In this paper we propose the parallelization of Ehrlich-Aberth method using two parallel programming paradigms OpenMP and MPI on CUDA multi-GPU platforms. Our CUDA/MPI and CUDA/OpenMP codes are the first implementations of Ehrlich-Aberth method with multiple GPUs for finding roots of polynomials. Our major contributions include:
\LZK{Pourquoi la méthode Ehrlich-Aberth et pas autres? the Ehrlich-Aberth have very good convergence and it is suitable to be implemented in parallel computers.}
\begin{itemize}
- \item An improvements for the Ehrlich-Aberth method using the exponential logarithm in order to be able to solve sparse and full polynomial of degree up to 1, 000, 000.
- \item A parallel implementation of Ehrlich-Aberth method on single GPU with CUDA.
+ \item An improvements for the Ehrlich-Aberth method using the exponential logarithm in order to be able to solve sparse and full polynomial of degree up to 1, 000, 000.\RC{j'ai envie de virer ca, car c'est pas la nouveauté dans ce papier}
+ \item A parallel implementation of Ehrlich-Aberth method on single GPU with CUDA.\RC{idem}
\item The parallel implementation of Ehrlich-Aberth algorithm on a multi-GPU platform with a shared memory using OpenMP API. It is based on threads created from the same system process, such that each thread is attached to one GPU. In this case the communications between GPUs are done by OpenMP threads through shared memory.
\item The parallel implementation of Ehrlich-Aberth algorithm on a multi-GPU platform with a distributed memory using MPI API, such that each GPU is attached and managed by a MPI process. The GPUs exchange their data by message-passing communications. This latter approach is more used on clusters to solve very complex problems that are too large for traditional supercomputers, which are very expensive to build and run.
\end{itemize}
%This paper is organized as follows. In Section~\ref{sec2} we recall the Ehrlich-Aberth method. In section~\ref{sec3} we present EA algorithm on single GPU. In section~\ref{sec4} we propose the EA algorithm implementation on Multi-GPU for (OpenMP-CUDA) approach and (MPI-CUDA) approach. In sectioné\ref{sec5} we present our experiments and discus it. Finally, Section~\ref{sec6} concludes this paper and gives some hints for future research directions in this topic.}
The paper is organized as follows. In Section~\ref{sec2} we present three different parallel programming models OpenMP, MPI and CUDA. In Section~\ref{sec3} we present the implementation of the Ehrlich-Aberth algorithm on a single GPU. In Section~\ref{sec4} we present the parallel implementations of the Ehrlich-Aberth algorithm on Multi-GPU using the OpenMP and MPI approaches. In section\ref{sec5} we present our experiments and discus it. Finally, Section~\ref{sec6} concludes this paper and gives some hints for future research directions in this topic.
-\LZK{A revoir toute cette organization: je viens de la revoir}
+%\LZK{A revoir toute cette organization: je viens de la revoir}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{1-\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}\sum_{j=1,j\neq i}^{j=n}{\frac{1}{(z_{i}^{k}-z_{j}^{k})}}}, i=1,. . . .,n
\end{equation}
-contain 4 steps, start from the initial approximations of all the roots of the polynomial,the second step initialize the solution vector $Z$ using the Guggenheimer~\cite{Gugg86} method to assure the distinction of the initial vector roots,
+This methods contains 4 steps. The first step consists of the initial
+approximations of all the roots of the polynomial. The second step
+initializes the solution vector $Z$ using the Guggenheimer
+method~\cite{Gugg86} to ensure the distinction of the initial vector
+roots. In step 3, the iterative function based on the Newton's
+method~\cite{newt70} and Weiestrass operator~\cite{Weierstrass03} is
+applied. With this step the computation of roots will converge,
+provided that all roots are different.
- than in step 3 we apply the the iterative function based on the Newton's method~\cite{newt70} and Weiestrass operator~\cite{Weierstrass03}, wich will make it possible to converge to the roots solution, provided that all the root are different.
-
- At the end of each application of the iterative function, a stop condition is verified consists in stopping the iterative process when the whole of the modules of the roots are lower than a fixed value $\xi$.
+In order to stop the iterative function, a stop condition is
+applied. This condition checks that all the root modules are lower
+than a fixed value $\xi$.
\begin{equation}
\label{eq:Aberth-Conv-Cond}
\forall i \in [1,n];\vert\frac{z_{i}^{k}-z_{i}^{k-1}}{z_{i}^{k}}\vert<\xi
\end{equation}
\subsection{Improving Ehrlich-Aberth method}
-With high degree polynomial, the Ehrlich-Aberth method suffer from overflow because the limited number in the mantissa of floating points representations, which makes the computation of $p(z)$ wrong when z is large.
+With high degree polynomials, the Ehrlich-Aberth method suffers from
+floating point overflows due to the mantissa of floating points
+representations. This induces errors in the computation of $p(z)$ when
+$z$ is large.
-Experimentally, it is very difficult to solve polynomials with Ehrlich-Aberth method and have roots which except the circle of unit, represented by the radius $r$ evaluated as:
-
-\begin{equation}
-\label{R.EL}
-R = exp(log(DBL\_MAX)/(2*n) );
-\end{equation}
-
+%Experimentally, it is very difficult to solve polynomials with the Ehrlich-Aberth method and have roots which except the circle of unit, represented by the radius $r$ evaluated as:
%\begin{equation}
-
-%R = \exp( \log(DBL\_MAX) / (2*n) )
+%\label{R.EL}
+%R = exp(log(DBL\_MAX)/(2*n) );
%\end{equation}
- where \verb=DBL_MAX= stands for the maximum representable \verb=double= value.
-
-In order to hold into account the limit of size of floats, we propose to modifying the iterative function and compute the logarithm of:
-\begin{equation}
-EA: z^{k+1}_{i}=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}
-{1-\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}\sum_{j=1,j\neq i}^{j=n}{\frac{1}{(z_{i}^{k}-z_{j}^{k})}}}, i=1,. . . .,n
-\end{equation}
-This method allows, indeed, to exceed the computation of the polynomials of degree 100,000 and to reach a degree upper to 1,000,000. For that purpose, it is necessary to use the logarithm and the exponential of a complex. The iterative function of Ehrlich-Aberth method with exponential and logarithm is given as following:
+
+% where \verb=DBL_MAX= stands for the maximum representable \verb=double= value.
+
+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. This new
+version of the Ehrlich-Aberth method with exponential and logarithm is
+defined as follows:
\begin{equation}
\label{Log_H2}
where:
-\begin{equation}
+\begin{eqnarray}
\label{Log_H1}
Q(z^{k}_{i})=\exp\left( \ln (p(z^{k}_{i}))-\ln(p'(z^{k}_{i}))+\ln \left(
-\sum_{i\neq j}^{n}\frac{1}{z^{k}_{i}-z^{k}_{j}}\right)\right)i=1,...,n,
-\end{equation}
+\sum_{i\neq j}^{n}\frac{1}{z^{k}_{i}-z^{k}_{j}}\right)\right) \nonumber \\
+i=1,...,n
+\end{eqnarray}
%We propose to use the logarithm and the exponential of a complex in order to compute the power at a high exponent.
-Using the logarithm and the exponential operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations manipulate lower absolute values and the roots for large polynomial degrees can be looked for successfully~\cite{Karimall98}.
+Using the logarithm and the exponential operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations manipulate lower absolute values~\cite{Karimall98}.
%This problem was discussed earlier in~\cite{Karimall98} for the Durand-Kerner method. The authors
%propose to use the logarithm and the exponential of a complex in order to compute the power at a high exponent. Using the logarithm and the exponential operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations manipulate lower absolute values and the roots for large polynomial degrees can be looked for successfully~\cite{Karimall98}.
\subsection{Ehrlich-Aberth parallel implementation on CUDA}
-We introduced three paradigms of parallel programming. Our objective consists in implementing a root finding polynomial algorithm on multiple GPUs. To this end, it is primordial to know how to manage CUDA contexts of different GPUs. A direct method for controlling the various GPUs is to use as many threads or processes as GPU devices. We can choose the GPU index based on the identifier of OpenMP thread or the rank of the MPI process. Both approaches will be investigated.
-
-
-
-
-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 specify the organization of parallel threads, by specifying the dimension of the grid Dimgrid, the number of blocks per grid DimBlock and the number of threads per block.
-
-The code is organized by what is named kernels, portions code that are run on GPU devices. For step 3, there are two kernels, the
-first named \textit{save} is used to save vector $Z^{K-1}$ and the seconde one is named
-\textit{update} and is used to update the $Z^{K}$ vector. For step 4, a kernel
-tests the convergence of the method. In order to
-compute the function H, we have two possibilities: either to use
-the Jacobi mode, or the Gauss-Seidel mode of iterating which uses the
-most recent computed roots. It is well known that the Gauss-
-Seidel mode converges more quickly. So, we 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
+%We introduced three paradigms of parallel programming.
+
+Our objective consists in implementing a root finding polynomial
+algorithm on multiple GPUs. To this end, it is primordial to know how
+to manage CUDA contexts of different GPUs. A direct method for
+controlling the various GPUs is to use as many threads or processes as
+GPU devices. We can choose the GPU index based on the identifier of
+OpenMP thread or the rank of the MPI process. Both approaches will be
+investigated.
+
+
+
+
+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 specify the organization of parallel
+threads, by specifying the dimension of the grid Dimgrid, the number
+of blocks per grid DimBlock and the number of threads per block.
+
+The code is organized kernels which are part of code that are run on
+GPU devices. For step 3, there are two kernels, the first named
+\textit{save} is used to save vector $Z^{K-1}$ and the second one is
+named \textit{update} and is used to update the $Z^{K}$ vector. For
+step 4, a kernel tests the convergence of the method. In order to
+compute the function H, we have two possibilities: either to use the
+Jacobi mode, or the Gauss-Seidel mode of iterating which uses the most
+recent computed roots. It is well known that the Gauss-Seidel mode
+converges more quickly. So, we use Gauss-Seidel iterations. To
+parallelize the code, we create kernels and many functions to be
+executed on the GPU for all the operations dealing with the
computation on complex numbers and the evaluation of the
-polynomials. As said previously, we 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~\ref{alg1-cuda} shows the GPU parallel implementation of Ehrlich-Aberth method.
+polynomials. As said previously, we manage both functions of
+evaluation: the normal method, based on the method of
+Horner and the method based on the logarithm of the polynomial. All
+these methods were rather long to implement, as the development of
+corresponding kernels with CUDA is longer than on a CPU host. This
+comes in particular from the fact that it is very difficult to debug
+CUDA running threads like threads on a CPU host. In the following
+paragraph Algorithm~\ref{alg1-cuda} shows the GPU parallel
+implementation of Ehrlich-Aberth method.
\begin{enumerate}
\begin{algorithm}[htpb]
