DK: z_i^{k+1}=z_{i}^{k}-\frac{P(z_i^{k})}{\prod_{i\neq j}(z_i^{k}-z_j^{k})}, i = 1, . . . , n,
\end{equation}
%%\end{center}
-where $z_i^k$ is the $i^{th}$ root of the polynomial $P$ at the
+where $z_i^k$ is the $i^{th}$ root of the polynomial $p$ at the
iteration $k$.
EA: z_i^{k+1}=z_i^{k}-\frac{1}{{\frac {P'(z_i^{k})} {P(z_i^{k})}}-{\sum_{i\neq j}\frac{1}{(z_i^{k}-z_j^{k})}}}, i = 1, . . . , n,
\end{equation}
%%\end{center}
-where $P'(z)$ is the polynomial derivative of $P$ evaluated in the
+where $p'(z)$ is the polynomial derivative of $p$ evaluated in the
point $z$.
Aberth, Ehrlich and Farmer-Loizou~\cite{Loizou83} have proved that
diverges, but the first two methods have speed-up equal to 5.5. Later,
Freeman and Bane~\cite{Freemanall90} considered asynchronous
algorithms, in which each processor continues to update its
-approximations even though the latest values of other $z_i((k))$
+approximations even though the latest values of other $z_i^{k}$
have not been received from the other processors, in contrast with synchronous algorithms where it would wait those values before making a new iteration.
Couturier and al.~\cite{Raphaelall01} proposed two methods of parallelization for
a shared memory architecture and for distributed memory one. They were able to
-compute the roots of sparse polynomials of degree 10000 in 430 seconds with only 8
+compute the roots of sparse polynomials of degree 10,000 in 430 seconds with only 8
personal computers and 2 communications per iteration. Comparing to the sequential implementation
-where it takes up to 3300 seconds to obtain the same results, the authors show an interesting speedup.
+where it takes up to 3,300 seconds to obtain the same results, the authors show an interesting speedup.
Very few works had been performed since this last work until the appearing of
the Compute Unified Device Architecture (CUDA)~\cite{CUDA10}, a
Durand-Kerner method on GPU. Their main
result showed that a parallel CUDA implementation is about 10 times faster than
the sequential implementation on a single CPU for sparse
-polynomials of degree 48000.
+polynomials of degree 48,000.
In this paper, we focus on the implementation of the Ehrlich-Aberth
\ref{secStateofArt}. In Section~\ref{sec5} we propose a parallel
implementation of the Ehrlich-Aberth method on GPU and discuss
it. Section~\ref{sec6} presents and investigates our implementation
-and experimental study results. Finally, Section~\ref{sec7} 6 concludes
+and experimental study results. Finally, Section~\ref{sec7} concludes
this paper and gives some hints for future research directions in this
topic.
\section{Ehrlich-Aberth method}
\label{sec1}
A cubically convergent iteration method for finding zeros of
-polynomials was proposed by O. Aberth~\cite{Aberth73}. In the
-following we present the main stages of our implementation the Ehrlich-Aberth method.
+polynomials was proposed by O. Aberth~\cite{Aberth73}. The Ehrlich-Aberth method contain 4 main steps, presented in the following.
%The Aberth method is a purely algebraic derivation.
%To illustrate the derivation, we let $w_{i}(z)$ be the product of linear factors
\subsection{Polynomials Initialization}
-The initialization of a polynomial p(z) is done by setting each of the $n$ complex coefficients $a_{i}$:
+The initialization of a polynomial $p(z)$ is done by setting each of the $n$ complex coefficients $a_{i}$:
\begin{equation}
\label{eq:SimplePolynome}
\end{equation}
-\subsection{Vector $z^{(0)}$ Initialization}
+\subsection{Vector $Z^{(0)}$ Initialization}
\label{sec:vec_initialization}
-As for any iterative method, we need to choose $n$ initial guess points $z^{(0)}_{i}, i = 1, . . . , n.$
+As for any iterative method, we need to choose $n$ initial guess points $z^{0}_{i}, i = 1, . . . , n.$
The initial guess is very important since the number of steps needed by the iterative method to reach
a given approximation strongly depends on it.
In~\cite{Aberth73} the Ehrlich-Aberth iteration is started by selecting $n$
\begin{equation}
\label{Eq:Hi}
-EA2: z^{k+1}=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=0,. . . .,n
+EA2: 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}
It can be noticed that this equation is equivalent to Eq.~\ref{Eq:EA},
but we prefer the latter one because we can use it to improve the
as well as the Durand-Kerner implement, suffers from overflow problems. This
situation occurs, for instance, in the case where a polynomial
having positive coefficients and a large degree is computed at a
-point $\xi$ where $|\xi| > 1$, where $|x|$ stands for the modolus of a complex $x$. Indeed, the limited number in the
-mantissa of floating points representations makes the computation of p(z) wrong when z
+point $\xi$ where $|\xi| > 1$, where $|z|$ stands for the modolus of a complex $z$. Indeed, the limited number in the
+mantissa of floating points representations makes the computation of $p(z)$ wrong when z
is large. For example $(10^{50}) +1+ (- 10^{50})$ will give the wrong result
of $0$ instead of $1$. Consequently, we can not compute the roots
for large degrees. This problem was early discussed in
%%$$ \exp \bigl( \ln(p(z)_{k})-ln(\ln(p(z)_{k}^{'}))- \ln(1- \exp(\ln(p(z)_{k})-ln(\ln(p(z)_{k}^{'})+\ln\sum_{i\neq j}^{n}\frac{1}{z_{k}-z_{j}})$$
\begin{equation}
\label{Log_H2}
-EA.EL: z^{k+1}=z_{i}^{k}-\exp \left(\ln \left(
+EA.EL: z^{k+1}_{i}=z_{i}^{k}-\exp \left(\ln \left(
p(z_{i}^{k})\right)-\ln\left(p'(z^{k}_{i})\right)- \ln
\left(1-Q(z^{k}_{i})\right)\right),
\end{equation}
\begin{equation}
\label{Log_H1}
Q(z^{k}_{i})=\exp\left( \ln (p(z^{k}_{i}))-\ln(p'(z^{k}_{i}))+\ln \left(
-\sum_{k\neq j}^{n}\frac{1}{z^{k}_{i}-z^{k}_{j}}\right)\right).
+\sum_{i\neq j}^{n}\frac{1}{z^{k}_{i}-z^{k}_{j}}\right)\right)i=1,...,n,
\end{equation}
-This solution is applied when the root except the circle unit, represented by the radius $R$ evaluated in C language as:
-
-\begin{verbatim}
+This solution is applied when the root except the circle unit, represented by the radius $R$ evaluated in C language as :
+\begin{equation}
+\label{R.EL}
R = exp(log(DBL_MAX)/(2*n) );
-\end{verbatim}
+\end{equation}
+
%\begin{equation}
finding algorithms. The first approach aims at reducing the total number of
iterations as by Miranker
~\cite{Mirankar68,Mirankar71}, Schedler~\cite{Schedler72} and
-Winogard~\cite{Winogard72}. The second approach aims at reducing the
+Winograd~\cite{Winogard72}. The second approach aims at reducing the
computation time per iteration, as reported
in~\cite{Benall68,Jana06,Janall99,Riceall06}.
in~\cite{Mirankar71} proposed two versions of parallel algorithm
for the Durand-Kerner method, and Ehrlich-Aberth method on a model of
Optoelectronic Transpose Interconnection System (OTIS).The
-algorithms are mapped on an OTIS-2D torus using N processors. This
-solution needs N processors to compute N roots, which is not
+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 did not mention implementations
%able to compute the roots of large degree polynomials (higher then
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.
+polynomials of 48,000.
%In this paper we present a parallel implementation of Ehrlich-Aberth
%method on GPUs for sparse and full polynomials with high degree (up
%to $1,000,000$).
In order to implement the Ehrlich-Aberth method in CUDA, it is
possible to use the Jacobi scheme or the Gauss Seidel one. With the
Jacobi iteration, at iteration $k+1$ we need all the previous values
-$z^{(k)}_{i}$ to compute the new values $z^{(k+1)}_{i}$, that is :
+$z^{k}_{i}$ to compute the new values $z^{k+1}_{i}$, that is :
\begin{equation}
-EAJ: z^{k+1}_{i}=\frac{p(z^{k}_{i})}{p'(z^{k}_{i})-p(z^{k}_{i})\sum^{n}_{j=1 j\neq i}\frac{1}{z^{k}_{i}-z^{k}_{j}}}, i=1,...,n.
+EAJ: 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}
With the Gauss-Seidel iteration, we have:
+%\begin{equation}
+%\label{eq:Aberth-H-GS}
+%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}
+
\begin{equation}
\label{eq:Aberth-H-GS}
-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.
+EAGS: 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^{i-1}_{j=1}\frac{1}{z^{k}_{i}-z^{k+1}_{j}}+\sum_{j=1,j\neq i}^{j=n}{\frac{1}{(z_{i}^{k}-z_{j}^{k})}})}, i=1,. . . .,n
\end{equation}
-%%Here a finiched my revision %%
+
Using Eq.~\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 $z^{k+1}_{i}$.
%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.
+Algorithm~\ref{alg2-cuda} shows sketch of the Ehrlich-Aberth algorithm using CUDA.
+\begin{enumerate}
\begin{algorithm}[H]
\label{alg2-cuda}
%\LinesNumbered
\caption{CUDA Algorithm to find roots with the Ehrlich-Aberth method}
\KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (error tolerance
- threshold), P(Polynomial to solve), Pu (the derivative of P), $n$ (Polynomial's degrees), $\Delta z_{max}$ (maximum value of stop condition)}
+ threshold), P (Polynomial to solve), Pu (Derivative of P), $n$ (Polynomial's degrees), $\Delta z_{max}$ (Maximum value of stop condition)}
-\KwOut {$Z$ (The solution root's vector), $ZPrec$ (the previous solution root's vector)}
+\KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}
\BlankLine
-Initialization of the of P\;
-Initialization of the of Pu\;
-Initialization of the solution vector $Z^{0}$\;
-Allocate and copy initial data to the GPU global memory\;
-k=0\;
+\item Initialization of the of P\;
+\item Initialization of the of Pu\;
+\item Initialization of the solution vector $Z^{0}$\;
+\item Allocate and copy initial data to the GPU global memory\;
+\item k=0\;
\While {$\Delta z_{max} > \epsilon$}{
- Let $\Delta z_{max}=0$\;
-$ kernel\_save(ZPrec,Z)$\;
-k=k+1\;
-$ kernel\_update(Z,P,Pu)$\;
-$kernel\_testConverge(\Delta z_{max},Z,ZPrec)$\;
+\item Let $\Delta z_{max}=0$\;
+\item $ kernel\_save(ZPrec,Z)$\;
+\item k=k+1\;
+\item $ kernel\_update(Z,P,Pu)$\;
+\item $kernel\_testConverge(\Delta z_{max},Z,ZPrec)$\;
}
-Copy results from GPU memory to CPU memory\;
+\item Copy results from GPU memory to CPU memory\;
\end{algorithm}
+\end{enumerate}
~\\
-After the initialization 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
-$d\_Z$, according to Algorithm~\ref{alg3-update}. We notice that the
-update kernel is called in two forms, separated with the value of
+After the initialization step, all data of the root finding problem
+must be copied from the CPU memory to the GPU global memory. Next, all
+the data-parallel arithmetic operations inside the main loop
+\verb=(while(...))= are executed as kernels by the GPU. The
+first kernel named \textit{save} in line 7 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
+10, Algorithm~\ref{alg2-cuda}).
+
+The second kernel executes the iterative function and updates
+$Z$, according to Algorithm~\ref{alg3-update}. We notice that the
+update kernel is called in two forms, according to the value
\emph{R} which determines the radius beyond which we apply the
exponential logarithm algorithm.
\caption{Kernel update}
\eIf{$(\left|Z\right|<= R)$}{
-$kernel\_update((Z,P,Pu)$\;}
+$kernel\_update(Z,P,Pu)$\;}
{
-$kernel\_update\_ExpoLog((Z,P,Pu))$\;
+$kernel\_update\_ExpoLog(Z,P,Pu)$\;
}
\end{algorithm}
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.
+(with Eq.~\ref{deflncomplex}, Eq.~\ref{defexpcomplex}). The radius $R$ is evaluated as in Eq.~\ref{R.EL}.
The last kernel checks the convergence of the roots after each update
-of $Z^{(k)}$, according to formula Eq.~\ref{eq:Aberth-Conv-Cond}. We used the functions of the CUBLAS Library (CUDA Basic Linear Algebra Subroutines) to implement this kernel.
+of $Z^{k}$, according to formula Eq.~\ref{eq:Aberth-Conv-Cond}. We used the functions of the CUBLAS Library (CUDA Basic Linear Algebra Subroutines) to implement this kernel.
The kernel terminates its computations when all the roots have
converged. It should be noticed that, as blocks of threads are
%polynomials. The execution time remains the
%element-key which justifies our work of parallelization.
For our tests, a CPU Intel(R) Xeon(R) CPU
-E5620@2.40GHz and a GPU K40 (with 6 Go of ram) is used.
+E5620@2.40GHz and a GPU K40 (with 6 Go of ram) are used.
%\subsection{Comparative study}
on a CPU with OpenMP (1 core, 4 cores) and on a Tesla GPU}
\label{fig:01}
\end{figure}
-%%Figure 1 %%show a comparison of execution time between the parallel and sequential version of the Ehrlich-Aberth algorithm with sparse polynomial exceed 100000,
-In Figure~\ref{fig:01}, we report respectively the execution time of the Ehrlich-Aberth method implemented initially on one core of the Quad-Core Xeon E5620 CPU than on four cores of the same machine with \textit{OpenMP} platform and the execution time of the same method implemented on one Nvidia Tesla K40c GPU, with sparse polynomial degrees ranging from 100,000 to 1,000,000. We can see that the method implemented on the GPU are faster than those implemented on the CPU (4 cores). This is due to the GPU ability to compute the data-parallel functions faster than its CPU counterpart. However, the execution time for the CPU(4 cores) implementation exceed 5,000 s for 250,000 degrees polynomials, in counterpart the GPU implementation for the same polynomials not reach 100 s, more than again, with an execution time under to 2,500 s CPU (4 cores) implementation can resolve polynomials degrees of only 200,000, whereas GPU implementation can resolve polynomials more than 1,000,000 degrees. We can also notice that the GPU implementation are almost 47 faster then those implementation on the CPU(4 cores). However the CPU(4 cores) implementation are almost 4 faster then his implementation on CPU (1 core). Furthermore, we verify that the number of iterations and the convergence precision is the same for the both CPU and GPU implementation. %This reduction of time allows us to compute roots of polynomial of more important degree at the same time than with a CPU.
+%%Figure 1 %%show a comparison of execution time between the parallel
+%%and sequential version of the Ehrlich-Aberth algorithm with sparse
+%%polynomial exceed 100000,
+
+In Figure~\ref{fig:01}, we report the execution times of the
+Ehrlich-Aberth method on one core of a Quad-Core Xeon E5620 CPU, on
+four cores on the same machine with \textit{OpenMP} and on a Nvidia
+Tesla K40 GPU. We chose different sparse polynomials with degrees
+ranging from 100,000 to 1,000,000. We can see that the implementation
+on the GPU is faster than those implemented on the CPU.
+However, the execution time for the
+CPU (4 cores) implementation exceed 5,000s for 250,000 degrees
+polynomials. In counterpart, the GPU implementation for the same
+polynomials do not take more 100s. With the GPU
+we can solve high degrees polynomials very quickly up to degree
+ of 1,000,000. We can also notice that the GPU implementation are
+ almost 40 faster then those implementation on the CPU (4 cores).
+
+
+
+
+%This reduction of time allows us to compute roots of polynomial of more important degree at the same time than with a CPU.
%We notice that the convergence precision is a round $10^{-7}$ for the both implementation on CPU and GPU. Consequently, we can conclude that Ehrlich-Aberth on GPU are faster and accurately then CPU implementation.
\subsection{Influence of the number of threads on the execution times of different polynomials (sparse and full)}
-To optimize the performances of an algorithm on a GPU, it is necessary to maximize the use of cores GPU (maximize the number of threads executed in parallel) and to optimize the use of the various memoirs GPU. In fact, it is interesting to see the influence of the number of threads per block on the execution time of Ehrlich-Aberth algorithm.
-For that, we notice that the maximum number of threads per block for the Nvidia Tesla K40 GPU is 1024, so we varied the number of threads per block from 8 to 1,024. We took into account the execution time for both sparse and full of 10 different polynomials of size 50,000 and 10 different polynomials of size 500,000 degrees.
+To optimize the performances of an algorithm on a GPU, it is necessary to maximize the use of cores GPU (maximize the number of threads executed in parallel). In fact, it is interesting to see the influence of the number of threads per block on the execution time of Ehrlich-Aberth algorithm.
+For that, we notice that the maximum number of threads per block for the Nvidia Tesla K40 GPU is 1,024, so we varied the number of threads per block from 8 to 1,024. We took into account the execution time for both sparse and full of 10 different polynomials of size 50,000 and 10 different polynomials of size 500,000 degrees.
\begin{figure}[htbp]
\centering
\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\textwidth]{figures/sparse_full_explog}
-\caption{The impact of exp.log solution to compute very high degrees of polynomial.}
+\caption{The impact of exp-log solution to compute very high degrees of polynomial.}
\label{fig:03}
\end{figure}
with full and sparse polynomials degrees. We can see that the
execution times for both algorithms are the same with full polynomials
degrees less than 4,000 and sparse polynomials less than 150,000. We
-also clearly show that the classical version (without log-exp) of
+also clearly show that the classical version (without exp-log) of
Ehrlich-Aberth algorithm do not converge after these degree with
sparse and full polynomials. In counterpart, the new version of
Ehrlich-Aberth algorithm with the exp-log solution can solve very
high degree polynomials.
-%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 .
+%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 exp-log 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 .
\label{fig:04}
\end{figure}
-\begin{figure}[htbp]
-\centering
- \includegraphics[width=0.8\textwidth]{figures/EA_DK1}
-\caption{Execution times of the Durand-Kerner and the Ehrlich-Aberth methods on GPU}
-\label{fig:0}
-\end{figure}
-
Figure~\ref{fig:04} shows the execution times of both methods with
sparse polynomial degrees ranging from 1,000 to 1,000,000. We can see
that the Ehrlich-Aberth algorithm is faster than Durand-Kerner
In future works, we plan to investigate the possibility of using
several multiple GPUs simultaneously, either with multi-GPU machine or
-with cluster of GPUs.
+with cluster of GPUs. It may also be interesting to study the
+implementation of other root finding polynomial methods on GPU.
