%%\usepackage[french]{babel}
\usepackage{amsmath,amsfonts,amssymb}
\usepackage[ruled,vlined]{algorithm2e}
+%\usepackage[french,boxed,linesnumbered]{algorithm2e}
\usepackage{array,multirow,makecell}
\setcellgapes{1pt}
\makegapedcells
~\\
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.
-There exists two ways to execute the iterative function that we call a Jacobi one and a 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 $z^{(k+1)}_{i}$, taht is :
+There exists two ways to execute the iterative function that we call a Jacobi one and a 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 :
\begin{equation}
H(i,z^{k+1})=\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.
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}
+\section{Experimental study}
-\subsubsection{Definition of the polynomial used}
-We use a polynomial of the following form for which the
-roots are distributed on 2 distinct circles:
+\subsection{Definition of the polynomial used}
+We use two forms of polynomials:
+\paragraph{sparse polynomial}:
+in this following form, the roots are distributed on 2 distinct circles:
\begin{equation}
\forall \alpha_{1} \alpha_{2} \in C,\forall n_{1},n_{2} \in N^{*}; P(z)= (z^{n^{1}}-\alpha_{1})(z^{n^{2}}-\alpha_{2})
\end{equation}
This form makes it possible to associate roots having two
different modules and thus to work on a polynomial constitute
of four non zero terms.
-\\
- An other form of the polynomial to obtain a full polynomial is:
+
+\paragraph{Full polynomial}:
+ the second form used to obtain a full polynomial is:
%%\begin{equation}
%%\forall \alpha_{i} \in C,\forall n_{i}\in N^{*}; P(z)= \sum^{n}_{i=1}(z^{n^{i}}.a_{i})
%%\end{equation}
\begin{equation}
- {\Large \forall a_{i} \in C; p(x)=\sum^{n-1}_{i=1} a_{i}.x^{i}}
+ {\Large \forall a_{i} \in C, i\in N; p(x)=\sum^{n-1}_{i=1} a_{i}.x^{i}}
\end{equation}
-with this formula, we can have until \textit{n} non zero terms.
+with this form, we can have until \textit{n} non zero terms.
-\subsubsection{The study condition}
+\subsection{The study condition}
In order to have representative average values, for each
point of our curves we measured the roots finding of 10
different polynomials.
polynomials. The execution time remains the
element-key which justifies our work of parallelization.
For our tests we used a CPU Intel(R) Xeon(R) CPU
-E5620@2.40GHz and a GPU Tesla C2070 (with 6 Go of ram)
+E5620@2.40GHz and a GPU K40 (with 6 Go of ram)
-\subsubsection{Comparative study}
-We initially carried out the convergence of Aberth algorithm with various sizes of polynomial, in second we evaluate the influence of the size of the threads per block....
-\paragraph{Aberth algorithm on CPU and GPU}
+\subsection{Comparative study}
+We initially carried out the convergence of Aberth algorithm with various sizes of polynomial, in second we evaluate the influence of the size of the threads per block....
-\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}
+\subsubsection{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}
-\paragraph{The impact of the thread's number into the convergence of Aberth algorithm}
+\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}
-\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}
+
+\subsubsection{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{figure}[htbp]
-\paragraph{A comparative study between Aberth and Durand-kerner algorithm}
+\subsubsection{A comparative study between Aberth and Durand-kerner algorithm}
\begin{table}[htbp]
\centering
\begin{tabular} {|R{2cm}|L{2.5cm}|L{2.5cm}|L{1.5cm}|L{1.5cm}|}
