbBlocks= \frac{N+Nbthreads-1}{Nbthreads} where N: the number of root$
+%the such that each thread in grid is in charge of the computation of one root.
-\section{The EA algorithm on Multiple GPUs}
+The development of this code is a rather long task due to the
+development of all the kernels that compute the parts ported on the
+GPU. 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 section the GPU parallel implementation of the
+Ehrlich-Aberth method with OpenMP and MPI is presented.
+
+\section{The Ehrlich-Aberth algorithm on multiple GPUs}
\label{sec4}
-\subsection{an OpenMP-CUDA approach}
+\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. 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 will be responsible to
-update its owns part of the vector Z. This part is call $Z_{loc}$ in
-the following. 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.
+OpenMP and CUDA programming model. This algorithm is presented in
+Algorithm~\ref{alg2-cuda-openmp}. All the data are shared with OpenMP
+among all the OpenMP threads. The shared data are the solution vector
+$Z$, the polynomial to solve $P$, its derivative $P'$, and the error
+vector $error$. 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 will be
+responsible for updating its own part of the vector $Z$. This part is
+called $Z_{loc}$ in the following. 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 and the
-computation of the local size for each GPU is performed (lines 5-7 in
-Algo\ref{alg2-cuda-openmp}). Each thread starts by copying all the
-previous roots inside its GPU (line 9). Then each GPU will copy the
-previous roots (line 10) and it will compute an iteration of the EA
-method on its own roots (line 11). For that all the other roots are
-used. The convergence is checked on the new roots (line 12). At the end
-of an iteration, the updated roots are copied from the GPU to the
-CPU (line 14) by direcly updating 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
- % \includegraphics[angle=-90,width=0.5\textwidth]{OpenMP-CUDA}
-%\caption{The OpenMP-CUDA architecture}
-%\label{fig:03}
-%\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:
-
-%% \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{algorithm}[h]
-\label{alg2-cuda-openmp}
-\LinesNumbered
-\SetAlgoNoLine
-\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)}
+followings steps. First, roots are shared with OpenMP and the
+computation of the local size for each GPU is performed (line 4). Each
+thread starts by copying all the previous roots inside its GPU (line
+5). At each iteration, the following operations are performed. First
+the vector $Z$ is transferred from the CPU to the GPU (line 7). Each
+GPU copies the previous roots (line 8) and it computes an iteration of
+the EA method on its own roots (line 9). For that all the other roots
+are used. The local error is computed on the new roots (line 10) and
+the maximum of the local errors is computed on all OpenMP threads (line 11). At
+the end of an iteration, the updated roots are copied from the GPU to
+the CPU (line 12) and each CPU directly updates its own roots in the shared
+memory arrays containing all the roots.
-\KwOut {$Z$ ( Root's vector), $ZPrec$ (Previous root's vector)}
-\BlankLine
-Initialization of P\;
-Initialization of Pu\;
-Initialization of the solution vector $Z^{0}$\;
-Start of a parallel part with OpenMP (Z, $\Delta z$, P are shared variables)\;
-gpu\_id=cudaGetDevice()\;
-Allocate memory on GPU\;
-Compute local size and offet according to gpu\_id\;
-\While {$error > \epsilon$}{
- copy Z from CPU to GPU\;
-$ ZPrec_{loc}=kernel\_save(Z_{loc})$\;
-$ Z_{loc}=kernel\_update(Z,P,Pu)$\;
-$\Delta z[gpu\_id] = kernel\_testConv(Z_{loc},ZPrec_{loc})$\;
-$ error= Max(\Delta z)$\;
- copy $Z_{loc}$ from GPU to Z in CPU
+\begin{algorithm}[htpb]
+\label{alg2-cuda-openmp}
+\LinesNumbered
+\SetAlgoNoLine
+\caption{Finding roots of polynomials with the Ehrlich-Aberth method on multiple GPUs using OpenMP}
+\KwIn{ $\epsilon$ (tolerance threshold)}
+\KwOut{$Z$ (solution vector of roots)}
+Initialize the polynomial $P$ and its derivative $P'$\;
+Set the initial values of vector $Z$\;
+Start of a parallel part with OpenMP ($Z$, $error$, $P$, $P'$ are shared variables)\;
+Determine the local part of the OpenMP thread\;
+Copy $P$, $P'$ from CPU to GPU\;
+\While{$error > \epsilon$}{
+ Copy $Z$ from CPU to GPU\;
+ $Z^{prev}_{loc}$ = KernelSave($Z_{loc}$)\;
+ $Z_{loc}$ = KernelUpdate($P,P',Z$)\;
+ $error_{loc}$ = KernelComputeError($Z_{loc},Z^{prev}_{loc}$)\;
+ $error = max(error_{loc})$\;
+ Copy $Z_{loc}$ from GPU to $Z$ in CPU\;
}
\end{algorithm}
-\subsection{an MPI-CUDA approach}
-%\begin{figure}[htbp]
-%\centering
- % \includegraphics[angle=-90,width=0.2\textwidth]{MPI-CUDA}
-%\caption{The MPI-CUDA architecture }
-%\label{fig:03}
-%\end{figure}
-Our parallel implementation of EA to find root of polynomials using a CUDA-MPI approach is a data parallel approach. It splits input data of the polynomial to solve among MPI processes. In Algorithm \ref{alg2-cuda-mpi}, input data are the polynomial to solve $P$, the solution vector $Z$, the previous solution vector $ZPrev$, and the value of errors of stop condition $\Delta z$. Let $p$ denote the number of MPI processes on and $n$ the degree of the polynomial to be solved. The algorithm performs a simple data partitioning by creating $p$ portions, of at most $\lceil n/p \rceil$ roots to find per MPI process, for each $Z$ and $ZPrec$. Consequently, each MPI process of rank $k$ will have its own solution vector $Z_{k}$ and $ZPrec$, the error related to the stop condition $\Delta z_{k}$, enabling each MPI process to compute $\lceil n/p \rceil$ roots.
-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.
+
+\subsection{A MPI-CUDA approach}
+Our parallel implementation of EA to find the roots of polynomials using a
+CUDA-MPI approach follows a similar approach to the one used in
+CUDA-OpenMP. Each processor is responsible for computing its own part of
+roots using all the roots computed by other processors at the previous
+iteration. The difference between both approaches lies in the way
+processors communicate and exchange data. With MPI, processors need to
+send and receive data explicitly. So in
+Algorithm~\ref{alg2-cuda-mpi}, after the initialization phase all the
+processors have the same $Z$ vector. Then they need to compute the
+parameters used by the $MPI\_AlltoAll$ routines (line 4). In practice,
+each processor needs to compute its offset and its local
+size. Processors need to allocate memory on their GPU and need to copy
+their data on the GPU (line 5). At the beginning of each iteration, a
+processor starts by transferring the whole vector $Z$ from the CPU to the
+GPU (line 7). Only the local part of $Z^{prev}$ is saved (line
+8). After that, a processor is able to compute an updated version of
+its own roots (line 9) with the EA method. The local error is computed
+(line 10) and the global error is also computed using $MPI\_Reduce$ (line 11). Then
+the local roots are transferred from the GPU memory to the CPU memory
+(line 12) before being exchanged between all processors (line 13) in
+order to give to all processors the last version of the roots (with
+the $MPI\_AlltoAll$ routine). If the convergence is not satisfied, a
+new iteration is executed.
\begin{algorithm}[htpb]
\label{alg2-cuda-mpi}
-%\LinesNumbered
-\caption{CUDA-MPI Algorithm to find roots with the Ehrlich-Aberth method}
+\LinesNumbered
+\SetAlgoNoLine
+\caption{Finding roots of polynomials with the Ehrlich-Aberth method on multiple GPUs using MPI}
-\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)}
+\KwIn{ $\epsilon$ (tolerance threshold)}
-\KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}
+\KwOut {$Z$ (solution vector of roots)}
\BlankLine
-Initialization of P\;
-Initialization of Pu\;
-Initialization of the solution vector $Z^{0}$\;
-Distribution of Z\;
-Allocate memory to GPU\;
+Initialize the polynomial $P$ and its derivative $P'$\;
+Set the initial values of vector $Z$\;
+Determine the local part of the MPI process\;
+Computation of the parameters for the $MPI\_AlltoAll$\;
+Copy $P$, $P'$ from CPU to GPU\;
\While {$error > \epsilon$}{
-copy Z from CPU to GPU\;
-$ZPrec_{loc}=kernel\_save(Z_{loc})$\;
-$Z_{loc}=kernel\_update(Z,P,Pu)$\;
-$\Delta z=kernel\_testConv(Z_{loc},ZPrec_{loc})$\;
-$error=MPI\_Reduce(\Delta z)$\;
-Copy $Z_{loc}$ from GPU to CPU\;
-$Z=MPI\_AlltoAll(Z_{loc})$\;
+ Copy $Z$ from CPU to GPU\;
+ $Z^{prev}_{loc}$ = KernelSave($Z_{loc}$)\;
+ $Z_{loc}$ = KernelUpdate($P,P',Z$)\;
+ $error_{loc}$ = KernelComputeError($Z_{loc},Z^{prev}_{loc}$)\;
+ $error=MPI\_Reduce(error_{loc})$\;
+ Copy $Z_{loc}$ from GPU to CPU\;
+ $Z=MPI\_AlltoAll(Z_{loc})$\;
}
\end{algorithm}
We study two categories of polynomials: sparse polynomials and full polynomials.\\
{\it A sparse polynomial} is a polynomial for which only some coefficients are not null. In this paper, we consider sparse polynomials for which the roots are distributed on 2 distinct circles:
\begin{equation}
- \forall \alpha_{1} \alpha_{2} \in C,\forall n_{1},n_{2} \in N^{*}; P(z)= (z^{n_{1}}-\alpha_{1})(z^{n_{2}}-\alpha_{2})
+ \forall \alpha_{1} \alpha_{2} \in \mathbb{C},\forall n_{1},n_{2} \in \mathbb{N}^{*}; p(z)= (z^{n_{1}}-\alpha_{1})(z^{n_{2}}-\alpha_{2})
\end{equation}\noindent
{\it A full polynomial} is, in contrast, a polynomial for which all the coefficients are not null. A full polynomial is defined by:
-%%\begin{equation}
- %%\forall \alpha_{i} \in C,\forall n_{i}\in N^{*}; P(z)= \sum^{n}_{i=1}(z^{n^{i}}.a_{i})
-%%\end{equation}
\begin{equation}
- {\Large \forall a_{i} \in C, i\in N; p(x)=\sum^{n}_{i=0} a_{i}.x^{i}}
+ {\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 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
+\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
-sizes. All experimental results obtained are perfomed with double
-precision float data and the convergence threshold of the EA method is
+degrees. All experimental results obtained are performed with double
+precision floating-point 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}.
-
+the methods are given by the Guggenheimer method~\cite{Gugg86}.
-\subsection{Evaluation of the CUDA-OpenMP approach}
+\subsection{Evaluation of the multi-GPUs approaches}
+In this part, we evaluate the performances of the CUDA-OpenMP and
+CUDA-MPI approaches of the EA algorithm on different GPU platforms
+composed each of 1, 2, 3 or 4 GPUs. In this experiments we report the
+experimental results of the EA algorithms to find the roots of different sparse and full polynomials of high degrees ranging from 100,000 to 1,400,000. Figures~\ref{fig:01} and~\ref{fig:02} show the execution times to solve, respectively, sparse and full polynomials with the CUDA-OpenMP algorithm, and Figures~\ref{fig:03} and~\ref{fig:04} show those to solve, respectively, sparse and full polynomials with the CUDA-MPI algorithm.
-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-GPUs with (2,3,4) GPUs, for different sparse polynomial degrees ranging from 100,000 to 1,400,000.
+All these figures show that the CUDA-OpenMP and the CUDA-MPI approaches of the EA algorithm, compared to the single GPU version, are efficient and scale well with multiple GPUs. Both approaches allow us to solve sparse and full polynomials of very high degrees. Using 4 GPUs allows us to achieve a quasi-linear speedup.
\begin{figure}[htbp]
\centering
- \includegraphics[angle=-90,width=0.5\textwidth]{Sparse_omp}
-\caption{Execution time in seconds of the Ehrlich-Aberth method to
- solve sparse polynomials on multiple GPUs with CUDA-OpenMP.}
+\includegraphics[angle=-90,width=0.5\textwidth]{Sparse_omp}
+\caption{Execution times in seconds of the Ehrlich-Aberth method to solve sparse polynomials on multiple GPUs with CUDA-OpenMP.}
\label{fig:01}
\end{figure}
-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 times of the EA method to solve full polynomials on multiple GPUs}
-
-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 to
- solve full polynomials on multiple GPUs with CUDA-OpenMP.}
+\includegraphics[angle=-90,width=0.5\textwidth]{Full_omp}
+\caption{Execution times in seconds of the Ehrlich-Aberth method to solve full polynomials on multiple GPUs with CUDA-OpenMP.}
\label{fig:02}
\end{figure}
-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{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 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 to
- solve sparse polynomials on multiple GPUs with CUDA-MPI.}
+ \caption{Execution times in seconds of the Ehrlich-Aberth method to solve sparse polynomials on multiple GPUs with CUDA-MPI.}
\label{fig:03}
-\end{figure}
-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}
+ \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 multiple GPUs with CUDA-MPI.}
-\label{fig:04}
-\end{figure}
+ \centering
+ \includegraphics[angle=-90,width=0.5\textwidth]{Full_mpi}
+ \caption{Execution times in seconds of the Ehrlich-Aberth method for full polynomials on multiple GPUs with CUDA-MPI.}
+ \label{fig:04}
+ \end{figure}
-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{Comparison of the CUDA-OpenMP and the CUDA-MPI approaches}
+\subsection{Comparison between the CUDA-OpenMP and the CUDA-MPI approaches}
+In the previous section we saw that both approaches are very efficient to reduce the execution times to solve sparse and full polynomials. In this section we try to compare these two approaches. In this experiment three sparse polynomials and three full polynomials of degrees 200,000, 800,000 and 1,400,000 are investigated. Figures~\ref{fig:05} and~\ref{fig:06} show the comparison between CUDA-OpenMP and CUDA-MPI algorithms of the EA method to solve sparse and full polynomials, respectively.
-In the previuos section we saw that both approches are very effecient
-to reduce the execution times the sparse and full polynomials. In
-this section we try to compare these two approaches.
-
-\subsubsection{Solving sparse polynomials}
-In this experiment three sparse polynomials of size 200K, 800K and 1,4M are investigated.
\begin{figure}[htbp]
\centering
\includegraphics[angle=-90,width=0.5\textwidth]{Sparse}
-\caption{Execution times to solvs sparse polynomials of three
- distinct sizes on multiple GPUs using MPI and OpenMP with the
- Ehrlich-Aberth method}
+\caption{Execution times to solve sparse polynomials of three distinct degrees on multiple GPUs using OpenMP and MPI with the Ehrlich-Aberth method}
\label{fig:05}
\end{figure}
-In Figure~\ref{fig:05} there is one curve for CUDA-MPI and another one
-for CUDA-OpenMP. We can see that the results are quite similar between
-OpenMP and MPI for the polynomials size of 200K. For the size of 800K,
-the MPI version is a little bit 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]
\centering
\includegraphics[angle=-90,width=0.5\textwidth]{Full}
-\caption{Execution time for solving full polynomials of three distinct sizes on multiple GPUs using MPI and OpenMP approaches using Ehrlich-Aberth}
+\caption{Execution times to solve full polynomials of three distinct degrees on multiple GPUs using OpenMP and MPI with the Ehrlich-Aberth method}
\label{fig:06}
\end{figure}
-In Figure~\ref{fig:06}, we can see that when it comes to full polynomials, both approaches are almost equivalent.
-\subsubsection{Solving sparse and full polynomials of the same size with CUDA-MPI}
+In Figure~\ref{fig:05} there is one curve for CUDA-OpenMP and another one for CUDA-MPI for each polynomial investigated. We can see that the results are quite similar between OpenMP and MPI for the polynomial degree of 200K. For the degree of 800K, the MPI version is a little bit slower than the OpenMP version but for the degree of 1,4 million, there is a slight advantage for the MPI version. In Figure~\ref{fig:06}, we can see that when it comes to full polynomials, both approaches are almost equivalent.
+
+
+\subsection{Solving sparse and full polynomials of the same degree on multiple GPUs}
+In this experiment we compare the execution times of the EA algorithm
+according to the number of GPUs to solve sparse and full polynomials
+on multiple GPUs using OpenMP or MPI approaches. We chose three sparse
+and three full polynomials of degrees 200,000, 800,000 and
+1,400,000. Figures~\ref{fig:07} and~\ref{fig:08} show the execution
+times to solve sparse and full polynomials of the same degrees with
+the CUDA-OpenMP version and the CUDA-MPI version, respectively.
-In this experiment we compare the execution time of the EA algorithm
-according to the number of GPUs to solve sparse and full
-polynomials on multiples GPUs using MPI. We chose three sparse and full
-polynomials of size 200K, 800K and 1,4M.
\begin{figure}[htbp]
\centering
- \includegraphics[angle=-90,width=0.5\textwidth]{MPI}
-\caption{Execution times to solve sparse and full polynomials of three distinct sizes on multiple GPUs using MPI.}
+ \includegraphics[angle=-90,width=0.5\textwidth]{OMP}
+\caption{Execution times to solve sparse and full polynomials of three distinct degrees on multiple GPUs using OpenMP.}
\label{fig:07}
\end{figure}
-In Figure~\ref{fig:07} we can see that CUDA-MPI can solve sparse and
-full polynomials of high degrees, the execution times with sparse
-polynomial are very low compared to full polynomials. With sparse
-polynomials the number of monomials is reduced, consequently the number
-of operations is reduced and the execution time decreases.
-
-\subsubsection{Solving sparse and full polynomials of the same size with CUDA-OpenMP}
\begin{figure}[htbp]
\centering
- \includegraphics[angle=-90,width=0.5\textwidth]{OMP}
-\caption{Execution time for solving sparse and full polynomials of three distinct sizes on multiple GPUs using OpenMP}
+ \includegraphics[angle=-90,width=0.5\textwidth]{MPI}
+\caption{Execution times to solve sparse and full polynomials of three distinct degrees on multiple GPUs using MPI.}
\label{fig:08}
\end{figure}
-Figure ~\ref{fig:08} shows the impact of sparsity on the effectiveness of the CUDA-OpenMP approach. We can see that the impact follows the same pattern, a difference in execution time in favor of the sparse polynomials.
+In Figure~\ref{fig:07} the execution times of the CUDA-OpenMP version to solve sparse polynomials are very low compared to those to solve full polynomials. With sparse polynomials the number of monomials is reduced, consequently the number of operations is reduced and the execution time decreases. Figure~\ref{fig:08} shows the impact of sparsity on the efficiency of the CUDA-MPI approach. We can see that the impact follows the same pattern, a difference in execution times in favor of the sparse polynomials.
+
\subsection{Scalability of the EA method on multiple GPUs to solve very high degree polynomials}
-These experiments report the execution times of the EA method for
-sparse and full polynomials ranging from 1,000,000 to 5,000,000.
+These experiments report the execution times of the EA method for sparse and full polynomials of high degrees ranging from 1,000,000 to 5,000,000. In Figure~\ref{fig:09} we can see that both approaches (CUDA-OpenMP and CUDA-MPI) are scalable and can solve very high degree polynomials. In addition, with full polynomial as well as sparse ones, both approaches give very similar results.
+
\begin{figure}[htbp]
\centering
\includegraphics[angle=-90,width=0.5\textwidth]{big}
- \caption{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials of high degree on 4 GPUs for sizes ranging from 1M to 5M}
+ \caption{Execution times in seconds of the Ehrlich-Aberth method to solve sparse and full polynomials of high degree on 4 GPUs for degrees ranging from 1M to 5M}
\label{fig:09}
\end{figure}
-In Figure~\ref{fig:09} we can see that both approaches are scalable
-and can solve very high degree polynomials. In addition, with full polynomial as well as sparse ones, both
-approaches give very similar results.
-
-%SIDER JE viens de virer \c ca For sparse polynomials here are a noticeable difference in favour of MPI when the degree is
-%above 4 millions. Between 1 and 3 millions, OpenMP is more effecient.
-%Under 1 million, OpenMPI and MPI are almost equivalent.
-
-%SIDER : il faut une explication sur les différences ici aussi.
-%for sparse and full polynomials
-% An example of a floating figure using the graphicx package.
-% Note that \label must occur AFTER (or within) \caption.
-% For figures, \caption should occur after the \includegraphics.
-% Note that IEEEtran v1.7 and later has special internal code that
-% is designed to preserve the operation of \label within \caption
-% even when the captionsoff option is in effect. However, because
-% of issues like this, it may be the safest practice to put all your
-% \label just after \caption rather than within \caption{}.
-%
-% Reminder: the "draftcls" or "draftclsnofoot", not "draft", class
-% option should be used if it is desired that the figures are to be
-% displayed while in draft mode.
-%
-%\begin{figure}[!t]
-%\centering
-%\includegraphics[width=2.5in]{myfigure}
-% where an .eps filename suffix will be assumed under latex,
-% and a .pdf suffix will be assumed for pdflatex; or what has been declared
-% via \DeclareGraphicsExtensions.
-%\caption{Simulation results for the network.}
-%\label{fig_sim}
-%\end{figure}
-
-% Note that the IEEE typically puts floats only at the top, even when this
-% results in a large percentage of a column being occupied by floats.
-
-
-% An example of a double column floating figure using two subfigures.
-% (The subfig.sty package must be loaded for this to work.)
-% The subfigure \label commands are set within each subfloat command,
-% and the \label for the overall figure must come after \caption.
-% \hfil is used as a separator to get equal spacing.
-% Watch out that the combined width of all the subfigures on a
-% line do not exceed the text width or a line break will occur.
-%
-%\begin{figure*}[!t]
-%\centering
-%\subfloat[Case I]{\includegraphics[width=2.5in]{box}%
-%\label{fig_first_case}}
-%\hfil
-%\subfloat[Case II]{\includegraphics[width=2.5in]{box}%
-%\label{fig_second_case}}
-%\caption{Simulation results for the network.}
-%\label{fig_sim}
-%\end{figure*}
-%
-% Note that often IEEE papers with subfigures do not employ subfigure
-% captions (using the optional argument to \subfloat[]), but instead will
-% reference/describe all of them (a), (b), etc., within the main caption.
-% Be aware that for subfig.sty to generate the (a), (b), etc., subfigure
-% labels, the optional argument to \subfloat must be present. If a
-% subcaption is not desired, just leave its contents blank,
-% e.g., \subfloat[].
-
-
-% An example of a floating table. Note that, for IEEE style tables, the
-% \caption command should come BEFORE the table and, given that table
-% captions serve much like titles, are usually capitalized except for words
-% such as a, an, and, as, at, but, by, for, in, nor, of, on, or, the, to
-% and up, which are usually not capitalized unless they are the first or
-% last word of the caption. Table text will default to \footnotesize as
-% the IEEE normally uses this smaller font for tables.
-% The \label must come after \caption as always.
-%
-%\begin{table}[!t]
-%% increase table row spacing, adjust to taste
-%\renewcommand{\arraystretch}{1.3}
-% if using array.sty, it might be a good idea to tweak the value of
-% \extrarowheight as needed to properly center the text within the cells
-%\caption{An Example of a Table}
-%\label{table_example}
-%\centering
-%% Some packages, such as MDW tools, offer better commands for making tables
-%% than the plain LaTeX2e tabular which is used here.
-%\begin{tabular}{|c||c|}
-%\hline
-%One & Two\\
-%\hline
-%Three & Four\\
-%\hline
-%\end{tabular}
-%\end{table}
-
-
-% Note that the IEEE does not put floats in the very first column
-% - or typically anywhere on the first page for that matter. Also,
-% in-text middle ("here") positioning is typically not used, but it
-% is allowed and encouraged for Computer Society conferences (but
-% not Computer Society journals). Most IEEE journals/conferences use
-% top floats exclusively.
-% Note that, LaTeX2e, unlike IEEE journals/conferences, places
-% footnotes above bottom floats. This can be corrected via the
-% \fnbelowfloat command of the stfloats package.
-
-
-
\section{Conclusion}
\label{sec6}
-In this paper, we have presented a parallel implementation of
-Ehrlich-Aberth algorithm to solve full and sparse polynomials, on
-single GPU with CUDA and on multiple GPUs using two parallel
-paradigms: shared memory with OpenMP and distributed memory with
-MPI. These architectures were addressed by a CUDA-OpenMP approach and
-CUDA-MPI approach, respectively. Experiments show that, using
-parallel programming model like (OpenMP, MPI). We can efficiently
-manage multiple graphics cards to solve the same
-problem and accelerate the parallel execution with 4 GPUs and solve a
-polynomial of degree up to 5,000,000, four times faster than on single
-GPU.
-
-
-%In future, we will evaluate our parallel implementation of Ehrlich-Aberth algorithm on other parallel programming model
-
-Our next objective is to extend the model presented here with clusters
-of GPU nodes, with a three-level scheme: inter-node communication via
-MPI processes (distributed memory), management of multi-GPU node by
-OpenMP threads (shared memory).
-
-%present a communication approach between multiple GPUs. The comparison between MPI and OpenMP as GPUs controllers shows that these
-%solutions can effectively manage multiple graphics cards to work together
-%to solve the same problem
-
-
- %than we have presented two communication approach between multiple GPUs.(CUDA-OpenMP) approach and (CUDA-MPI) approach, in the objective to manage multiple graphics cards to work together and solve the same problem. in the objective to manage multiple graphics cards to work together and solve the same problem.
-
-
+In this paper, we have presented parallel implementations of the Ehrlich-Aberth algorithm to solve full and sparse polynomials, on a single GPU with CUDA and on multiple GPUs using two parallel paradigms: shared memory with OpenMP and distributed memory with MPI. These architectures were addressed by a CUDA-OpenMP approach and CUDA-MPI approach, respectively. Experiments show that, using parallel programming model like OpenMP or MPI, we can efficiently manage multiple graphics cards to solve the same problem and accelerate the parallel execution with 4 GPUs and solve a polynomial of degree up-to 5,000,000 four times faster than on a single GPU.
+Our next objective is to extend the model presented here to clusters of GPU nodes, with a three-level scheme: inter-node communications via MPI processes (distributed memory), management of multi-GPU nodes by OpenMP threads (shared memory). Actual platforms may probably also contain purely multi-core nodes without any GPU. This heterogeneous setting may lead to the integration of load balancing algorithms so as to allow an optimal use of hardware resources.
-% conference papers do not normally have an appendix
-
-% use section* for acknowledgment
\section*{Acknowledgment}
+This paper is partially funded by the Labex ACTION program (contract
+ANR-11-LABX-01-01). Computations have been performed on the supercomputer facilities of the Mésocentre de calcul de Franche-Comté. We also would like to thank Nvidia for hardware donation under CUDA Research Center 2014.
-Computations have been performed on the supercomputer facilities of
-the Mésocentre de calcul de Franche-Comté. We also would like to thank
-Nvidia for hardware donation under CUDA Research Center 2014.
-
-
-
-
-
-
-% trigger a \newpage just before the given reference
-% number - used to balance the columns on the last page
-% adjust value as needed - may need to be readjusted if
-% the document is modified later
-%\IEEEtriggeratref{8}
-% The "triggered" command can be changed if desired:
-%\IEEEtriggercmd{\enlargethispage{-5in}}
-% references section
-
-% can use a bibliography generated by BibTeX as a .bbl file
-% BibTeX documentation can be easily obtained at:
-% http://mirror.ctan.org/biblio/bibtex/contrib/doc/
-% The IEEEtran BibTeX style support page is at:
-% http://www.michaelshell.org/tex/ieeetran/bibtex/
-%\bibliographystyle{IEEEtran}
-% argument is your BibTeX string definitions and bibliography database(s)
-%\bibliography{IEEEabrv,../bib/paper}
-%\bibliographystyle{./IEEEtran}
\bibliography{mybibfile}
-%
-% <OR> manually copy in the resultant .bbl file
-% set second argument of \begin to the number of references
-% (used to reserve space for the reference number labels box)
-%\begin{thebibliography}{1}
-
-%\bibitem{IEEEhowto:kopka}
-%H.~Kopka and P.~W. Daly, \emph{A Guide to \LaTeX}, 3rd~ed.\hskip 1em plus
- % 0.5em minus 0.4em\relax Harlow, England: Addison-Wesley, 1999.
-
-%\bibitem{IEEEhowto:NVIDIA12}
- %NVIDIA Corporation, \textit{Whitepaper NVIDA’s Next Generation CUDATM Compute
-%Architecture: KeplerTM }, 1st ed., 2012.
-
-%\end{thebibliography}
-
-
-
-
-% that's all folks
\end{document}
