X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/kahina_paper1.git/blobdiff_plain/6f76482b569516829ba1ec3f0358dec6c479c001..5b5a5f8b12045a82b01725b4d0de867b8b1123f2:/paper.tex?ds=sidebyside

diff --git a/paper.tex b/paper.tex
index 9897244..5f12b9d 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}
@@ -53,7 +54,7 @@
 
 \begin{frontmatter}
 
-\title{Parallel polynomial root finding  using GPU}
+\title{Rapid solution of very high degree polynomials root finding using GPU}
 
 %% Group authors per affiliation:
 \author{Elsevier\fnref{myfootnote}}
@@ -78,14 +79,12 @@
 \address[mysecondaryaddress]{FEMTO-ST Institute, University of Franche-Compté }
 
 \begin{abstract}
-in this article we present a parallel implementation
-of the Aberth algorithm for the problem root finding for
-high degree polynomials on GPU architecture (Graphics
-Processing Unit).
+Polynomials are mathematical algebraic structures that play a great role in science and engineering. But the process of solving them  for high and large degrees is computationally demanding and still not solved. In this paper, we present the results of a parallel implementation of the Ehrlish-Aberth algorithm for the problem root finding for
+high degree polynomials on GPU architectures (Graphics Processing Unit). The main result of this work is to be able to solve high and very large degree polynomials (up to 100000) very efficiently. We also compare the results with a sequential implementation and the Durand-Kerner method on full and sparse polynomials.
 \end{abstract}
 
 \begin{keyword}
-root finding of polynomials, high degree, iterative methods, Durant-Kerner, GPU, CUDA, CPU , Parallelization
+root finding of polynomials, high degree, iterative methods, Ehrlish-Aberth, Durant-Kerner, GPU, CUDA, CPU , Parallelization
 \end{keyword}
 
 \end{frontmatter}
@@ -93,14 +92,19 @@ root finding of polynomials, high degree, iterative methods, Durant-Kerner, GPU,
 \linenumbers
 
 \section{The problem of finding roots of a polynomial}
-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 $\alpha_{i},\textit{i=1,...,n}$ 
+Polynomials are mathematical algebraic structures used in science and engineering to capture physical phenomenons and to express any outcome in the form of a function of some unknown variables. Formally speaking,  a polynomial $p(x)$ of degree \textit{n} having $n$ coefficients in the complex plane \textit{C} is :
 %%\begin{center}
 \begin{equation}
-     {\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}.
+     {\Large p(x)=\sum_{i=0}^{n}{a_{i}x^{i}}}.
 \end{equation}
 %%\end{center}
 
-The root finding problem consists in finding the values of all the $n$ values of the variable $x$ for which \textit{p(x)} is nullified. Such values are called zeroes of $p$. The problem of finding a root is equivalent to that of solving a fixed-point problem. To see this, consider the fixed-point problem of finding the $n$-dimensional
+The root finding problem consists in finding the values of all the $n$ values of the variable $x$ for which \textit{p(x)} is nullified. Such values are called zeroes of $p$. If zeros are $\alpha_{i},\textit{i=1,...,n}$ the $p(x)$ can be written as :
+\begin{equation}
+     {\Large p(x)=a_{n}\prod_{i=1}^{n}(x-\alpha_{i}), a_{0} a_{n}\neq 0}.
+\end{equation}
+
+The problem of finding a root is equivalent to that of solving a fixed-point problem. To see this, consider the fixed-point problem of finding the $n$-dimensional
 vector $x$ such that
 \begin{center}
 $x=g(x)$
@@ -588,7 +592,6 @@ We study two forms of  polynomials the sparse polynomials and the full polynomia
 \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.
@@ -636,7 +639,7 @@ 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}
@@ -665,14 +668,15 @@ 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}
 
-\begin{figure}[htbp]
+\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.}
@@ -680,18 +684,14 @@ We initially carried out the convergence of Aberth algorithm with various sizes
 \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}