spell checked

[kahina_paper2.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index c593d34d06505b112c59b80d2b2c991065b59835..0c1a417a55746967570e28255164a75e86f01925 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -39,17 +39,17 @@ Email: zianekhodja.lilia@gmail.com\\ raphael.couturier@univ-fcomte.fr}}
 \maketitle
 
 \begin{abstract}
 \maketitle
 
 \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}
 
 \begin{IEEEkeywords}
 \end{abstract}
 
 \begin{IEEEkeywords}


@@ -64,7 +64,7 @@ of polynomials of degree up-to 5 millions.
 \section{Introduction}
 
 
 \section{Introduction}
 
 

-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:
+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}{\alpha_ix^i},\alpha_n\neq 0, 
 \end{equation}
 \begin{equation}
 p(x) = \displaystyle\sum^n_{i=0}{\alpha_ix^i},\alpha_n\neq 0, 
 \end{equation}


@@ -73,10 +73,25 @@ where $\{\alpha_i\}_{0\leq i\leq n}$ are complex coefficients and $n$ is a high
  p(x)=\alpha_n\displaystyle\prod_{i=1}^n(x-z_i), \alpha_n\neq 0.
 \end{equation}
 
  p(x)=\alpha_n\displaystyle\prod_{i=1}^n(x-z_i), \alpha_n\neq 0.
 \end{equation}
 

-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,cs01:nj,Couturier02}, using several paradigms of parallelization (synchronous or asynchronous computations, mechanism of shared or distributed memory, etc). However, they have treated only polynomials not exceeding degrees of 20,000.
+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
 
 %The main problem of the simultaneous methods is that the necessary
 %time needed for the convergence increases with the increasing of the


@@ -100,21 +115,25 @@ The convergence time of simultaneous methods drastically increases with the incr
 %roots of sparse polynomials of degree 10,000. The authors showed an interesting
 %speedup that is 20 times as fast as the sequential implementation. 
 
 %roots of sparse polynomials of degree 10,000. The authors showed an interesting
 %speedup that is 20 times as fast as the sequential implementation. 
 

-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. 
+With 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  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 adopts a totally new computing architecture to use the hardware resources provided by the GPU in order to offer a stronger computing ability to the massive data computing. Ghidouche et al.~\cite{Kahinall14} proposed an implementation of the Durand-Kerner method on a single GPU. Their main results showed that a parallel CUDA implementation is about 10 times faster than the sequential implementation on a single CPU for sparse polynomials of degree 48,000. 

 
 

-In this paper we propose the parallelization of Ehrlich-Aberth method 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:
+In this paper we propose the parallelization of the 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 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}
 
  \begin{itemize}
 

-\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
+\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
   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.
+  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
 \item
   Our method is efficient to compute the roots of sparse and full

-  polynomials of degree up to 5 millions.
+  polynomials of degree up to 5 million.

  \end{itemize}
  \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.
+

 
 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. 
 
 
 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. 
 


@@ -124,16 +143,61 @@ The paper is organized as follows. In Section~\ref{sec2} we present three differ
  
 \section{Parallel programming models}
 \label{sec2}
  
 \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}
  
 \subsection{OpenMP}

-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} 
 
 \subsection{MPI} 

-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}
  
 \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.
+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.

 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


@@ -158,7 +222,7 @@ 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
 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
+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 $\epsilon$.
 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 $\epsilon$.


@@ -169,13 +233,13 @@ root modules are lower than a fixed value $\epsilon$.
 \end{equation}
 
 \subsection{Improving Ehrlich-Aberth method}
 \end{equation}
 
 \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.
+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
  
 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
 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 $i=1,\dots,n$:
+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}
 
 \begin{equation}
 \label{Log_H2}


@@ -193,27 +257,27 @@ 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
 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:
+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
 \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.
+\verb=double= value and $n$ is the degree of the polynomial.

 
 
 \subsection{The Ehrlich-Aberth parallel implementation on CUDA}
 
 
 \subsection{The Ehrlich-Aberth parallel implementation on CUDA}

-The code is organized as kernels which are parts of code that are run
+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
 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
+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
 the GPU) (line 3). Then at each iteration, if the error is greater

-than a threshold, the following operations are performed. The previous
+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
 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
+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
 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


@@ -244,7 +308,7 @@ 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
 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
+section the GPU parallel implementation of the Ehrlich-Aberth method with

 OpenMP and MPI is presented.
 
 
 OpenMP and MPI is presented.
 
 


@@ -262,13 +326,13 @@ $Z$, 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
 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
+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
 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
+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
 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


@@ -276,9 +340,9 @@ 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
 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 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 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
+the CPU (line 12) and each CPU directly updates its own roots in the shared

 memory arrays containing all the roots.
 
 
 memory arrays containing all the roots.
 
 


@@ -310,14 +374,14 @@ Copy $P$, $P'$ from CPU to GPU\;
 
 
 \subsection{A MPI-CUDA approach}
 
 
 \subsection{A MPI-CUDA approach}

-Our parallel implementation of EA to find roots of polynomials using a
+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-MPI approach follows a similar approach to the one used in

-CUDA-OpenMP. Each process is responsible to compute its own part of
+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
 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
+processors communicate and exchange data. With MPI, processors need to

 send and receive data explicitly. So in
 send and receive data explicitly. So in

-Algorithm~\ref{alg2-cuda-mpi}, after the initialization all the
+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
 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


@@ -327,11 +391,11 @@ 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
 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
+(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 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
+the $MPI\_AlltoAll$ routine). If the convergence is not satisfied, a

 new iteration is executed.
 
 \begin{algorithm}[htpb]
 new iteration is executed.
 
 \begin{algorithm}[htpb]


@@ -375,12 +439,22 @@ We study two categories of polynomials: sparse polynomials and full polynomials.
      {\Large \forall \alpha_{i} \in \mathbb{C}, i\in \mathbb{N};  p(x)=\sum^{n}_{i=0} \alpha_{i}.x^{i}} 
 \end{equation}
 
      {\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, 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}.
+For our tests, 4 GPU cards 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
+degrees.  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 the Guggenheimer method~\cite{Gugg86}.

 
 \subsection{Evaluation of the multi-GPUs approaches}
 
 \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.
+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.

 
 

-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.
+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
 
 \begin{figure}[htbp]
 \centering


@@ -428,11 +502,17 @@ In the previous section we saw that both approaches are very efficient to reduce
 \label{fig:06}
 \end{figure}
 
 \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.
+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}
 
 
 \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.
+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.

 
 \begin{figure}[htbp]
 \centering
 
 \begin{figure}[htbp]
 \centering


@@ -448,7 +528,7 @@ In this experiment we compare the execution times of the EA algorithm according
 \label{fig:08}
 \end{figure}
 
 \label{fig:08}
 \end{figure}
 

-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.
+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}
 
 
 \subsection{Scalability of the EA method on multiple GPUs to solve very high degree polynomials}


@@ -464,9 +544,9 @@ These experiments report the execution times of the EA method for sparse and ful
 
 \section{Conclusion}
 \label{sec6}
 
 \section{Conclusion}
 \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. 
+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 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).
+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 resource's. 

 
 
 \section*{Acknowledgment}
 
 
 \section*{Acknowledgment}