X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/kahina_paper1.git/blobdiff_plain/c63517ec92fa2ee92e2b058daad0608142c0f617..d2eaa5b6b9f4ba47dbfa2aeaae51427945accf72:/paper.tex

diff --git a/paper.tex b/paper.tex
index fa4f62a..04e1316 100644
--- a/paper.tex
+++ b/paper.tex
@@ -1,7 +1,7 @@
 \documentclass[review]{elsarticle}
 
 \usepackage{lineno,hyperref}
-%%\usepackage[utf8]{inputenc}
+\usepackage[utf8]{inputenc}
 %%\usepackage[T1]{fontenc}
 %%\usepackage[french]{babel}
 \usepackage{float} 
@@ -54,37 +54,47 @@
 
 \begin{frontmatter}
 
-\title{Rapid solution of very high degree polynomials root finding using GPU}
+\title{Efficient high degree polynomial root finding using GPU}
 
 %% Group authors per affiliation:
-\author{Elsevier\fnref{myfootnote}}
-\address{Radarweg 29, Amsterdam}
-\fntext[myfootnote]{Since 1880.}
+%\author{Elsevier\fnref{myfootnote}}
+%\address{Radarweg 29, Amsterdam}
+%\fntext[myfootnote]{Since 1880.}
 
 %% or include affiliations in footnotes:
-\author[mymainaddress]{Ghidouche Kahina\corref{mycorrespondingauthor}}
+\author[mymainaddress]{Kahina Ghidouche}
 %%\ead[url]{kahina.ghidouche@univ-bejaia.dz}
 \cortext[mycorrespondingauthor]{Corresponding author}
 \ead{kahina.ghidouche@univ-bejaia.dz}
 
-\author[mysecondaryaddress]{Couturier Raphael\corref{mycorrespondingauthor}}
+\author[mysecondaryaddress]{Raphaël Couturier\corref{mycorrespondingauthor}}
 %%\cortext[mycorrespondingauthor]{Corresponding author}
 \ead{raphael.couturier@univ-fcomte.fr}
 
-\author[mymainaddress]{Abderrahmane Sider\corref{mycorrespondingauthor}}
+\author[mymainaddress]{Abderrahmane Sider}
 %%\cortext[mycorrespondingauthor]{Corresponding author}
 \ead{ar.sider@univ-bejaia.dz}
 
-\address[mymainaddress]{Laboratoire LIMED,Faculté des sciences exactes,Université de Bejaia,06000,Algeria}
-\address[mysecondaryaddress]{FEMTO-ST Institute,Université de Franche-Compté }
+\address[mymainaddress]{Laboratoire LIMED, Faculté des sciences
+  exactes, Université de Bejaia, 06000, Algeria}
+\address[mysecondaryaddress]{FEMTO-ST Institute, University of
+  Bourgogne Franche-Comte, France }
 
 \begin{abstract}
-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 Ehrlich-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.
+Polynomials are mathematical algebraic structures that play a great
+role in science and engineering. Finding roots of high degree
+polynomials is computationally demanding. In this paper, we present
+the results of a parallel implementation of the Ehrlich-Aberth
+algorithm for the root finding problem for high degree polynomials on
+GPU architectures. The main result of this
+work is to be able to solve high degree polynomials (up
+to 1,000,000) 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, Ehrlich-Aberth, Durant-Kerner, GPU, CUDA, CPU , Parallelization
+Polynomial root finding, Iterative methods, Ehrlich-Aberth, Durand-Kerner, GPU
 \end{keyword}
 
 \end{frontmatter}
@@ -124,22 +134,25 @@ Generally speaking, algorithms for solving problems can be divided into
 two main groups: direct methods and iterative methods.
 \\
 Direct methods exist only for $n \leq 4$, solved in closed form by G. Cardano
-in the mid-16th century. However, N.H. Abel in the early 19th
+in the mid-16th century. However, N. H. Abel in the early 19th
 century showed that polynomials of degree five or more could not
-be solved by  directs methods. Since then, mathmathicians have
+be solved by  direct methods. Since then, mathmathicians have
 focussed on numerical (iterative) methods such as the famous
-Newton's method, Bernoulli's method of the 18th, and Graeffe's.
+Newton method, the Bernoulli method of the 18th, and the Graeffe method.
 
-Later on, with the advent of electronic computers, other methods has
-been developed such as the Jenkins-Traub method, Larkin's method,
-Muller's method, and several methods for simultaneous
+Later on, with the advent of electronic computers, other methods have
+been developed such as the Jenkins-Traub method, the Larkin method,
+the Muller method, and several methods for simultaneous
 approximation of all the roots, starting with the Durand-Kerner (DK)
-method :
+method:
 %%\begin{center}
 \begin{equation}
- Z_{i}=Z_{i}-\frac{P(Z_{i})}{\prod_{i\neq j}(z_{i}-z_{j})}
+ Z_i^{k+1}=Z_{i}^k-\frac{P(Z_i^k)}{\prod_{i\neq j}(Z_i^k-Z_j^k)}
 \end{equation}
 %%\end{center}
+where $Z_i^k$ is the $i^{th}$ root of the polynomial $P$ at the
+iteration $k$.
+
 
 This formula is mentioned for the first time by
 Weiestrass~\cite{Weierstrass03} as part of the fundamental theorem
@@ -151,9 +164,11 @@ in the following form by Ehrlich~\cite{Ehrlich67} and
 Aberth~\cite{Aberth73} uses a different iteration formula given as fellows :
 %%\begin{center}
 \begin{equation}
- Z_{i}=Z_{i}-\frac{1}{{\frac {P'(Z_{i})} {P(Z_{i})}}-{\sum_{i\neq j}(z_{i}-z_{j})}}.
+ Z_i^{k+1}=Z_i^k-\frac{1}{{\frac {P'(Z_i^k)} {P(Z_i^k)}}-{\sum_{i\neq j}(Z_i^k-Z_j^k)}}.
 \end{equation}
 %%\end{center}
+where $P'(Z)$ is the polynomial derivative of $P$ evaluated in the
+point $Z$.
 
 Aberth, Ehrlich and Farmer-Loizou~\cite{Loizon83} have proved that
 the Ehrlich-Aberth method (EA) has a cubic order of convergence for simple roots whereas the Durand-Kerner has a quadratic order of convergence.
@@ -702,9 +717,20 @@ This figure show the execution time of the both algorithm EA and DK with sparse
 \label{fig:01}
 \end{figure}
 
-\bibliography{mybibfile}
+\subsubsection{The execution time of Ehrlich-Aberth algorithm on OpenMP(1 core, 4 cores) vs. on a Tesla GPU}
+
+\begin{figure}[H]
+\centering
+  \includegraphics[width=0.8\textwidth]{figures/openMP-GPU}
+\caption{The execution time of Ehrlich-Aberth algorithm on OpenMP(1core, 4cores) and GPU(Tesla k40)}
+\label{fig:01}
+\end{figure}
+
+
 
 
 \section{Conclusion and perspective}
 
+\bibliography{mybibfile}
+
 \end{document}