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26 \title{Two parallel implementations of Ehrlich-Aberth algorithm for root-finding of polynomials on multiple GPUs with OpenMP and MPI}
28 \author{\IEEEauthorblockN{Kahina Ghidouche, Abderrahmane Sider }
29 \IEEEauthorblockA{Laboratoire LIMED\\
30 Faculté des sciences exactes\\
31 Université de Bejaia, 06000, Algeria\\
32 Email: \{kahina.ghidouche,ar.sider\}@univ-bejaia.dz}
34 \IEEEauthorblockN{Lilia Ziane Khodja, Raphaël Couturier}
35 \IEEEauthorblockA{FEMTO-ST Institute\\
36 University Bourgogne Franche-Comte, France\\
37 Email: zianekhodja.lilia@gmail.com\\ raphael.couturier@univ-fcomte.fr}}
43 Finding the roots of polynomials is a very important part of solving
44 real-life problems but the higher the degree of the polynomials is,
45 the less easy it becomes. In this paper, we present two different
46 parallel algorithms of the Ehrlich-Aberth method to find roots of
47 sparse and fully defined polynomials of high degrees. Both algorithms
48 are based on CUDA technology to be implemented on multi-GPU computing
49 platforms but each use different parallel paradigms: OpenMP or
50 MPI. The experiments show a quasi-linear speedup by using up-to 4 GPU
51 devices compared to 1 GPU to find the roots of polynomials of degree up-to
52 1.4 million. Moreover, other experiments show it is possible to find the
53 roots of polynomials of degree up-to 5 million.
57 root finding method, Ehrlich-Aberth method, GPU, MPI, OpenMP
60 \IEEEpeerreviewmaketitle
63 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
64 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
65 \section{Introduction}
68 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:
70 p(x) = \displaystyle\sum^n_{i=0}{\alpha_ix^i},\alpha_n\neq 0,
72 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 :
74 p(x)=\alpha_n\displaystyle\prod_{i=1}^n(x-z_i), \alpha_n\neq 0.
77 Most of the numerical methods that deal with the polynomial
78 root-finding problems are simultaneous methods, \textit{i.e.} the
79 iterative methods to find simultaneous approximations of the $n$
80 polynomial roots. These methods start from the initial approximation
81 of all $n$ polynomial roots and give a sequence of approximations that
82 converge to the roots of the polynomial. Two examples of well-known
83 simultaneous methods for root-finding problem of polynomials are
84 the Durand-Kerner method~\cite{Durand60,Kerner66} and the Ehrlich-Aberth method~\cite{Ehrlich67,Aberth73}.
87 The convergence time of simultaneous methods drastically increases
88 with the increasing of the polynomial's degree. The great challenge
89 with simultaneous methods is to parallelize them and to improve their
90 convergence. Many authors have proposed parallel simultaneous
91 methods~\cite{Freeman89,Loizou83,Freemanall90,bini96,cs01:nj,Couturier02},
92 using several paradigms of parallelization (synchronous or
93 asynchronous computations, mechanism of shared or distributed memory,
94 etc). However, so far until now, only polynomials not exceeding
95 degrees of less than 100,000 have been solved.
97 %The main problem of the simultaneous methods is that the necessary
98 %time needed for the convergence increases with the increasing of the
99 %polynomial's degree. Many authors have treated the problem of
100 %implementing simultaneous methods in
101 %parallel. Freeman~\cite{Freeman89} implemented and compared
102 %Durand-Kerner method, Ehrlich-Aberth method and another method of the
103 %fourth order of convergence proposed by Farmer and
104 %Loizou~\cite{Loizou83} on a 8-processor linear chain, for polynomials
105 %of degree up-to 8. The method of Farmer and Loizou~\cite{Loizou83}
106 %often diverges, but the first two methods (Durand-Kerner and
107 %Ehrlich-Aberth methods) have a speed-up equals to 5.5. Later, Freeman
108 %and Bane~\cite{Freemanall90} considered asynchronous algorithms in
109 %which each processor continues to update its approximations even
110 %though the latest values of other approximations $z^{k}_{i}$ have not
111 %been received from the other processors, in contrast with synchronous
112 %algorithms where it would wait those values before making a new
113 %iteration. Couturier and al.~\cite{cs01:nj} proposed two methods
114 %of parallelization for a shared memory architecture with OpenMP and
115 %for a distributed memory one with MPI. They are able to compute the
116 %roots of sparse polynomials of degree 10,000. The authors showed an interesting
117 %speedup that is 20 times as fast as the sequential implementation.
119 The recent advent of the Compute Unified Device Architecture
120 (CUDA)~\cite{CUDA15}, a programming
121 model and a parallel computing architecture developed by NVIDIA, has revived parallel programming interest in
122 this problem. Indeed, the computing power of GPUs (Graphics Processing
123 Units) has exceeded that of traditional CPUs processors, which makes
124 it very appealing to the research community to investigate new
125 parallel implementations for a whole set of scientific problems in the
126 reasonable hope to solve bigger instances of well known
127 computationally demanding issues such as the one beforehand. However,
128 CUDA provides an efficient massive data computing model which is
129 suited to GPU architectures. Ghidouche et al.~\cite{Kahinall14} proposed an implementation of the Durand-Kerner method on a single GPU. Their main results showed that a parallel CUDA implementation is about 10 times faster than the sequential implementation on a single CPU for sparse polynomials of degree 48,000.
131 In this paper we propose the parallelization of the Ehrlich-Aberth
132 (EA) method which has a much better cubic convergence rate than the
133 quadratic rate of the Durand-Kerner method that has already been investigated in \cite{Kahinall14}. In the other hand, EA is suitable to be implemented in parallel computers according to the data-parallel paradigm. In this model, computing elements carry computations on the data they are assigned and communicate with other computing elements in order to get fresh data or to synchronize. Classically, two parallel programming paradigms OpenMP and MPI are used to code such solutions. But in our case, computing elements are CUDA multi-GPU platforms. This architectural setting poses new programming challenges but offers also new opportunities to efficiently solve huge problems, otherwise considered intractable until recently. To the best of our knowledge, our CUDA-MPI and CUDA-OpenMP codes are the first implementations of EA method with multiple GPUs for finding roots of polynomials. Our major contributions include:
136 \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.
137 \item The parallel implementation of EA algorithm on a
138 multi-GPU platform with a distributed memory using MPI API, such
139 that each GPU is attached and managed by a MPI process. The GPUs
140 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.
142 Our method is efficient to compute the roots of sparse and full
143 polynomials of degree up to 5 million.
147 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.
150 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
151 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
153 \section{Parallel programming models}
155 Our objective consists in implementing a root-finding algorithm of
156 polynomials on multiple GPUs. To this end, it is essential to know how
157 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.
160 OpenMP (Open Multi-processing) is an application programming interface
161 for parallel programming~\cite{openmp13}. It is a portable approach
162 based on the multithreading designed for shared memory computers,
163 where a master thread forks a number of slave threads which execute
164 blocks of code in parallel. An OpenMP program alternates sequential
165 regions and parallel regions of code, where the sequential regions are
166 executed by the master thread and the parallel ones may be executed by
167 multiple threads. During the execution of an OpenMP program the
168 threads communicate their data (read and modified) in the shared
169 memory. One advantage of OpenMP is the global view of the memory
170 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.
173 MPI (Message Passing Interface) is a portable message passing style of
174 the parallel programming designed specifically for distributed memory
175 architectures~\cite{Peter96}. In most MPI implementations, a
176 computation contains a fixed set of processes created at the
177 initialization of the program in such a way that one process is
178 created per processor. The processes synchronize their computations
179 and communicate by sending/receiving messages to/from other
180 processes. In this case, the data are explicitly exchanged by message
181 passing while the data exchanges are implicit in a multithread
182 programming model like OpenMP and Pthreads. However in the MPI
183 programming model, the processes may either execute different programs
184 referred to as multiple program multiple data (MPMD) or every process
185 executes the same program (SPMD). The MPI approach is one of the most used HPC programming model to solve large scale and complex applications.
188 CUDA (Compute Unified Device Architecture) is a parallel computing
189 architecture developed by NVIDIA~\cite{CUDA15} for GPUs. It provides a
190 high level GPGPU-based programming model to program GPUs for general
191 purpose computations. The GPU is viewed as an accelerator such that
192 data-parallel operations of a CUDA program running on a CPU are
193 off-loaded onto GPU and executed by this latter. The data-parallel
194 operations executed by GPUs are called kernels. The same kernel is
195 executed in parallel by a large number of threads organized in grids
196 of thread blocks, such that each GPU multiprocessor executes one or
197 more thread blocks in SIMD fashion (Single Instruction, Multiple Data)
198 and in turn each core of the multiprocessor executes one or more
199 threads within a block. Threads within a block can cooperate by
200 sharing data through a fast shared memory and coordinate their
201 execution through synchronization points. In contrast, within a grid
202 of thread blocks, there is no synchronization at all between
203 blocks. The GPU only works on data filled in the global memory and the
204 final results of the kernel executions must be transferred out of the
205 GPU. In the GPU, the global memory has lower bandwidth than the shared
206 memory associated to each multiprocessor. Thus with CUDA programming,
207 it is necessary to design carefully the arrangement of the thread
208 blocks in order to ensure a low latency and a proper use of the shared
209 memory. As for the global memory accesses, it should also be minimized.
212 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
213 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
215 \section{The Ehrlich-Aberth algorithm on a GPU}
218 \subsection{The Ehrlich-Aberth method}
220 The Ehrlich-Aberth method is a simultaneous method~\cite{Aberth73} using the following iteration
223 z^{k+1}_{i}=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}
224 {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
227 This method contains 4 steps. The first step consists in the
228 initializing the polynomial. The second step initializes the solution
229 vector $Z$ using the Guggenheimer method~\cite{Gugg86} to ensure that
230 initial roots are all distinct from each other. In step 3, the
231 iterative function based on the Newton's method~\cite{newt70} and
232 Weiestrass operator~\cite{Weierstrass03} is applied. In our case, the
233 Ehrlich-Aberth is applied as in~(\ref{Eq:EA1}). Iterations of the
234 EA method will converge to the roots of the considered
235 polynomial. In order to stop the iterative function, a stop condition
236 is applied, this is the 4th step. This condition checks that all the
237 root modules are lower than a fixed value $\epsilon$.
240 \label{eq:Aberth-Conv-Cond}
241 \forall i\in[1,n],~\vert\frac{z_i^k-z_i^{k-1}}{z_i^k}\vert<\epsilon
244 \subsection{Improving Ehrlich-Aberth method}
245 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.
247 In order to solve this problem, we propose to modify the iterative
248 function by using the logarithm and the exponential of a complex and
249 we propose a new version of the EA method. This method
250 allows us to exceed the computation of the polynomials of degree
251 100,000 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 $i=1,\dots,n$:
255 z^{k+1}_i = z_i^k - \exp(\ln(p(z_i^k)) - \ln(p'(z^k_i)) - \ln(1-Q(z^k_i))),
262 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})).
265 Using the logarithm and the exponential operators, we can replace any
266 multiplications and divisions with additions and
267 subtractions. Consequently, computations manipulate lower values in
268 absolute values~\cite{Karimall98}. In practice, the exponential and
269 logarithm mode is used when a root is outside the circle unit represented by the radius $R$ evaluated in C language with:
272 R = exp(log(DBL\_MAX)/(2*n) );
274 where \verb=DBL_MAX= stands for the maximum representable
275 \verb=double= value and $n$ is the degree of the polynomial.
278 \subsection{The Ehrlich-Aberth parallel implementation on CUDA}
279 The algorithm ~\ref{alg1-cuda} shows sketch of the Ehrlich-Aberth method using CUDA.
280 The first step consists in the initialization of the input data, for
281 exemple the polynomial P, the derivative of P and the vector solution Z. Then, all data of the root finding problem
282 are transfered from the CPU memory to the GPU global memory, because
283 the GPUs only work on the data filled in their memories.
284 Next, all the data-parallel arithmetic operations inside the main loop
285 \verb=(while(...))= are executed as kernels by the GPU. The
286 first kernel named \textit{Kernelsave} in line 5 of Algorithm~\ref{alg1-cuda} consists in saving the vector of polynomial roots found at the previous time-step in GPU memory, in
287 order to check the convergence of the roots after each iteration (line
288 7, Algorithm~\ref{alg1-cuda}). Then the new roots with the
289 new iterations are computed using the EA method with a Gauss-Seidel
290 iteration mode in order to use the latest updated roots (line
291 6). This improves the convergence compared to the Jacobi method. This kernel is, in practice, very
292 long since it performs all the operations with complex numbers with
293 the normal mode of the EA method as in Eq.~\ref{Eq:EA1} but also with the logarithm-exponential one as in Eq.(~\ref{Log_H1},~\ref{Log_H2}). The last kernel checks the convergence of the roots after each update of $Z^{k}$, according to formula Eq.~\ref{eq:Aberth-Conv-Cond} line (7). We used the functions of the CUBLAS Library (CUDA Basic Linear Algebra Subroutines) to implement this kernel. The algorithm terminates its computations when all the roots have
295 %The code is organized as kernels which are parts of codes that are run
296 %on GPU devices. Algorithm~\ref{alg1-cuda} describes the CUDA
297 %implementation of the Ehrlich-Aberth on a GPU. This algorithms starts
298 %by initializing the polynomial and its derivative (line 1). The
299 %initialization of the roots is performed (line 2). Data are transferred
300 %%from the CPU to the GPU (after the allocation of the required memory on
301 %the GPU) (line 3). Then at each iteration, if the error is greater
302 %%than the threshold, the following operations are performed. The previous
303 %roots are saved using a kernel (line 5). Then the new roots with the
304 %new iterations are computed using the EA method with a Gauss-Seidel
305 %iteration mode in order to use the latest updated roots (line
306 %6). This improves the convergence. This kernel is, in practice, very
307 %long since it performs all the operations with complex numbers with
308 %the normal mode of the EA method but also with the
309 %logarithm-exponential one. Then the error is computed with a final
310 %kernel (line 7). Finally when the EA method has converged, the roots
311 %are transferred from the GPU to the CPU.
312 \begin{algorithm}[htpb]
316 \caption{Finding roots of polynomials with the Ehrlich-Aberth method on a GPU}
317 \KwIn{ $\epsilon$ (tolerance threshold)}
318 \KwOut{$Z$ (solution vector of roots)}
319 Initialize the polynomial $P$ and its derivative $P'$\;
320 Set the initial values of vector $Z$\;
321 Copy $P$, $P'$ and $Z$ from CPU to GPU\;
322 \While{$error > \epsilon$}{
323 $Z^{prev}$ = KernelSave($Z$)\;
324 $Z$ = KernelUpdate($P,P',Z$)\;
325 $error$ = KernelComputeError($Z,Z^{prev}$)\;
327 Copy $Z$ from GPU to CPU\;
330 Listing~\ref{lst:01} shows the second kernel code. As can be seen this
331 kernel calls multiple kernels, all the kernals for complex numbers and
332 kernels for the evaluation of a polynomial are not detailed.
335 \lstinputlisting[label=lst:01,caption=Kernels to update the roots]{code.c}
338 %\begin{figure}[htbp]
340 %\includegraphics[angle=+0,width=0.4\textwidth]{code}
341 %\caption{The Kernel Update code}
345 %We noticed that the code is executed by a large number of GPU threads organized as grid of to dimension (Number of block per grid (NbBlock), number of threads per block(Nbthread)), the Nbthread is fixed initially, the NbBlock is computed as fallow:
346 %$ NbBlocks= \frac{N+Nbthreads-1}{Nbthreads} where N: the number of root$
347 %the such that each thread in grid is in charge of the computation of one root.
349 The development of this code is a rather long task due to the
350 development of all the kernels that compute the parts ported on the
351 GPU. This comes in particular from the fact that it is very difficult
352 to debug CUDA running threads like threads on a CPU host. In the
353 following section the GPU parallel implementation of the
354 Ehrlich-Aberth method with OpenMP and MPI is presented.
356 \section{The Ehrlich-Aberth algorithm on multiple GPUs}
358 In order to manage the CUDA contexts of different GPUs, two parallel
359 paradigms are investigated: OpenMP and MPI. In this section we present
360 the \textit{OpenMP-CUDA} and the \textit{MPI-CUDA} approaches used to implement the Ehrlich-Aberth algorithm on multiple GPUs.
362 \subsection{An OpenMP-CUDA approach}
363 Our OpenMP-CUDA implementation of EA algorithm is based on the hybrid
364 OpenMP and CUDA programming model. This algorithm is presented in
365 Algorithm~\ref{alg2-cuda-openmp}. All the data are shared with OpenMP
366 among all the OpenMP threads. The shared data are the solution vector
367 $Z$, the polynomial to solve $P$, its derivative $P'$, and the error
368 vector $error$. The number of OpenMP threads is equal to the number of
369 GPUs, each OpenMP thread binds to one GPU, and it controls a part of
370 the shared memory. More precisely each OpenMP thread will be
371 responsible for updating its own part of the vector $Z$. This part is
372 called $Z_{loc}$ in the following. Then all GPUs will have a grid of
373 computation organized according to the device performance and the size
374 of data on which it runs the computation kernels.
376 To compute one iteration of the EA method each GPU performs the
377 followings steps. First, roots are shared with OpenMP and the
378 computation of the local size for each GPU is performed (line 4). Each
379 thread starts by copying all the previous roots inside its GPU (line
380 5). At each iteration, the following operations are performed. First
381 the vector $Z$ is transferred from the CPU to the GPU (line 7). Each
382 GPU copies the previous roots (line 8) and it computes an iteration of
383 the EA method on its own roots (line 9). For that all the other roots
384 are used. The local error is computed on the new roots (line 10) and
385 the maximum of the local errors is computed on all OpenMP threads (line 11). At
386 the end of an iteration, the updated roots are copied from the GPU to
387 the CPU (line 12) and each CPU directly updates its own roots in the shared
388 memory arrays containing all the roots.
392 \begin{algorithm}[htpb]
393 \label{alg2-cuda-openmp}
396 \caption{Finding roots of polynomials with the Ehrlich-Aberth method on multiple GPUs using OpenMP}
397 \KwIn{ $\epsilon$ (tolerance threshold)}
398 \KwOut{$Z$ (solution vector of roots)}
399 Initialize the polynomial $P$ and its derivative $P'$\;
400 Set the initial values of vector $Z$\;
401 Start of a parallel part with OpenMP ($Z$, $error$, $P$, $P'$ are shared variables)\;
402 Determine the local part of the OpenMP thread\;
403 Copy $P$, $P'$ from CPU to GPU\;
404 \While{$error > \epsilon$}{
405 Copy $Z$ from CPU to GPU\;
406 $Z^{prev}_{loc}$ = KernelSave($Z_{loc}$)\;
407 $Z_{loc}$ = KernelUpdate($P,P',Z$)\;
408 $error_{loc}$ = KernelComputeError($Z_{loc},Z^{prev}_{loc}$)\;
409 $error = max(error_{loc})$\;
410 Copy $Z_{loc}$ from GPU to $Z$ in CPU\;
418 \subsection{A MPI-CUDA approach}
419 Our parallel implementation of EA to find the roots of polynomials using a
420 CUDA-MPI approach follows a similar approach to the one used in
421 CUDA-OpenMP. Each processor is responsible for computing its own part of
422 roots using all the roots computed by other processors at the previous
423 iteration. The difference between both approaches lies in the way
424 processors communicate and exchange data. With MPI, processors need to
425 send and receive data explicitly. So in
426 Algorithm~\ref{alg2-cuda-mpi}, after the initialization phase all the
427 processors have the same $Z$ vector. Then they need to compute the
428 parameters used by the $MPI\_AlltoAll$ routines (line 4). In practice,
429 each processor needs to compute its offset and its local
430 size. Processors need to allocate memory on their GPU and need to copy
431 their data on the GPU (line 5). At the beginning of each iteration, a
432 processor starts by transferring the whole vector $Z$ from the CPU to the
433 GPU (line 7). Only the local part of $Z^{prev}$ is saved (line
434 8). After that, a processor is able to compute an updated version of
435 its own roots (line 9) with the EA method. The local error is computed
436 (line 10) and the global error is also computed using $MPI\_Reduce$ (line 11). Then
437 the local roots are transferred from the GPU memory to the CPU memory
438 (line 12) before being exchanged between all processors (line 13) in
439 order to give to all processors the last version of the roots (with
440 the $MPI\_AlltoAll$ routine). If the convergence is not satisfied, a
441 new iteration is executed.
443 \begin{algorithm}[htpb]
444 \label{alg2-cuda-mpi}
447 \caption{Finding roots of polynomials with the Ehrlich-Aberth method on multiple GPUs using MPI}
449 \KwIn{ $\epsilon$ (tolerance threshold)}
451 \KwOut {$Z$ (solution vector of roots)}
454 Initialize the polynomial $P$ and its derivative $P'$\;
455 Set the initial values of vector $Z$\;
456 Determine the local part of the MPI process\;
457 Computation of the parameters for the $MPI\_AlltoAll$\;
458 Copy $P$, $P'$ from CPU to GPU\;
459 \While {$error > \epsilon$}{
460 Copy $Z$ from CPU to GPU\;
461 $Z^{prev}_{loc}$ = KernelSave($Z_{loc}$)\;
462 $Z_{loc}$ = KernelUpdate($P,P',Z$)\;
463 $error_{loc}$ = KernelComputeError($Z_{loc},Z^{prev}_{loc}$)\;
464 $error=MPI\_Reduce(error_{loc})$\;
465 Copy $Z_{loc}$ from GPU to CPU\;
466 $Z=MPI\_AlltoAll(Z_{loc})$\;
471 \section{Experiments}
473 We study two categories of polynomials: sparse polynomials and full polynomials.\\
474 {\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:
476 \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})
477 \end{equation}\noindent
478 {\it A full polynomial} is, in contrast, a polynomial for which all the coefficients are not null. A full polynomial is defined by:
481 {\Large \forall \alpha_{i} \in \mathbb{C}, i\in \mathbb{N}; p(x)=\sum^{n}_{i=0} \alpha_{i}.x^{i}}
484 For our tests, a CPU Intel(R) Xeon(R) CPU
485 X5650@2.40GHz and 4 GPUs cards Tesla Kepler K40, are used with CUDA version 7.5.
487 In order to evaluate both the GPU and Multi-GPU approaches, we performed a set of
488 experiments on a single GPU and multiple GPUs using OpenMP or MPI with
489 the EA algorithm, for both sparse and full polynomials of different
490 degrees. All experimental results obtained are performed with double
491 precision floating-point data and the convergence threshold of the EA method is
492 set to $10^{-7}$. The initialization values of the vector solution of
493 the methods are given by the Guggenheimer method~\cite{Gugg86}.
495 \subsection{Evaluation of the multi-GPUs approaches}
496 In this part, we evaluate the performances of the CUDA-OpenMP and
497 CUDA-MPI approaches of the EA algorithm on different GPU platforms
498 composed each of 1, 2, 3 or 4 GPUs. In this experiments we report the
499 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.
501 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.
505 \includegraphics[angle=-90,width=0.5\textwidth]{Sparse_omp}
506 \caption{Execution times in seconds of the Ehrlich-Aberth method to solve sparse polynomials on multiple GPUs with CUDA-OpenMP.}
512 \includegraphics[angle=-90,width=0.5\textwidth]{Full_omp}
513 \caption{Execution times in seconds of the Ehrlich-Aberth method to solve full polynomials on multiple GPUs with CUDA-OpenMP.}
519 \includegraphics[angle=-90,width=0.5\textwidth]{Sparse_mpi}
520 \caption{Execution times in seconds of the Ehrlich-Aberth method to solve sparse polynomials on multiple GPUs with CUDA-MPI.}
526 \includegraphics[angle=-90,width=0.5\textwidth]{Full_mpi}
527 \caption{Execution times in seconds of the Ehrlich-Aberth method for full polynomials on multiple GPUs with CUDA-MPI.}
532 \subsection{Comparison between the CUDA-OpenMP and the CUDA-MPI approaches}
533 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.
537 \includegraphics[angle=-90,width=0.5\textwidth]{Sparse}
538 \caption{Execution times to solve sparse polynomials of three distinct degrees on multiple GPUs using OpenMP and MPI with the Ehrlich-Aberth method}
544 \includegraphics[angle=-90,width=0.5\textwidth]{Full}
545 \caption{Execution times to solve full polynomials of three distinct degrees on multiple GPUs using OpenMP and MPI with the Ehrlich-Aberth method}
549 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.
552 \subsection{Solving sparse and full polynomials of the same degree on multiple GPUs}
553 In this experiment we compare the execution times of the EA algorithm
554 according to the number of GPUs to solve sparse and full polynomials
555 on multiple GPUs using OpenMP or MPI approaches. We chose three sparse
556 and three full polynomials of degrees 200,000, 800,000 and
557 1,400,000. Figures~\ref{fig:07} and~\ref{fig:08} show the execution
558 times to solve sparse and full polynomials of the same degrees with
559 the CUDA-OpenMP version and the CUDA-MPI version, respectively.
563 \includegraphics[angle=-90,width=0.5\textwidth]{OMP}
564 \caption{Execution times to solve sparse and full polynomials of three distinct degrees on multiple GPUs using OpenMP.}
570 \includegraphics[angle=-90,width=0.5\textwidth]{MPI}
571 \caption{Execution times to solve sparse and full polynomials of three distinct degrees on multiple GPUs using MPI.}
575 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.
578 \subsection{Scalability of the EA method on multiple GPUs to solve very high degree polynomials}
579 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.
583 \includegraphics[angle=-90,width=0.5\textwidth]{big}
584 \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}
591 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.
593 Our next objective is to extend the model presented here to clusters of GPU nodes, with a three-level scheme: inter-node communications via MPI processes (distributed memory), management of multi-GPU nodes by OpenMP threads (shared memory). Actual platforms may probably also contain purely multi-core nodes without any GPU. This heterogeneous setting may lead to the integration of load balancing algorithms so as to allow an optimal use of hardware resources.
596 \section*{Acknowledgment}
597 This paper is partially funded by the Labex ACTION program (contract
598 ANR-11-LABX-01-01) and the Franche-Comte regional council. 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.
601 \bibliography{mybibfile}
