:
\begin{equation}
+\label{eq:SimplePolynome}
p(z)=\sum{a_{i}z^{n-i}} , a_{n} \neq 0,a_{0}=1, a_{i}\subset C
\end{equation}
circles and relies on the result of~\cite{Ostrowski41}.
\begin{equation}
+\label{eq:radiusR}
%%\begin{align}
\sigma_{0}=\frac{u+v}{2};u=\frac{\sum_{i=1}^{n}u_{i}}{n.max_{i=1}^{n}u_{i}};
v=\frac{\sum_{i=0}^{n-1}v_{i}}{n.min_{i=0}^{n-1}v_{i}};\\
}
\end{algorithm}
-The first form executes the formula (8) if the modulus is of the current complex is less than the radius i.e. ($ |z^{k}_{i}|<= R$), else the kernel executes formulas (13,14). The radius R is evaluated as :
+The first form executes formula \ref{eq:SimplePolynome} if the modulus of the current complex is less than the a certain value called the radius i.e. ($ |z^{k}_{i}|<= R$), else the kernel executes formulas (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.
The last kernel verifies the convergence of the roots after each update of $Z^{(k)}$, according to formula. We used the functions of the CUBLAS Library (CUDA Basic Linear Algebra Subroutines) to implement this kernel.
-The kernels terminates it computations when all the root are converged. Finally, the solution of the root finding problem is copied back from the GPU global memory to the CPU memory. We use the communication functions of CUDA for the memory allocations in the GPU \verb=(cudaMalloc())= and the data transfers from the CPU memory to the GPU memory \verb=(cudaMemcpyHostToDevice)=
-or from the GPU memory to the CPU memory \verb=(cudaMemcpyDeviceToHost))=.
+The kernels terminate it computations when all the roots converge. Finally, the solution of the root finding problem is copied back from GPU global memory to CPU memory. We use the communication functions of CUDA for the memory allocation in the GPU \verb=(cudaMalloc())= and for data transfers from the CPU memory to the GPU memory \verb=(cudaMemcpyHostToDevice)=
+or from GPU memory to CPU memory \verb=(cudaMemcpyDeviceToHost))=.
%%HIER END MY REVISIONS (SIDER)
\subsection{Experimental study}
\paragraph{Aberth algorithm on CPU and GPU}
-\begin{table}[!ht]
- \centering
- \begin{tabular} {|R{2cm}|L{2.5cm}|L{2.5cm}|L{1.5cm}|L{1.5cm}|}
- \hline Polynomial's degrees & $T_{exe}$ on CPU & $T_{exe}$ on GPU & CPU iteration & GPU iteration\\
- \hline 5000 & 1.90 & 0.40 & 18 & 17\\
- \hline 10000 & 172.723 & 0.59 & 21 & 24\\
- \hline 20000 & 172.723 & 1.52 & 21 & 25\\
- \hline 30000 & 172.723 & 2.77 & 21 & 33\\
- \hline 50000 & 172.723 & 3.92 & 21 & 18\\
- \hline 500000 & $>$1h & 497.109 & & 24\\
- \hline 1000000 & $>$1h & 1,524.51& & 24\\
- \hline
- \end{tabular}
- \caption{the convergence of Aberth algorithm}
- \label{tab:theConvergenceOfAberthAlgorithm}
-\end{table}
+%\begin{table}[!ht]
+% \centering
+% \begin{tabular} {|R{2cm}|L{2.5cm}|L{2.5cm}|L{1.5cm}|L{1.5cm}|}
+% \hline Polynomial's degrees & $T_{exe}$ on CPU & $T_{exe}$ on GPU & CPU iteration & GPU iteration\\
+% \hline 5000 & 1.90 & 0.40 & 18 & 17\\
+% \hline 10000 & 172.723 & 0.59 & 21 & 24\\
+% \hline 20000 & 172.723 & 1.52 & 21 & 25\\
+% \hline 30000 & 172.723 & 2.77 & 21 & 33\\
+% \hline 50000 & 172.723 & 3.92 & 21 & 18\\
+% \hline 500000 & $>$1h & 497.109 & & 24\\
+% \hline 1000000 & $>$1h & 1,524.51& & 24\\
+% \hline
+% \end{tabular}
+% \caption{the convergence of Aberth algorithm}
+% \label{tab:theConvergenceOfAberthAlgorithm}
+%\end{table}
+\begin{figure}[htbp]
+\centering
+ \includegraphics[width=0.8\textwidth]{figures/Compar_EA_algorithm_CPU_GPU}
+\caption{Aberth algorithm on CPU and GPU}
+\label{fig:01}
+\end{figure}
+
+
\paragraph{The impact of the thread's number into the convergence of Aberth algorithm}
-\begin{table}[!h]
- \centering
- \begin{tabular} {|R{2.5cm}|L{2.5cm}|L{2.5cm}|}
- \hline Thread's numbers & Execution time &Number of iteration\\
- \hline 1024 & 523 & 27\\
- \hline 512 & 449.426 & 24\\
- \hline 256 & 440.805 & 24\\
- \hline 128 & 456.175 & 22\\
- \hline 64 & 472.862 & 23\\
- \hline 32 & 830.152 & 24\\
- \hline 8 & 2632.78 & 23 \\
- \hline
- \end{tabular}
- \caption{The impact of the thread's number into the convergence of Aberth algorithm}
- \label{tab:Theimpactofthethread'snumberintotheconvergenceofAberthalgorithm}
-
-\end{table}
+%\begin{table}[!h]
+% \centering
+% \begin{tabular} {|R{2.5cm}|L{2.5cm}|L{2.5cm}|}
+% \hline Thread's numbers & Execution time &Number of iteration\\
+% \hline 1024 & 523 & 27\\
+% \hline 512 & 449.426 & 24\\
+% \hline 256 & 440.805 & 24\\
+% \hline 128 & 456.175 & 22\\
+% \hline 64 & 472.862 & 23\\
+% \hline 32 & 830.152 & 24\\
+% \hline 8 & 2632.78 & 23 \\
+% \hline
+% \end{tabular}
+% \caption{The impact of the thread's number into the convergence of Aberth algorithm}
+% \label{tab:Theimpactofthethread'snumberintotheconvergenceofAberthalgorithm}
+%
+%\end{table}
\begin{figure}[htbp]
