X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/kahina_paper1.git/blobdiff_plain/f3cfcb3a5c68d2bc48c1087c56d50165364c133e..cfd82bbfb39a9364876ec4ae2e03ec4877c7cda1:/paper.tex

diff --git a/paper.tex b/paper.tex
index c8a6698..b9ed2ff 100644
--- a/paper.tex
+++ b/paper.tex
@@ -4,6 +4,7 @@
 %%\usepackage[utf8]{inputenc}
 %%\usepackage[T1]{fontenc}
 %%\usepackage[french]{babel}
+\usepackage{float} 
 \usepackage{amsmath,amsfonts,amssymb}
 \usepackage[ruled,vlined]{algorithm2e}
 %\usepackage[french,boxed,linesnumbered]{algorithm2e}
@@ -96,7 +97,7 @@ root finding of polynomials, high degree, iterative methods, Durant-Kerner, GPU,
 Polynomials are algebraic structures used in mathematics that capture physical phenomenons and that express the outcome in the form of a function of some unknown variable. Formally speaking,  a polynomial $p(x)$ of degree \textit{n} having $n$ coefficients in the complex plane \textit{C} and zeros $\alpha_{i},\textit{i=1,...,n}$ 
 %%\begin{center}
 \begin{equation}
-     {\Large p(x)=\sum{a_{i}x^{i}}=a_{n}\prod(x-\alpha_{i}),a_{0} a_{n}\neq 0}.
+     {\Large p(x)=\sum_{i=0}^{n}{a_{i}x^{i}}=a_{n}\prod_{i=1}^{n}(x-\alpha_{i}), a_{0} a_{n}\neq 0}.
 \end{equation}
 %%\end{center}
 
@@ -583,33 +584,26 @@ or from GPU memory to CPU memory \verb=(cudaMemcpyDeviceToHost))=.
 \section{Experimental study}
 
 \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:
+We study two forms of  polynomials the sparse polynomials and the full 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})
+	\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.
 
-\paragraph{Full polynomial}:
- the second form used 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, i\in N; p(x)=\sum^{n-1}_{i=1} a_{i}.x^{i}} 
+     {\Large \forall a_{i} \in C, i\in N;  p(x)=\sum^{n}_{i=0} a_{i}.x^{i}} 
 \end{equation}
 with this form, we can have until \textit{n} non zero terms.
 
 \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.
-
 The our experiences results concern two parameters which are
 the polynomial degree and the execution time of our program
 to converge on the solution. The polynomial degree allows us
@@ -617,7 +611,7 @@ 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
-E5620@2.40GHz and a GPU K40 (with 6 Go of ram)
+E5620@2.40GHz and a GPU K40 (with 6 Go of ram).
 
 
 \subsection{Comparative study}
@@ -642,7 +636,7 @@ We initially carried out the convergence of Aberth algorithm with various sizes
 %	\label{tab:theConvergenceOfAberthAlgorithm}
 %\end{table}
  
-\begin{figure}[htbp]
+\begin{figure}[H]
 \centering
   \includegraphics[width=0.8\textwidth]{figures/Compar_EA_algorithm_CPU_GPU}
 \caption{Aberth algorithm on CPU and GPU}
@@ -671,28 +665,30 @@ We initially carried out the convergence of Aberth algorithm with various sizes
 %\end{table}
 
 
-\begin{figure}[htbp]
+\begin{figure}[H]
 \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}
 
-
+\subsubsection{The impact of exp-log solution to compute very high degrees of  polynomial}
+\begin{figure}[H]
+\centering
+  \includegraphics[width=0.8\textwidth]{figures/log_exp}
+\caption{The impact of exp-log solution to compute very high degrees of  polynomial.}
+\label{fig:01}
+\end{figure}
 
 \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}|}
-			\hline Polynomial's degrees & Aberth $T_{exe}$ & D-Kerner $T_{exe}$ & Aberth iteration & D-Kerner iteration\\
-			\hline 5000 &  0.40 & 3.42 & 17 & 138 \\
-			\hline 50000 & 3.92 & 385.266 & 17 & 823\\
-			\hline 500000 & 497.109 & 4677.36 & 24 & 214\\
-			\hline					
-					\end{tabular}
-	\caption{Aberth algorithm compare to Durand-Kerner algorithm}
-	\label{tab:AberthAlgorithCompareToDurandKernerAlgorithm}
-\end{table}
+
+
+\begin{figure}[H]
+\centering
+  \includegraphics[width=0.8\textwidth]{figures/EA_DK}
+\caption{Ehrlisch-Aberth and Durand-Kerner algorithm on GPU}
+\label{fig:01}
+\end{figure}
 
 
 \bibliography{mybibfile}