where $P'(z)$ is the polynomial derivative of $P$ evaluated in the
point $z$.
-Aberth, Ehrlich and Farmer-Loizou~\cite{Loizon83} have proved that
+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.
Many authors have dealt with the parallelization of
simultaneous methods, i.e. that find all the zeros simultaneously.
Freeman~\cite{Freeman89} implemented and compared DK, EA and another method of the fourth order proposed
-by Farmer and Loizou~\cite{Loizon83}, on a 8-processor linear
+by Farmer and Loizou~\cite{Loizou83}, 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~\cite{Freemanall90} considered asynchronous
\begin{equation}
\label{eq:Aberth-Conv-Cond}
-\forall i \in
-[1,n];\frac{z_{i}^{k}-z_{i}^{k-1}}{z_{i}^{k}}<\xi
+\forall i \in [1,n];\vert\frac{z_{i}^{k}-z_{i}^{k-1}}{z_{i}^{k}}\vert<\xi
\end{equation}
\begin{align}
\label{defexpcomplex}
\forall(x,y)\in R^{*2}; \exp(x+i.y) & = \exp(x).\exp(i.y)\\
- & =\exp(x).\cos(y)+i.\exp(x).\sin(y)\label{defexpcomplex}
+ & =\exp(x).\cos(y)+i.\exp(x).\sin(y)\label{defexpcomplex1}
\end{align}
%%\end{equation}
There are many schemes for the simultaneous approximation of all roots of a given
polynomial. Several works on different methods and issues of root
-finding have been reported in~\cite{Azad07, Gemignani07, Kalantari08, Skachek08, Zhancall08, Zhuall08}. However, Durand-Kerner and Ehrlich-Aberth methods are the most practical choices among
+finding have been reported in~\cite{Azad07, Gemignani07, Kalantari08, Zhancall08, Zhuall08}. However, Durand-Kerner and Ehrlich-Aberth methods are the most practical choices among
them~\cite{Bini04}. These two methods have been extensively
-studied for parallelization due to their intrinsics, i.e. the
+studied for parallelization due to their intrinsics parallelism, i.e. the
computations involved in both methods has some inherent
parallelism that can be suitably exploited by SIMD machines.
Moreover, they have fast rate of convergence (quadratic for the
algorithms are mapped on an OTIS-2D torus using N processors. This
solution needs N processors to compute N roots, which is not
practical for solving polynomials with large degrees.
-Until very recently, the literature doen not mention implementations able to compute the roots of
-large degree polynomials (higher then 1000) and within small or at least tractable times. Finding polynomial roots rapidly and accurately is the main objective of our work.
+%Until very recently, the literature did not mention implementations
+%able to compute the roots of large degree polynomials (higher then
+%1000) and within small or at least tractable times.
+
+Finding polynomial roots rapidly and accurately is the main objective of our work.
With the advent of CUDA (Compute Unified Device
Architecture), finding the roots of polynomials receives a new attention because of the new possibilities to solve higher degree polynomials in less time.
In~\cite{Kahinall14} we already proposed the first implementation
of a root finding method on GPUs, that of the Durand-Kerner method. The main result showed
that a parallel CUDA implementation is 10 times as fast as the
sequential implementation on a single CPU for high degree
-polynomials of 48000. In this paper we present a parallel implementation of Ehlisch-Aberth method on
-GPUs, which details are discussed in the sequel.
+polynomials of 48000.
+%In this paper we present a parallel implementation of Ehrlich-Aberth
+%method on GPUs for sparse and full polynomials with high degree (up
+%to $1,000,000$).
\section {A CUDA parallel Ehrlich-Aberth method}
In the following, we describe the parallel implementation of Ehrlich-Aberth method on GPU
-for solving high degree polynomials. First, the hardware and software of the GPUs are presented. Then, a CUDA parallel Ehrlich-Aberth method are presented.
+for solving high degree polynomials (up to $1,000,000$). First, the hardware and software of the GPUs are presented. Then, the CUDA parallel Ehrlich-Aberth method is presented.
\subsection{Background on the GPU architecture}
A GPU is viewed as an accelerator for the data-parallel and
texture space, which reside in external DRAM, and are accessed via
read-only caches.
-\section{ The implementation of Ehrlich-Aberth method on GPU}
+\section{ Implementation of Ehrlich-Aberth method on GPU}
\label{sec5}
%%\subsection{A CUDA implementation of the Aberth's method }
%%\subsection{A GPU implementation of the Aberth's method }
-\subsection{A sequential Ehrlich-Aberth algorithm}
+\subsection{Sequential Ehrlich-Aberth algorithm}
The main steps of Ehrlich-Aberth method are shown in Algorithm.~\ref{alg1-seq} :
-
+%\LinesNumbered
\begin{algorithm}[H]
\label{alg1-seq}
-%\LinesNumbered
+
\caption{A sequential algorithm to find roots with the Ehrlich-Aberth method}
-\KwIn{$Z^{0}$(Initial root's vector),$\varepsilon$ (error tolerance threshold),P(Polynomial to solve)}
-\KwOut {Z(The solution root's vector)}
+\KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (error tolerance
+ threshold), P (Polynomial to solve), $\Delta z_{max}$ (maximum value
+ of stop condition), k (number of iteration), n (Polynomial's degrees)}
+\KwOut {Z (The solution root's vector), ZPrec (the previous solution root's vector)}
\BlankLine
Initialization of the coefficients of the polynomial to solve\;
Initialization of the solution vector $Z^{0}$\;
+$\Delta z_{max}=0$\;
+ k=0\;
-\While {$\Delta z_{max}\succ \epsilon$}{
+\While {$\Delta z_{max} > \varepsilon$}{
Let $\Delta z_{max}=0$\;
\For{$j \gets 0 $ \KwTo $n$}{
-$ZPrec\left[j\right]=Z\left[j\right]$\;
-$Z\left[j\right]=H\left(j,Z\right)$\;
+$ZPrec\left[j\right]=Z\left[j\right]$;// save Z at the iteration k.\
+
+$Z\left[j\right]=H\left(j,Z\right)$;//update Z with the iterative function.\
}
+k=k+1\;
\For{$i \gets 0 $ \KwTo $n-1$}{
-$c=\frac{\left|Z\left[i\right]-ZPrec\left[i\right]\right|}{Z\left[i\right]}$\;
+$c= testConverge(\Delta z_{max},ZPrec\left[j\right],Z\left[j\right])$\;
\If{$c > \Delta z_{max}$ }{
$\Delta z_{max}$=c\;}
}
+
}
\end{algorithm}
EAGS: z^{k+1}_{i}=\frac{p(z^{k}_{i})}{p'(z^{k}_{i})-p(z^{k}_{i})(\sum^{i-1}_{j=1}\frac{1}{z^{k}_{i}-z^{k+1}_{j}}+\sum^{n}_{j=i+1}\frac{1}{z^{k}_{i}-z^{k}_{j}})}, i=1,...,n.
\end{equation}
%%Here a finiched my revision %%
-Using Equation.~\ref{eq:Aberth-H-GS} to update the vector solution \textit{Z}, we expect the Gauss-Seidel iteration to converge more quickly because, just as its ancestor (for solving linear systems of equations), it uses the most fresh computed roots $z^{k+1}_{i}$.
+Using Equation.~\ref{eq:Aberth-H-GS} to update the vector solution
+\textit{Z}, we expect the Gauss-Seidel iteration to converge more
+quickly because, just as any Jacobi algorithm (for solving linear systems of equations), it uses the most fresh computed roots $z^{k+1}_{i}$.
The $4^{th}$ step of the algorithm checks the convergence condition using Equation.~\ref{eq:Aberth-Conv-Cond}.
Both steps 3 and 4 use 1 thread to compute all the $n$ roots on CPU, which is very harmful for performance in case of the large degree polynomials.
-\subsection{A Parallel implementation with CUDA }
+\subsection{Parallel implementation with CUDA }
On the CPU, both steps 3 and 4 contain the loop \verb=for= and a single thread executes all the instructions in the loop $n$ times. In this subsection, we explain how the GPU architecture can compute this loop and reduce the execution time.
-In the GPU, the schduler assigns the execution of this loop to a group of threads organised as a grid of blocks with block containing a number of threads. All threads within a block are executed concurrently in parallel. The instructions run on the GPU are grouped in special function called kernels. It's up to the programmer, to describe the execution context, that is the size of the Grid, the number of blocks and the number of threads per block upon the call of a given kernel, according to a special syntax defined by CUDA.
+In the GPU, the scheduler assigns the execution of this loop to a
+group of threads organised as a grid of blocks with block containing a
+number of threads. All threads within a block are executed
+concurrently in parallel. The instructions run on the GPU are grouped
+in special function called kernels. With CUDA, a programmer must
+describe the kernel execution context: the size of the Grid, the number of blocks and the number of threads per block.
-In CUDA programming, all the instructions of the \verb=for= loop are executed by the GPU as a kernel. A kernel is a function written in CUDA and defined by the \verb=__global__= qualifier added before a usual \verb=C= function, which instructs the compiler to generate appropriate code to pass it to the CUDA runtime in order to be executed on the GPU.
+%In CUDA programming, all the instructions of the \verb=for= loop are executed by the GPU as a kernel. A kernel is a function written in CUDA and defined by the \verb=__global__= qualifier added before a usual \verb=C= function, which instructs the compiler to generate appropriate code to pass it to the CUDA runtime in order to be executed on the GPU.
Algorithm~\ref{alg2-cuda} shows a sketch of the Ehrlich-Aberth algorithm using 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)}
+\KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (error tolerance threshold), P(Polynomial to solve), $\Delta z_{max}$ (maximum value of stop condition)}
-\KwOut {Z(The solution root's vector)}
+\KwOut {Z (The solution root's vector)}
\BlankLine
-Initialization of the coeffcients of the polynomial to solve\;
+Initialization of the coefficients of the polynomial to solve\;
Initialization of the solution vector $Z^{0}$\;
Allocate and copy initial data to the GPU global memory\;
-
+k=0\;
\While {$\Delta z_{max}\succ \epsilon$}{
Let $\Delta z_{max}=0$\;
-$ kernel\_save(d\_z^{k-1})$\;
-$ kernel\_update(d\_z^{k})$\;
-$kernel\_testConverge(\Delta z_{max},d_z^{k},d_z^{k-1})$\;
+$ kernel\_save(d\_Z^{k-1})$\;
+k=k+1\;
+$ kernel\_update(d\_Z^{k})$\;
+$kernel\_testConverge(\Delta z_{max},d\_Z^{k},d\_Z^{k-1})$\;
+
}
\end{algorithm}
~\\
After the initialisation step, all data of the root finding problem to be solved must be copied from the CPU memory to the GPU global memory, because the GPUs only access data already present in their memories. Next, all the data-parallel arithmetic operations inside the main loop \verb=(do ... while(...))= are executed as kernels by the GPU. The first kernel named \textit{save} in line 6 of Algorithm~\ref{alg2-cuda} consists in saving the vector of polynomial's root found at the previous time-step in GPU memory, in order to check the convergence of the roots after each iteration (line 8, Algorithm~\ref{alg2-cuda}).
-The second kernel executes the iterative function $H$ and updates $z^{k}$, according to Algorithm~\ref{alg3-update}. We notice that the update kernel is called in two forms, separated with the value of \emph{R} which determines the radius beyond which we apply the logarithm computation of the power of a complex.
+The second kernel executes the iterative function $H$ and updates
+$z^{k}$, according to Algorithm~\ref{alg3-update}. We notice that the
+update kernel is called in two forms, separated with the value of
+\emph{R} which determines the radius beyond which we apply the
+exponential logarithm algorithm.
\begin{algorithm}[H]
\label{alg3-update}
\eIf{$(\left|Z^{(k)}\right|<= R)$}{
$kernel\_update(d\_z^{k})$\;}
{
-$kernel\_update\_Log(d\_z^{k})$\;
+$kernel\_update\_ExpoLog(d\_z^{k})$\;
}
\end{algorithm}
-The first form executes formula \ref{eq:SimplePolynome} if the modulus of the current complex is less than the a certain value called the radius i.e. ($ |z^{k}_{i}|<= R$), else the kernel executes formulas (Eq.~\ref{deflncomplex},Eq.~\ref{defexpcomplex}). The radius $R$ is evaluated as :
+The first form executes formula \ref{eq:SimplePolynome} if the modulus
+of the current complex is less than the a certain value called the
+radius i.e. ($ |z^{k}_{i}|<= R$), else the kernel executes the EA.EL
+function Eq.~\ref{Log_H2}
+( with Eq.~\ref{deflncomplex}, Eq.~\ref{defexpcomplex}). The radius $R$ is evaluated as :
$$R = \exp( \log(DBL\_MAX) / (2*n) )$$ where $DBL\_MAX$ stands for the maximum representable double value.
\subsection{The execution time in seconds of Ehrlich-Aberth algorithm on CPU OpenMP (1 core, 4 cores) vs. on a Tesla GPU}
-%\begin{figure}[H]
-%\centering
- % \includegraphics[width=0.8\textwidth]{figures/Compar_EA_algorithm_CPU_GPU}
-%\caption{The execution time in seconds of Ehrlich-Aberth algorithm on CPU core vs. on a Tesla GPU}
-%\label{fig:01}
-%\end{figure}
\begin{figure}[H]
\centering
\centering
\includegraphics[width=0.8\textwidth]{figures/influence_nb_threads}
\caption{Influence of the number of threads on the execution times of different polynomials (sparse and full)}
-\label{fig:01}
+\label{fig:02}
\end{figure}
The figure 2 show that, the best execution time for both sparse and full polynomial are given when the threads number varies between 64 and 256 threads per bloc. We notice that with small polynomials the best number of threads per block is 64, Whereas, the large polynomials the best number of threads per block is 256. However,In the following experiments we specify that the number of thread by block is 256.
\centering
\includegraphics[width=0.8\textwidth]{figures/sparse_full_explog}
\caption{The impact of exp-log solution to compute very high degrees of polynomial.}
-\label{fig:01}
+\label{fig:03}
\end{figure}
The figure 3, show a comparison between the execution time of the Ehrlich-Aberth algorithm applying exp.log solution and the execution time of the Ehrlich-Aberth algorithm without applying exp.log solution, with full and sparse polynomials degrees. We can see that the execution time for the both algorithms are the same while the full polynomials degrees are less than 4000 and full polynomials are less than 150,000. After,we show clearly that the classical version of Ehrlich-Aberth algorithm (without applying log.exp) stop to converge and can not solving any polynomial sparse or full. In counterpart, the new version of Ehrlich-Aberth algorithm (applying log.exp solution) can solve very high and large full polynomial exceed 100,000 degrees.
in fact, when the modulus of the roots are up than \textit{R} given in ~\ref{R},this exceed the limited number in the mantissa of floating points representations and can not compute the iterative function given in ~\ref{eq:Aberth-H-GS} to obtain the root solution, who justify the divergence of the classical Ehrlich-Aberth algorithm. However, applying log.exp solution given in ~\ref{sec2} took into account the limit of floating using the iterative function in(Eq.~\ref{Log_H1},Eq.~\ref{Log_H2} and allows to solve a very large polynomials degrees .
-%\begin{figure}[H]
-\%centering
- %\includegraphics[width=0.8\textwidth]{figures/log_exp_Sparse}
-%\caption{The impact of exp-log solution to compute very high degrees of polynomial.}
-%\label{fig:01}
-%\end{figure}
-
-%we report the performances of the exp.log for the Ehrlich-Aberth algorithm for solving very high degree of polynomial.
-
\subsection{A comparative study between Ehrlich-Aberth algorithm and Durand-kerner algorithm}
In this part, we are interesting to compare the simultaneous methods, Ehrlich-Aberth and Durand-Kerner in parallel computer using GPU. We took into account the execution time, the number of iteration and the polynomial's size. for the both sparse and full polynomials.
\centering
\includegraphics[width=0.8\textwidth]{figures/EA_DK}
\caption{The execution time of Ehrlich-Aberth versus Durand-Kerner algorithm on GPU}
-\label{fig:01}
+\label{fig:04}
\end{figure}
This figure show the execution time of the both algorithm EA and DK with sparse polynomial degrees ranging from 1000 to 1000000. We can see that the Ehrlich-Aberth algorithm are faster than Durand-Kerner algorithm, with an average of 25 times as fast. Then, when degrees of polynomial exceed 500000 the execution time with EA is of the order 100 whereas DK passes in the order 1000. %with double precision not exceed $10^{-5}$.
\centering
\includegraphics[width=0.8\textwidth]{figures/EA_DK_nbr}
\caption{The iteration number of Ehrlich-Aberth versus Durand-Kerner algorithm}
-\label{fig:01}
+\label{fig:05}
\end{figure}
%\subsubsection{The execution time of Ehrlich-Aberth algorithm on OpenMP(1 core, 4 cores) vs. on a Tesla GPU}
