MAJ commentaire de la figure 1

[kahina_paper1.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index fa4f62aadd5d1fab4ff89c1accd0b7db4120d0ad..d9d72c31c05b570457dfed406a93e3964c5efa97 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -1,7 +1,7 @@
 \documentclass[review]{elsarticle}
 
 \usepackage{lineno,hyperref}
 \documentclass[review]{elsarticle}
 
 \usepackage{lineno,hyperref}

-%%\usepackage[utf8]{inputenc}
+\usepackage[utf8]{inputenc}

 %%\usepackage[T1]{fontenc}
 %%\usepackage[french]{babel}
 \usepackage{float} 
 %%\usepackage[T1]{fontenc}
 %%\usepackage[french]{babel}
 \usepackage{float} 


@@ -54,37 +54,47 @@
 
 \begin{frontmatter}
 
 
 \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:
 
 %% 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:
 
 %% 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}
 
 %%\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}
 
 %%\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}
 
 %%\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}
 
 \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}
 \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}
 \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
 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
 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
 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)
 approximation of all the roots, starting with the Durand-Kerner (DK)

-method :
+method:

 %%\begin{center}
 \begin{equation}
 %%\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}
 \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
 
 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}
 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}
 \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.
 
 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.


@@ -627,7 +642,7 @@ E5620@2.40GHz and a GPU K40 (with 6 Go of ram).
 \subsection{Comparative study}
 In this section, we discuss the performance Ehrlich-Aberth method  of root finding polynomials implemented on CPUs and on GPUs.
 
 \subsection{Comparative study}
 In this section, we discuss the performance Ehrlich-Aberth method  of root finding polynomials implemented on CPUs and on GPUs.
 

-We performed a set of experiments on the sequential and the parallel algorithms, for both sparse and full polynomials and different sizes. We took into account the execution time,the  polynomial size and the number of threads per block performed by sum or each experiment on CPUs and on GPUs.
+We performed a set of experiments on the sequential and the parallel algorithms, for both sparse and full polynomials and different sizes. We took into account the execution time, the  polynomial size and the number of threads per block performed by sum or each experiment on CPUs and on GPUs.

 
 All experimental results obtained from the simulations are made in double precision data, for a convergence tolerance of the methods set to $10^{-7}$. Since we were more interested in the comparison of the performance behaviors of Ehrlich-Aberth and Durand-Kerner methods on CPUs versus on GPUs. The initialization values of the vector solution of the Ehrlich-Aberth method are given in section 2.2. 
 \subsubsection{The execution time in seconds of Ehrlich-Aberth algorithm on CPU core vs. on a Tesla GPU}
 
 All experimental results obtained from the simulations are made in double precision data, for a convergence tolerance of the methods set to $10^{-7}$. Since we were more interested in the comparison of the performance behaviors of Ehrlich-Aberth and Durand-Kerner methods on CPUs versus on GPUs. The initialization values of the vector solution of the Ehrlich-Aberth method are given in section 2.2. 
 \subsubsection{The execution time in seconds of Ehrlich-Aberth algorithm on CPU core vs. on a Tesla GPU}


@@ -641,7 +656,8 @@ All experimental results obtained from the simulations are made in double precis
 \end{figure}
 
 Figure 1 %%show a comparison of execution time between the parallel and sequential version of the Ehrlich-Aberth algorithm with sparse polynomial exceed 100000, 
 \end{figure}
 
 Figure 1 %%show a comparison of execution time between the parallel and sequential version of the Ehrlich-Aberth algorithm with sparse polynomial exceed 100000, 

-We report the execution times of the Ehrlich-Aberth method implemented on one core of the Quad-Core Xeon E5620 CPU and those of the same methods implemented on one Nvidia Tesla K40c GPU, with sparse polynomial degrees ranging from 100,000 to 1,000,000. We can see that the methods implemented on the GPU are faster than those implemented on the CPU. This is due to the GPU ability to compute the data-parallel functions faster than its CPU counterpart. However, the execution time for the sequential implementation exceed 16,000 s for 450,000 degrees polynomials, in counterpart  the GPU implementation for the same polynomials need only 350 s, more than again, with 1,000,000 polynomials degrees GPU implementation not reach 2,300 s degrees. While CPU implementation need more than 10 hours. We can also notice that the GPU implementation are almost 47 faster then those implementation on the CPU. Furthermore, we verify that the number of iterations is the same. This reduction of time allows us to compute roots of polynomial of more important degree at the same time than with a CPU.
+We report the execution times of the Ehrlich-Aberth method implemented on one core of the Quad-Core Xeon E5620 CPU and those of the same methods implemented on one Nvidia Tesla K40c GPU, with sparse polynomial degrees ranging from 100,000 to 1,000,000. We can see that the method implemented on the GPU are faster than those implemented on the CPU. This is due to the GPU ability to compute the data-parallel functions faster than its CPU counterpart. However, the execution time for the CPU implementation exceed 5,000 s for 250,000 degrees polynomials, in counterpart  the GPU implementation for the same polynomials not reach 100 s, more than again, % with 1,000,000 polynomials degrees GPU implementation not reach 2,300 s degrees. While CPU implementation need more than 10 hours.
+with an execution time under to 2500 s CPU implementation can resolve polynomials degrees of only 200,000 s, whereas GPU implementation can resolve polynomials more than 1,000,000 degrees. We can also notice that the GPU implementation are almost 47 faster then those implementation on the CPU. Furthermore, we verify that the number of iterations is the same. This reduction of time allows us to compute roots of polynomial of more important degree at the same time than with a CPU.

  
 
 
  
 
 


@@ -702,9 +718,20 @@ This figure show the execution time of the both algorithm EA and DK with sparse
 \label{fig:01}
 \end{figure}
 
 \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 in seconds of Ehrlich-Aberth algorithm on OpenMP(1 core, 4 cores) vs. on a Tesla GPU}
+\label{fig:01}
+\end{figure}
+
+

 
 
 \section{Conclusion and perspective}
 
 
 
 \section{Conclusion and perspective}
 

+\bibliography{mybibfile}
+

 \end{document}
 \end{document}