-
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-% 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}
-\begin{document}
-%
-% paper title
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-% 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{A parallel implementation of Ehrlich-Aberth algorithm for root finding of polynomials
-on Multi-GPU with OpenMP/MPI}
-
-
-% author names and affiliations
-% use a multiple column layout for up to three different
-% affiliations
-\author{\IEEEauthorblockN{Michael Shell}
-\IEEEauthorblockA{School of Electrical and\\Computer Engineering\\
-Georgia Institute of Technology\\
-Atlanta, Georgia 30332--0250\\
-Email: http://www.michaelshell.org/contact.html}
-\and
-\IEEEauthorblockN{Homer Simpson}
-\IEEEauthorblockA{Twentieth Century Fox\\
-Springfield, USA\\
-Email: homer@thesimpsons.com}
-\and
-\IEEEauthorblockN{James Kirk\\ and Montgomery Scott}
-\IEEEauthorblockA{Starfleet Academy\\
-San Francisco, California 96678--2391\\
-Telephone: (800) 555--1212\\
-Fax: (888) 555--1212}}
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-% conference papers do not typically use \thanks and this command
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-%Homer Simpson\IEEEauthorrefmark{2},
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-%Eldon Tyrell\IEEEauthorrefmark{4}}
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-%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}
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-%Telephone: (800) 555--1212, Fax: (888) 555--1212}
-%\IEEEauthorblockA{\IEEEauthorrefmark{4}Tyrell Inc., 123 Replicant Street, Los Angeles, California 90210--4321}}
+\usepackage{amsfonts}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage[textsize=footnotesize]{todonotes}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\newcommand{\LZK}[2][inline]{%
+ \todo[color=red!10,#1]{\sffamily\textbf{LZK:} #2}\xspace}
+\newcommand{\RC}[2][inline]{%
+ \todo[color=blue!10,#1]{\sffamily\textbf{RC:} #2}\xspace}
+\newcommand{\KG}[2][inline]{%
+ \todo[color=green!10,#1]{\sffamily\textbf{KG:} #2}\xspace}
+\newcommand{\AS}[2][inline]{%
+ \todo[color=orange!10,#1]{\sffamily\textbf{AS:} #2}\xspace}
-% use for special paper notices
-%\IEEEspecialpapernotice{(Invited Paper)}
+\begin{document}
+\title{Two parallel implementations of Ehrlich-Aberth algorithm for root-finding of polynomials on multiple GPUs with OpenMP and MPI}
+\author{\IEEEauthorblockN{Kahina Ghidouche, Abderrahmane Sider }
+ \IEEEauthorblockA{Laboratoire LIMED\\
+ Faculté des sciences exactes\\
+ Université de Bejaia, 06000, Algeria\\
+Email: \{kahina.ghidouche,ar.sider\}@univ-bejaia.dz}
+\and
+\IEEEauthorblockN{Lilia Ziane Khodja, Raphaël Couturier}
+\IEEEauthorblockA{FEMTO-ST Institute\\
+ University of Bourgogne Franche-Comte, France\\
+Email: zianekhodja.lilia@gmail.com\\ raphael.couturier@univ-fcomte.fr}}
-% make the title area
\maketitle
-% As a general rule, do not put math, special symbols or citations
-% in the abstract
\begin{abstract}
-The abstract goes here.
+Finding roots of polynomials is a very important part of solving
+real-life problems but it is not so easy for polynomials of high
+degrees. In this paper, we present two different parallel algorithms
+of the Ehrlich-Aberth method to find roots of sparse and fully defined
+polynomials of high degrees. Both algorithms are based on CUDA
+technology to be implemented on multi-GPU computing platforms but each
+using different parallel paradigms: OpenMP or MPI. The experiments
+show a quasi-linear speedup by using up-to 4 GPU devices compared to 1
+GPU to find roots of polynomials of degree up-to 1.4
+million. Moreover, other experiments show it is possible to find roots
+of polynomials of degree up-to 5 millions.
\end{abstract}
-% no keywords
-
+\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 used in science and engineering to capture physical phenomena and to express any outcome in the form of a function of some unknown variables. Formally speaking, a polynomial $p(x)$ of degree \textit{n} having $n$ coefficients in the complex plane \textit{C} is :
-%%\begin{center}
-\begin{equation}
- {\Large p(x)=\sum_{i=0}^{n}{a_{i}x^{i}}}.
-\end{equation}
-%%\end{center}
-The root finding problem consists in finding the values of all the $n$ values of the variable $x$ for which \textit{p(x)} is nullified. Such values are called zeros of $p$. If zeros are $\alpha_{i},\textit{i=1,...,n}$ the $p(x)$ can be written as :
-\begin{equation}
- {\Large p(x)=a_{n}\prod_{i=1}^{n}(x-\alpha_{i}), a_{0} a_{n}\neq 0}.
-\end{equation}
-The problem of finding the roots of polynomials is encountered in different applications. Most of the numerical methods that deal with this problem are simultaneous ones. These methods start from the initial approximations of all the roots of the polynomial and give a sequence of approximations that converge to the roots of the polynomial. The first method of this group is Durand-Kerner method:
+Finding 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}
-\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, . . . , n,
+p(x) = \displaystyle\sum^n_{i=0}{\alpha_ix^i},\alpha_n\neq 0,
\end{equation}
-%%\end{center}
-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 and brought
-in the following form by Ehrlich~\cite{Ehrlich67} and
-Aberth~\cite{Aberth73} uses a different iteration formula given as:
-%%\begin{center}
+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}
-\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, . . . , n,
+ p(x)=\alpha_n\displaystyle\prod_{i=1}^n(x-z_i), \alpha_n\neq 0.
\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 is increased with the increasing of the degree of the polynomial. Many authors have treated the problem of implementation of simultaneous methods in parallel. Freeman [10] implemented and compared DK, EA and another method of the fourth order proposed by Farmer
-and Loizou [9], on a 8-processor linear chain, for polynomials of degree up to 8.
-The third method often diverges, but the first two methods have speed-up equal to 5.5. Later, Freeman and Bane [11] considered asynchronous algorithms, in which each processor continues to update its approximations even though the latest values of other $z^{k}_{i}$ 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. [12] proposed two methods of parallelization for a shared memory architecture with \textit{OpenMP} and for distributed memory one with \textit{MPI}. They were able to compute the roots of sparse polynomials of degree 10,000 in 116 seconds with \textit{OpenMP} and 135 seconds with \textit{MPI} only 8 personal computers and 2 communications per iteration. Comparing to the sequential implementation where it takes up to 3,300 seconds to obtain the same results, the authors show an interesting speedup.
-Very few works had been performed since this last work until the appearing of the Compute Unified Device Architecture (CUDA) [13], a parallel computing platform and a programming model invented by NVIDIA. The computing power of GPUs (Graphics Processing Unit) has exceeded that of CPUs. However, CUDA adopts a totally new computing architecture to use the hardware resources provided by GPU in order to offer a stronger computing ability to the massive data computing. Ghidouche and al [14] proposed an implementation of the Durand-Kerner method on GPU. Their main result 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 a parallel programming paradigms (OpenMP, MPI) on GPUs. We consider two architectures: Shared memory with OpenMP API based on threads from the same system process, which each thread is attached to one GPU and after the various memory allocation, each thread throws its part of calculation ( to do this you must first load on the GPU required data and after Suddenly repatriate the result on the host). Distributed memory with MPI: The MPI library is often used for parallel programming [11] in
-cluster systems because it is a message-passing programming language. Each GPU are attached to one process MPI, and a loop is in charge of the distribution of tasks between the MPI processes. this solution can be used on one GPU, or executed on a distributed cluster of GPUs, employing the Message Passing Interface (MPI) to communicate between separate CUDA cards. This solution permits scaling of the problem size to larger classes than would be possible on a single device and demonstrates the performance which users might expect from future
-HPC architectures where accelerators are deployed.
-
-This paper is organized as follows, in section 2 we recall the Ehrlich-Aberth method. In section 3 we present EA algorithm on single GPU. In section 4 we propose the EA algorithm implementation on MGPU for (OpenMP-CUDA) approach and (MPI-CUDA) approach. In section 5 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.
+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}.
+
+
+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, they have solved only polynomials not exceeding degrees of 20,000.
+
+%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.
+
+But the recent advent of the Compute Unified Device Architecture (CUDA)~\cite{CUDA15}, a parallel computing platform and a programming model invented by NVIDIA had revived parallel programming interest for this problem. Indeed, the computing power of GPUs (Graphics Processing Units) has exceeded that of traditional processors CPUs, 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 adopts a totally new computing architecture to use the hardware resources provided by the GPU in order to offer a stronger computing ability to the massive data computing. Ghidouche et al.~\cite{Kahinall14} proposed an implementation of the Durand-Kerner method on a single GPU. Their main results showed that a parallel CUDA implementation is about 10 times faster than the sequential implementation on a single CPU for sparse polynomials of degree 48,000.
+
+In this paper we propose the parallelization of Ehrlich-Aberth (EA) method which has a cubic convergence rate which is much better than the quadratic rate of the Durand-Kerner method which 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 synchronise. 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 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 millions.
+ \end{itemize}
+
+
+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.
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\section{Parallel Programmings Model}
+\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.
\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 allo-
-cated 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, in OpenMP a usage of threads ids and managing data explicitly as done in an MPI
-code can be considered, it defeats the advantages of OpenMP.
-
-%\subsection{OpenMP} %L'article en Français Programmation multiGPU – OpenMP versus MPI
-%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.
\subsection{MPI}
- The library MPI allows to use a distributed memory architecture. The various processes have their own environment of execution and execute their codes in a asynchronous way, according to the model MIMD (Multiple Instruction streams, Multiple Dated streams); they communicate and synchronize by exchanges of messages~\cite{Peter96}. MPI messages are explicitly sent, while the exchanges are implicit within the framework of a programming multi-thread (OpenMP/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.
-\subsection{CUDA}%L'article en anglais Multi-GPU and multi-CPU accelerated FDTD scheme for vibroacoustic applications
- CUDA (an acronym for Compute Unified Device Architecture) is a parallel computing architecture developed by NVIDIA~\cite{NVIDIA12}. The
-unit of execution in CUDA is called a thread. Each thread executes the kernel by the streaming processors in parallel. In CUDA,
-a group of threads that are executed together is called thread blocks, and the computational grid consists of a grid of thread
-blocks. Additionally, a thread block can use the shared memory on a single multiprocessor as while as 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.
-
-
-We introduced three paradigms of parallel programming. Our objective consist to implement an algorithm of root finding polynomial on multiple GPUs. It primordial to know how manage CUDA context of different GPUs. A direct method for controlling the various GPU is to use as many threads or processes that GPU. We can choose the GPU index based on the identifier of OpenMP thread or the rank of the MPI process. Both approaches will be created.
-
-\section{The EA algorithm on single GPU}
-\subsection{the EA method}
-
-A cubically convergent iteration method to find zeros of
-polynomials was proposed by O. Aberth~\cite{Aberth73}. The
-Ehrlich-Aberth method contains 4 main steps, presented in what
-follows.
+\subsection{CUDA}
+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 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.
-%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}
+\section{The Ehrlich-Aberth algorithm on a GPU}
+\label{sec3}
-%\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 $a_{i}$:
+\subsection{The Ehrlich-Aberth method}
+The Ehrlich-Aberth method is a simultaneous method~\cite{Aberth73} using the following iteration
\begin{equation}
-\label{eq:SimplePolynome}
- p(z)=\sum{a_{i}z^{n-i}} , a_{n} \neq 0,a_{0}=1, a_{i}\subset C
+\label{Eq:EA1}
+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,\ldots,n
\end{equation}
+This method contains 4 steps. The first step consists in the
+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
+Ehrlich-Aberth 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 $\epsilon$.
-\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-spaced 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}.
+\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}
-\subsubsection{Iterative Function}
-The operator used by the Aberth method is corresponding to the
-following equation~\ref{Eq:EA} which will enable the convergence towards
-polynomial solutions, provided all the roots are distinct.
-
-%Here we give a second form of the iterative function used by the Ehrlich-Aberth method:
+\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.
+
+In order to solve this problem, we propose to modify the iterative
+function by using the logarithm and the exponential of a complex and
+we propose a new version of the Ehrlich-Aberth method. This method
+allows us to exceed the computation of the polynomials of degree
+100,000 and to reach a degree up to more than 1,000,000. The reformulation of the iteration~(\ref{Eq:EA1}) of the Ehrlich-Aberth method with exponential and logarithm operators is defined as follows, for $i=1,\dots,n$:
\begin{equation}
-\label{Eq:EA}
-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
+\label{Log_H2}
+z^{k+1}_i = z_i^k - \exp(\ln(p(z_i^k)) - \ln(p'(z^k_i)) - \ln(1-Q(z^k_i))),
\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:
+where:
\begin{equation}
-\label{eq:Aberth-Conv-Cond}
-\forall i \in [1,n];\vert\frac{z_{i}^{k}-z_{i}^{k-1}}{z_{i}^{k}}\vert<\xi
+\label{Log_H1}
+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}
+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}. In practice, the exponential and
+logarithm mode is used when a root is outisde 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 polynimal.
+
+
+\subsection{The Ehrlich-Aberth parallel implementation on CUDA}
+The code is organized as kernels which are parts of code 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 allocation of the required memory on
+the GPU) (line 3). Then at each iteration, if the error is greater
+than a 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 lastest 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{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[...,...], wich will make it possible to converge to the roots solution, provided that all the root are different. At the end of each application of the iterative function, a stop condition is verified consists in stopping the iterative process when the whole of the modules of the roots
-%are lower than a fixed value $ε$
-
-
-\subsection{EA parallel implementation on CUDA}
-Like any parallel code, a GPU parallel implementation first
-requires to determine the sequential tasks and the
-parallelizable parts of the sequential version of the
-program/algorithm. In our case, all the operations that are easy
-to execute in parallel must be made by the GPU to accelerate
-the execution of the application, like the step 3 and step 4. On the other hand, all the
-sequential operations and the operations that have data
-dependencies between threads or recursive computations must
-be executed by only one CUDA or CPU thread (step 1 and step 2). Initially we specifies the organization of threads in parallel, need to specify the dimension of the grid Dimgrid: the number of block per grid and block by DimBlock: the number of threads per block required to process a certain task.
-
-we create the kernel, for step 3 we have two kernels, the
-first named \textit{save} is used to save vector $Z^{K-1}$ and the kernel
-\textit{update} is used to update the $Z^{K}$ vector. In step 4 a kernel is
-created to test the convergence of the method. In order to
-compute function H, we have two possibilities: either to use
-the Jacobi method, or the Gauss-Seidel method which uses the
-most recent computed roots. It is well known that the Gauss-
-Seidel mode converges more quickly. So, we used the Gauss-Seidel mode of iteration. To
-parallelize the code, we created kernels and many functions to
-be executed on the GPU for all the operations dealing with the
-computation on complex numbers and the evaluation of the
-polynomials. As said previously, we managed both functions
-of evaluation of a polynomial: the normal method, based on
-the method of Horner and the method based on the logarithm
-of the polynomial. All these methods were rather long to
-implement, as the development of corresponding kernels with
-CUDA is longer than on a CPU host. This comes in particular
-from the fact that it is very difficult to debug CUDA running
-threads like threads on a CPU host. In the following paragraph
-Algorithm 1 shows the GPU parallel implementation of Ehrlich-Aberth method.
-
-Algorithm~\ref{alg2-cuda} shows a sketch of the Ehrlich-Aberth method using CUDA.
-
-\begin{enumerate}
\begin{algorithm}[htpb]
\label{alg1-cuda}
-%\LinesNumbered
-\caption{CUDA Algorithm to find roots with the Ehrlich-Aberth method}
-
-\KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (Error tolerance
- threshold), P (Polynomial to solve), Pu (Derivative of P), $n$ (Polynomial degrees), $\Delta z_{max}$ (Maximum value of stop condition)}
+\LinesNumbered
+\SetAlgoNoLine
+\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 $Z$ from GPU to CPU\;
+\end{algorithm}
-\KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}
-%\BlankLine
+The development of this code is a rather long task, 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
+section the GPU parallel implementation of Ehrlich-Aberth method with
+OpenMP and MPI is presented.
-\item Initialization of the of P\;
-\item Initialization of the of Pu\;
-\item Initialization of the solution vector $Z^{0}$\;
-\item Allocate and copy initial data to the GPU global memory\;
-\item k=0\;
-\While {$\Delta z_{max} > \epsilon$}{
-\item Let $\Delta z_{max}=0$\;
-\item $ kernel\_save(ZPrec,Z)$\;
-\item k=k+1\;
-\item $ kernel\_update(Z,P,Pu)$\;
-\item $kernel\_testConverge(\Delta z_{max},Z,ZPrec)$\;
-}
-\item Copy results from GPU memory to CPU memory\;
-\end{algorithm}
-\end{enumerate}
-~\\
-\section{The EA algorithm on Multi-GPU}
-
-\subsection{MGPU (OpenMP-CUDA) approach}
-Our OpenMP-CUDA implementation of EA algorithm is based on the hybrid OpenMP and CUDA programming model. It works
-as follows.
-Based on the metadata, a shared memory is used to make data evenly shared among OpenMP threads. The shared data are the solution vector $Z$, the polynomial to solve $P$. vector of error of stop condition $\Delta z$. Let(T\_omp) number of OpenMP threads is equal to the number of GPUs, each threads OpenMP checks one GPU, and control a part of the shared memory, that is a part of the vector Z like: $(n/num\_gpu)$ roots, n: the polynomial's degrees, $num\_gpu$ the number of GPUs. Each OpenMP thread copies its data from host memory to GPU’s device memory.Than every GPU will have a grid of computation organized with its performances and the size of data of which it checks and compute kernels. %In principle a grid is set by two parameter DimGrid, the number of block per grid, DimBloc: the number of threads per block. The following schema shows the architecture of (CUDA,OpenMP).
-
-%\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:
-
-$num\_gpus$ thread OpenMP are created using \verb=omp_set_num_threads();=function (line,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}), than each OpenMP threads allocate and copy initial data from CPU memory to the GPU global memories, execute the kernels on GPU, and compute 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, OpenMP threads synchronize using \verb=#pragma omp barrier;= to recuperate all values of vector $\Delta z$, to compute the maximum stop condition in vector $\Delta z$(line 12, Algorithm \ref{alg2-cuda-openmp}).Finally,they copy the results from GPU memories to CPU memory. The OpenMP threads execute kernels until the roots converge sufficiently.
-\begin{enumerate}
-\begin{algorithm}[htpb]
-\label{alg2-cuda-openmp}
-%\LinesNumbered
-\caption{CUDA-OpenMP Algorithm to find roots with the Ehrlich-Aberth method}
-
-\KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (Error tolerance
- threshold), P (Polynomial to solve), Pu (Derivative of P), $n$ (Polynomial degrees), $\Delta z$ ( Vector of errors of stop condition), $num_gpus$ (number of OpenMP threads/ number of GPUs), $Size$ (number of roots)}
+\section{The Ehrlich-Aberth algorithm on multiple GPUs}
+\label{sec4}
+\subsection{An OpenMP-CUDA approach}
+Our OpenMP-CUDA implementation of EA algorithm is based on the hybrid
+OpenMP and CUDA programming model. This algorithm is presented in
+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 to update its owns 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 (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 max of the local errors is computed on all OpenMP threads (lien 11). At
+the end of an iteration, the updated roots are copied from the GPU to
+the CPU (line 12) by directly updating its own roots in the shared
+memory arrays containing all the roots.
-\KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}
-\BlankLine
-\item Initialization of the of P\;
-\item Initialization of the of Pu\;
-\item Initialization of the solution vector $Z^{0}$\;
-\verb=omp_set_num_threads(num_gpus);=
-\verb=#pragma omp parallel shared(Z,$\Delta$ z,P);=
-\verb=cudaGetDevice(gpu_id);=
-\item Allocate and copy initial data from CPU memory to the GPU global memories\;
-\item index= $Size/num\_gpus$\;
-\item k=0\;
-\While {$error > \epsilon$}{
-\item Let $\Delta z=0$\;
-\item $ kernel\_save(ZPrec,Z)$\;
-\item k=k+1\;
-\item $ kernel\_update(Z,P,Pu,index)$\;
-\item $kernel\_testConverge(\Delta z[gpu\_id],Z,ZPrec)$\;
-%\verb=#pragma omp barrier;=
-\item error= Max($\Delta z$)\;
+\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\;
}
-
-\item Copy results from GPU memories to CPU memory\;
\end{algorithm}
-\end{enumerate}
-~\\
-\subsection{Multi-GPU (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 the Ehrlich-Aberth method to find root polynomial using (CUDA-MPI) approach, splits input data of the polynomial to solve between MPI processes. From Algorithm 3, the 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 size of the polynomial to be solved. The algorithm performs a simple data partitioning by creating $p$ portions, of at most $⌈n/p⌉$ roots to find per MPI process, for each element mentioned above. Consequently, each MPI process $k$ will have its own solution vector $Z_{k}$,polynomial to be solved $p_{k}$, the error of stop condition $\Delta z_{k}$, Than each MPI processes compute only $⌈n/p⌉$ roots.
-Since a GPU works only on data of its memory, all local input data, $Z_{k}, p_{k}$ and $\Delta z_{k}$, must be transferred from CPU memories to the corresponding GPU memories. Afterward, the same EA algorithm (Algorithm 1) is run by all processes but on different sub-polynomial root $ p(x)_{k}=\sum_{i=k(\frac{n}{p})}^{k+1(\frac{n}{p})} a_{i}x^{i}, k=1,...,p$. Each processes MPI execute the loop \verb=(While(...)...do)= contain the kernels. Than each process MPI compute only his portion of roots indicated with variable \textit{index} initialized in (line 5, Algorithm \ref{alg2-cuda-mpi}), used as input data in the $kernel\_update$ (line 10, Algorithm \ref{alg2-cuda-mpi}). After each iteration, MPI processes synchronize using \verb=MPI_Allreduce= function, in order to compute the maximum error stops condition $\Delta z_{k}$ computed by each process MPI line (line, Algorithm\ref{alg2-cuda-mpi}), and copy the values of new roots computed from GPU memories to CPU memories, than communicate her results to the neighboring processes,using \verb=MPI_Alltoallv=. If maximum stop condition $error > \epsilon$ the processes stay to execute the loop \verb= while(...)...do= until all the roots converge sufficiently.
-\begin{enumerate}
+\subsection{A MPI-CUDA approach}
+Our parallel implementation of EA to find roots of polynomials using a
+CUDA-MPI approach follows a similar approach to the one used in
+CUDA-OpenMP. Each process is responsible to compute 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
+processes communicate and exchange data. With MPI, processors need to
+send and receive data explicitly. So in
+Algorithm~\ref{alg2-cuda-mpi}, after the initialization 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 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, an
+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
-\item Initialization of the P\;
-\item Initialization of the Pu\;
-\item Initialization of the solution vector $Z^{0}$\;
-\item Allocate and copy initial data from CPU memories to the GPU global memories\;
-\item $index= Size/num_gpus$\;
-\item k=0\;
+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 {$error > \epsilon$}{
-\item Let $\Delta z=0$\;
-\item $ kernel\_save(ZPrec,Z)$\;
-\item k=k+1\;
-\item $ kernel\_update(Z,P,Pu,index)$\;
-\item $kernel\_testConverge(\Delta z,Z,ZPrec)$\;
-\item ComputeMaxError($\Delta z$,error)\;
-\item Copy results from GPU memories to CPU memories\;
-\item Send $Z[id]$ to all neighboring processes\;
-\item Receive $Z[j]$ from neighboring process j\;
-
-
+ 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}
-\end{enumerate}
-~\\
-\section{experiments}
+
+\section{Experiments}
+\label{sec5}
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 tests, a CPU Intel(R) Xeon(R) CPU E5620@2.40GHz and a GPU K40 (with 6 Go of ram) are used.
-
-We performed a set of experiments on single GPU and Multi-GPU using (OpenMP/MPI) to find roots polynomials with EA algorithm, for both sparse and full polynomials of different sizes. We took into account the execution times and the polynomial size performed by sum or each experiment.
-All experimental results obtained from the simulations are made in
-double precision data, the convergence threshold of the methods is set
-to $10^{-7}$.
-%Since we were more interested in the comparison of the
-%performance behaviors of Ehrlich-Aberth and Durand-Kerner methods on
-%CPUs versus on GPUs.
-The initialization values of the vector solution
-of the methods are given in %Section~\ref{sec:vec_initialization}.
-
-\subsection{Test with Multi-GPU (CUDA OpenMP) approach}
-
-In this part we performed a set of experiments on Multi-GPU (CUDA OpenMP) approach for full and sparse polynomials of different degrees, compare it with Single GPU (CUDA).
- \subsubsection{Execution times in seconds of the Ehrlich-Aberth method for solving sparse polynomials on GPUs using shared memory paradigm with OpenMP}
-
- In this experiments we report the execution time of the EA algorithm, on single GPU and Multi-GPU with (2,3,4) GPUs, for different sparse polynomial degrees ranging from 100,000 to 1,400,000
-
-\begin{figure}[htbp]
-\centering
- \includegraphics[angle=-90,width=0.5\textwidth]{Sparse_omp}
-\caption{Execution times in seconds of the Ehrlich-Aberth method for solving sparse polynomials on GPUs using shared memory paradigm with OpenMP}
-\label{fig:01}
-\end{figure}
-This figure~\ref{fig:01} shows that (CUDA OpenMP) Multi-GPU approach reduce the execution time up to the scale 100 whereas single GPU is of scale 1000 for polynomial who exceed 1,000,000. It shows the advantage to use OpenMP parallel paradigm to connect the performances of several GPUs and solve a polynomial of high degrees.
+For our tests, 4 cards GPU Tesla Kepler K40 are used. In order to evaluate both the GPU and Multi-GPU approaches, we performed a set of experiments on a single GPU and multiple GPUs using OpenMP or MPI with the EA algorithm, for both sparse and full polynomials of different sizes. All experimental results obtained are performed with double precision float data and the convergence threshold of the EA method is set to $10^{-7}$. The initialization values of the vector solution of the methods are given by Guggenheimer method~\cite{Gugg86}.
-\subsubsection{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials on GPUs using shared memory paradigm with OpenMP}
+\subsection{Evaluation of the multi-GPUs approaches}
+Here 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 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.
-This experiments shows the execution time of the EA algorithm, on single GPU (CUDA) and Multi-GPU (CUDA OpenMP)approach for full polynomials of degrees ranging from 100,000 to 1,400,000
+All these figures show that the CUDA-OpenMP and 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]{Full_omp}
-\caption{Execution times in seconds of the Ehrlich-Aberth method for solving full polynomials on GPUs using shared memory paradigm with OpenMP}
-\label{fig:03}
+\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}
-The second test with full polynomial shows a very important saving of time, for a polynomial of degrees 1,4M (CUDA OpenMP) approach with 4 GPUs compute and solve it 4 times as fast as single GPU. We notice that curves are positioned one below the other one, more the number of used GPUs increases more the execution time decreases.
-
\begin{figure}[htbp]
\centering
- \includegraphics[angle=-90,width=0.5\textwidth]{Sparse_mpi}
-\caption{Execution times in seconds of the Ehrlich-Aberth method for solving sparse polynomials on GPUs using distributed memory paradigm with MPI}
+\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}
-
\begin{figure}[htbp]
\centering
+ \includegraphics[angle=-90,width=0.5\textwidth]{Sparse_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}
+
+\begin{figure}[htbp]
+ \centering
\includegraphics[angle=-90,width=0.5\textwidth]{Full_mpi}
-\caption{Execution times in seconds of the Ehrlich-Aberth method for full polynomials on GPUs using distributed memory paradigm with MPI}
-\label{fig:04}
-\end{figure}
+ \caption{Execution times in seconds of the Ehrlich-Aberth method for full polynomials on multiple GPUs with CUDA-MPI.}
+ \label{fig:04}
+ \end{figure}
+
+
+\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.
\begin{figure}[htbp]
\centering
- \includegraphics[angle=-90,width=0.5\textwidth]{Sparse}
-\caption{Comparaison between MPI and OpenMP versions of the Ehrlich-Aberth method for solving sparse plynomials on GPUs}
+ \includegraphics[angle=-90,width=0.5\textwidth]{Sparse}
+\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}
\begin{figure}[htbp]
\centering
- \includegraphics[angle=-90,width=0.5\textwidth]{Full}
-\caption{Comparaison between MPI and OpenMP versions of the Ehrlich-Aberth method for solving full polynomials on GPUs}
+ \includegraphics[angle=-90,width=0.5\textwidth]{Full}
+\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:05} there is one curve for CUDA-OpenMP and another one for CUDA-MPI. 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 millions, 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 CUDA-OpenMP version and CUDA-MPI version, respectively.
+
\begin{figure}[htbp]
\centering
- \includegraphics[angle=-90,width=0.5\textwidth]{MPI}
-\caption{Comparaison of execution times of the Ehrlich-Aberth method for solving sparse and full polynomials on GPUs with distributed memory paradigm 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}
\begin{figure}[htbp]
\centering
- \includegraphics[angle=-90,width=0.5\textwidth]{OMP}
-\caption{Comparaison of execution times of the Ehrlich-Aberth method for solving sparse and full polynomials on GPUs with shared memory paradigm 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}
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-% of issues like this, it may be the safest practice to put all your
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-% option should be used if it is desired that the figures are to be
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-%\begin{figure}[!t]
-%\centering
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-% 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}
-
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-% results in a large percentage of a column being occupied by floats.
-
-
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-% (The subfig.sty package must be loaded for this to work.)
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-% and the \label for the overall figure must come after \caption.
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-% 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*}
-%
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-% 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.
+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 effectiveness 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 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 to solve sparse and full polynomials of high degree on 4 GPUs for degrees ranging from 1M to 5M}
+\label{fig:09}
+\end{figure}
+
\section{Conclusion}
-The conclusion goes here~\cite{IEEEexample:bibtexdesign}.
-
-
-
+\label{sec6}
+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 single GPU.
-% conference papers do not normally have an appendix
+Our next objective is to extend the model presented here with 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).
-% 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.
-The authors would like to thank...
-
-
-
-
-
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-
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- % 0.5em minus 0.4em\relax Harlow, England: Addison-Wesley, 1999.
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-%\bibitem{IEEEhowto:NVIDIA12}
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-%Architecture: KeplerTM }, 1st ed., 2012.
-
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-% that's all folks
\end{document}
