\begin{equation}
\label{Eq:Hi}
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=0,. . . .,n
+{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
\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
\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
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 6 of
+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
-8, Algorithm~\ref{alg2-cuda}).
+10, Algorithm~\ref{alg2-cuda}).
-The second kernel executes the iterative function $H$ and updates
-Z, according to Algorithm~\ref{alg3-update}. We notice that the
+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.
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}
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 K40c GPU. We chose different sparse polynomials with degrees
+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
