ajout des pdf des figures

[kahina_paper1.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 7da8ca6f200d46e0ccc1338cac862e9f1dd2f3b2..62ded307955714b3c4f5254f8af27286f4e35126 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -232,6 +232,7 @@ The initialization of a polynomial p(z) is done by setting each of the $n$ compl
 :
 
 \begin{equation}
 :
 
 \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}
 
   p(z)=\sum{a_{i}z^{n-i}} , a_{n} \neq 0,a_{0}=1, a_{i}\subset C
 \end{equation}
 


@@ -248,6 +249,7 @@ performed this choice by selecting complex numbers along different
 circles and relies on the result of~\cite{Ostrowski41}.
 
 \begin{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}};\\
 %%\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}};\\


@@ -552,9 +554,9 @@ $kernel\_testConverge(\Delta z_{max},d_z^{k},d_z^{k-1})$\;
 \end{algorithm}
 ~\\ 
 
 \end{algorithm}
 ~\\ 
 

-After the initialisation 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 \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}).
+After the initialisation 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 $z^{k}$, according to Algorithm~\ref{alg3-update}. We notice that the update kernel is called in two forms, separated with thevalue of \emph{R} which determines the radius beyond which we apply the logarithm computation of the power of a complex. 
+The second kernel executes the iterative function $H$ and updates $z^{k}$, according to Algorithm~\ref{alg3-update}. We notice that the update kernel is called in two forms, separated with the value of \emph{R} which determines the radius beyond which we apply the logarithm computation of the power of a complex. 

 
 \begin{algorithm}[H]
 \label{alg3-update}
 
 \begin{algorithm}[H]
 \label{alg3-update}


@@ -568,14 +570,15 @@ $kernel\_update\_Log(d\_z^{k})$\;
 }
 \end{algorithm}
 
 }
 \end{algorithm}
 

-The first form execute the formula (8) if the modulus is of the current complex is less than the radius i.e. ($ |z^{k}_{i}|<= R$), else the kernel executes the formulas (13,14).the radius R was computed like:
+The first form executes formula \ref{eq:SimplePolynome} if the modulus of the current complex is less than the a certain value called the radius i.e. ($ |z^{k}_{i}|<= R$), else the kernel executes formulas (Eq.~\ref{deflncomplex},Eq.~\ref{defexpcomplex}). The radius $R$ is evaluated as :

 
 

-$$R = \exp( \log(DBL\_MAX) / (2*n) )$$
+$$R = \exp( \log(DBL\_MAX) / (2*n) )$$ where $DBL\_MAX$ stands for the maximum representable double value.

 
 

-The last kernel verify the convergence of the root after each update of $Z^{(k)}$, as formula(), we used the function of the CUBLAS Library (CUDA Basic Linear Algebra Subroutines) to implement this kernel. 
+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 its 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}
 
 \subsubsection{Definition of the polynomial used}
 \subsection{Experimental study}
 
 \subsubsection{Definition of the polynomial used}


@@ -618,42 +621,60 @@ We initially carried out the convergence of Aberth algorithm with various sizes
 
 \paragraph{Aberth algorithm on CPU and GPU}
 
 
 \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}
 
 \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]
+\centering
+  \includegraphics[width=0.8\textwidth]{figures/influence_nb_threads}
+\caption{Influence of the number of threads on the execution times of different polynomials (sparse and full)}
+\label{fig:01}
+\end{figure}
+
+

 
 \paragraph{A comparative study between Aberth and Durand-kerner algorithm}
 \begin{table}[htbp]
 
 \paragraph{A comparative study between Aberth and Durand-kerner algorithm}
 \begin{table}[htbp]