%% In this sequential algorithm, one CPU thread executes all the steps. Let us look to the $3^{rd}$ step i.e. the execution of the iterative function, 2 sub-steps are needed. The first sub-step \textit{save}s the solution vector of the previous iteration, the second sub-step \textit{update}s or computes the new values of the roots vector.
\subsection{Parallel implementation with CUDA }
-
+\KG{cette section a totalement modifié}
%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
If the modulus of the current complex is less than a given value called the
radius i.e. ($ |z^{k}_{i}|<= R$), then the classical form of the EA
function Eq.~\ref{Eq:Hi} is executed, else the EA.EL function Eq.~\ref{Log_H2} is executed.
-(with Eq.~\ref{deflncomplex}, Eq.~\ref{defexpcomplex}). The radius $R$ is evaluated as in Eq.~\ref{R.EL}. \KG{experimentally it is difficult to solve the high degree polynomial with the classical Ehrlich-Aberth method, so if the root are under to the unit circle ($R$)the kernel \textit{update} is called in the EA.EL function Eq.~\ref{Log_H2} (with Eq.~\ref{deflncomplex}, Eq.~\ref{defexpcomplex}) line 3, to put into account the limited of grander floating manipulated by processors and compute more roots}.
-\KG{} We notice that we used \verb= cuDoubleComplex= to exploit the complex number in CUDA, and the functions of the CUBLAS library to implement some vector operations on the GPU. We use the following functions:
+(with Eq.~\ref{deflncomplex}, Eq.~\ref{defexpcomplex}). The radius $R$ is evaluated as in Eq.~\ref{R.EL}. \KG{experimentally it is difficult to solve the high degree polynomial with the classical Ehrlich-Aberth method, so if the root are under to the unit circle ($R$)the kernel \textit{update} is called in the EA.EL function Eq.~\ref{Log_H2} (with Eq.~\ref{deflncomplex}, Eq.~\ref{defexpcomplex}) line 3, to put into account the limited of grander floating manipulated by processors and compute more roots}. We notice that we used \verb= cuDoubleComplex= to exploit the complex number in CUDA, and the functions of the CUBLAS library to implement some vector operations on the GPU. We use the following functions:
\begin{itemize}
\item \verb= cublasIdamax()= for the
Each GPU threads in grid compute one root en parallel, if the polynomial size exceed the capacity of the grid the G.S schema are finely executed, like the grid can only compute << Blocks,Threads>> roots at the same time, if we need to compute more roots, the grid can used the roots previously executed to compute other root ih the same iteration, like the following schema:
-%\begin{figure}[htbp]
-%\centering
- % \includegraphics[width=0.8\textwidth]{figures/G.S}
-%\caption{Gauss Seidel iteration}
-%\label{fig:08}
-
+\begin{figure}[htbp]
+\centering
+ \includegraphics[width=0.8\textwidth]{figures/GS}
+\caption{Gauss Seidel iteration}
+\label{fig:08}
+\end{figure}
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.
