MAJ figure

[kahina_paper1.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 24a86ca3bf779e5c3e173a728c631b2b0f84e621..c8a669875ef7f21354062b4e62411a0c98328977 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -6,6 +6,7 @@
 %%\usepackage[french]{babel}
 \usepackage{amsmath,amsfonts,amssymb}
 \usepackage[ruled,vlined]{algorithm2e}
 %%\usepackage[french]{babel}
 \usepackage{amsmath,amsfonts,amssymb}
 \usepackage[ruled,vlined]{algorithm2e}

+%\usepackage[french,boxed,linesnumbered]{algorithm2e}

 \usepackage{array,multirow,makecell}
 \setcellgapes{1pt}
 \makegapedcells
 \usepackage{array,multirow,makecell}
 \setcellgapes{1pt}
 \makegapedcells


@@ -232,6 +233,7 @@ 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 +250,7 @@ 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}};\\


@@ -476,7 +479,7 @@ $\Delta z_{max}$=c\;}
 
 ~\\ 
 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.
 
 ~\\ 
 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.
 
 \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.


@@ -552,9 +555,9 @@ $kernel\_testConverge(\Delta z_{max},d_z^{k},d_z^{k-1})$\;
 \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}


@@ -568,19 +571,21 @@ $kernel\_update\_Log(d\_z^{k})$\;
 }
 \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))=. 
-\subsection{Experimental study}
+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)
+\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}
 \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}


@@ -588,18 +593,19 @@ roots are distributed on 2 distinct circles:
 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.
 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}
 %%\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}
 \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.
 In order to have representative average values, for each
 point of our curves we measured the roots finding of 10
 different polynomials.


@@ -611,49 +617,58 @@ to validate that our algorithm is powerful with high degree
 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
 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]
 
 
 \begin{figure}[htbp]


@@ -665,7 +680,7 @@ We initially carried out the convergence of Aberth algorithm with various sizes
 
 
 
 
 
 

-\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}|}
 \begin{table}[htbp]
        \centering
                \begin{tabular} {|R{2cm}|L{2.5cm}|L{2.5cm}|L{1.5cm}|L{1.5cm}|}