\documentclass[review]{elsarticle}
\usepackage{lineno,hyperref}
-%%\usepackage[utf8]{inputenc}
+\usepackage[utf8]{inputenc}
%%\usepackage[T1]{fontenc}
%%\usepackage[french]{babel}
\usepackage{float}
\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}}
-%%\ead[url]{kahina.ghidouche@gmail.com}
+\author[mymainaddress]{Kahina Ghidouche}
+%%\ead[url]{kahina.ghidouche@univ-bejaia.dz}
\cortext[mycorrespondingauthor]{Corresponding author}
-\ead{kahina.ghidouche@gmail.com}
+\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]{Department of informatics,University of Bejaia,Algeria}
-\address[mysecondaryaddress]{FEMTO-ST Institute, University of 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}
\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
+vector $x$ such that :
\begin{center}
$x=g(x)$
\end{center}
where $g : C^{n}\longrightarrow C^{n}$. Usually, we can easily
rewrite this fixed-point problem as a root-finding problem by
setting $f(x) = x-g(x)$ and likewise we can recast the
-root-finding problem into a fixed-point problem by setting
+root-finding problem into a fixed-point problem by setting :
\begin{center}
$g(x)= f(x)-x$.
\end{center}
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})}, i = 1, . . . , n,
\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
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})}}.
+\label{Eq:EA}
+ z_i^{k+1}=z_i^{k}-\frac{1}{{\frac {P'(z_i^{k})} {P(z_i^{k})}}-{\sum_{i\neq j}\frac{1}{(z_i^{k}-z_j^{k})}}}, i = 1, . . . , n,
\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.
algorithms, in which each processor continues to update its
approximations even though the latest values of other $z_i((k))$
have not been received from the other processors, in contrast with synchronous algorithms where it would wait those values before making a new iteration.
-Couturier et al. ~\cite{Raphaelall01} proposed two methods of parallelisation for
+Couturier and al~\cite{Raphaelall01} proposed two methods of parallelisation for
a shared memory architecture and for distributed memory one. They were able to
compute the roots of polynomials of degree 10000 in 430 seconds with only 8
personal computers and 2 communications per iteration. Comparing to the sequential implementation
Very few works had been since this last work until the appearing of
the Compute Unified Device Architecture (CUDA)~\cite{CUDA10}, a
parallel computing platform and a programming model invented by
-NVIDIA. The computing power of GPUs (Graphics Processing Unit) has exceeded that of
-of CPUs. However, CUDA adopts a totally new computing architecture to use the
+NVIDIA. The computing power of GPUs (Graphics Processing Unit) has exceeded that of CPUs. However, CUDA adopts a totally new computing architecture to use the
hardware resources provided by GPU in order to offer a stronger
computing ability to the massive data computing.
-Ghidouche et al. ~\cite{Kahinall14} proposed an implementation of the
+Ghidouche and al~\cite{Kahinall14} proposed an implementation of the
Durand-Kerner method on GPU. Their main
result showed that a parallel CUDA implementation is 10 times as fast as
the sequential implementation on a single CPU for high degree
polynomials of about 48000. To our knowledge, it is the first time such high degree polynomials are numerically solved.
-In this paper, we focus on the implementation of the Aberth method for
-high degree polynomials on GPU. The paper is organised as fellows. Initially, we recall the Aberth method in Section.\ref{sec1}. Improvements for the Aberth method are proposed in Section.\ref{sec2}. Related work to the implementation of simultaneous methods using a parallel approach is presented in Section.\ref{secStateofArt}.
-In Section.4 we propose a parallel implementation of the Aberth method on GPU and discuss it. Section 5 presents and investigates our implementation and experimental study results. Finally, Section 6 concludes this paper and gives some hints for future research directions in this topic.
+In this paper, we focus on the implementation of the Ehrlich-Aberth method for
+high degree polynomials on GPU. The paper is organized as fellows. Initially, we recall the Ehrlich-Aberth method in Section \ref{sec1}. Improvements for the Ehrlich-Aberth method are proposed in Section \ref{sec2}. Related work to the implementation of simultaneous methods using a parallel approach is presented in Section \ref{secStateofArt}.
+In Section \ref{sec5} we propose a parallel implementation of the Ehrlich-Aberth method on GPU and discuss it. Section \ref{sec6} presents and investigates our implementation and experimental study results. Finally, Section\ref{sec7} 6 concludes this paper and gives some hints for future research directions in this topic.
\section{The Sequential Aberth method}
\label{sec1}
A cubically convergent iteration method for finding zeros of
-polynomials was proposed by O.Aberth~\cite{Aberth73}. The Aberth
-method is a purely algebraic derivation. To illustrate the
-derivation, we let $w_{i}(z)$ be the product of linear factors
+polynomials was proposed by O. Aberth~\cite{Aberth73}. In the fellowing we present the main stages of the running of the Aberth method.
+%The Aberth method is a purely algebraic derivation.
+%To illustrate the derivation, we let $w_{i}(z)$ be the product of linear factors
-\begin{equation}
-w_{i}(z)=\prod_{j=1,j \neq i}^{n} (z-x_{j})
-\end{equation}
+%\begin{equation}
+%w_{i}(z)=\prod_{j=1,j \neq i}^{n} (z-x_{j})
+%\end{equation}
-And let a rational function $R_{i}(z)$ be the correction term of the
-Weistrass method~\cite{Weierstrass03}
+%And let a rational function $R_{i}(z)$ be the correction term of the
+%Weistrass method~\cite{Weierstrass03}
-\begin{equation}
-R_{i}(z)=\frac{p(z)}{w_{i}(z)} , i=1,2,...,n.
-\end{equation}
-
-Differentiating the rational function $R_{i}(z)$ and applying the
-Newton method, we have:
+%\begin{equation}
+%R_{i}(z)=\frac{p(z)}{w_{i}(z)} , i=1,2,...,n.
+%\end{equation}
-\begin{equation}
-\frac{R_{i}(z)}{R_{i}^{'}(z)}= \frac{p(z)}{p^{'}(z)-p(z)\frac{w_{i}(z)}{w_{i}^{'}(z)}}= \frac{p(z)}{p^{'}(z)-p(z) \sum _{j=1,j \neq i}^{n}\frac{1}{z-x_{i}}}, i=1,2,...,n
-\end{equation}
+%Differentiating the rational function $R_{i}(z)$ and applying the
+%Newton method, we have:
-Substituting $x_{j}$ for z we obtain the Aberth iteration method.
+%\begin{equation}
+%\frac{R_{i}(z)}{R_{i}^{'}(z)}= \frac{p(z)}{p^{'}(z)-p(z)\frac{w_{i}(z)}{w_{i}^{'}(z)}}= \frac{p(z)}{p^{'}(z)-p(z) \sum _{j=1,j \neq i}^{n}\frac{1}{z-x_{j}}}, i=1,2,...,n
+%\end{equation}
+%where R_{i}^{'}(z)is the rational function derivative of F evaluated in the point z
+%Substituting $x_{j}$ for $z_{j}$ we obtain the Aberth iteration method.%
-In the fellowing we present the main stages of the running of the Aberth method.
\subsection{Polynomials Initialization}
-The initialization of a polynomial p(z) is done by setting each of the $n$ complex coefficients $a_{i}$
-:
+The initialization of a polynomial p(z) is done by setting each of the $n$ complex coefficients $a_{i}$:
\begin{equation}
\label{eq:SimplePolynome}
a given approximation strongly depends on it.
In~\cite{Aberth73} the Aberth iteration is started by selecting $n$
equi-spaced points on a circle of center 0 and radius r, where r is
-an upper bound to the moduli of the zeros. Later, Bini et al.~\cite{Bini96}
+an upper bound to the moduli of the zeros. Later, Bini and al.~\cite{Bini96}
performed this choice by selecting complex numbers along different
circles and relies on the result of~\cite{Ostrowski41}.
v_{i}=\frac{|\frac{a_{n}}{a_{i}}|^{\frac{1}{n-i}}}{2}.
\end{equation}
-\subsection{Iterative Function $H_{i}$}
+\subsection{Iterative Function $H_{i}(z^{k})$}
The operator used by the Aberth method is corresponding to the
-following equation which will enable the convergence towards
+following equation~\ref{Eq:EA} which will enable the convergence towards
polynomial solutions, provided all the roots are distinct.
\begin{equation}
-H_{i}(z)=z_{i}-\frac{1}{\frac{p^{'}(z_{i})}{p(z_{i})}-\sum_{j\neq
-i}{\frac{1}{z_{i}-z_{j}}}}
+\label{Eq:Hi}
+H_{i}(z^{k+1})=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}
+{1-\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}\sum_{j=1,j\neq i}^{j=n}{\frac{1}{(z_{i}^{k}-z_{j}^{k})}}}, i=0,. . . .,n
\end{equation}
-
+we notice that the function iterative $H_{i}$ in Eq.~\ref{Eq:Hi} it the same those presented in Eq.~\ref{Eq:EA}, but we prefer used the last one seen the advantage of its use to improve the Aberth method. More detail in the section ~\ref{sec2}.
\subsection{Convergence Condition}
The convergence condition determines the termination of the algorithm. It consists in stopping from running
the iterative function $H_{i}(z)$ when the roots are sufficiently stable. We consider that the method
\begin{equation}
\label{eq:Aberth-Conv-Cond}
\forall i \in
-[1,n];\frac{z_{i}^{(k)}-z_{i}^{(k-1)}}{z_{i}^{(k)}}<\xi
+[1,n];\frac{z_{i}^{k}-z_{i}^{k-1}}{z_{i}^{k}}<\xi
\end{equation}
Using the logarithm (eq.~\ref{deflncomplex}) and the exponential (eq.~\ref{defexpcomplex}) operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations
manipulate lower absolute values and the roots for large polynomial's degrees can be looked for successfully~\cite{Karimall98}.
-Applying this solution for the Aberth method we obtain the
+Applying this solution for the Ehrlich-Aberth method we obtain the
iteration function with logarithm:
%%$$ \exp \bigl( \ln(p(z)_{k})-ln(\ln(p(z)_{k}^{'}))- \ln(1- \exp(\ln(p(z)_{k})-ln(\ln(p(z)_{k}^{'})+\ln\sum_{i\neq j}^{n}\frac{1}{z_{k}-z_{j}})$$
\begin{equation}
\label{Log_H2}
-H_{i}(z)=z_{i}^{k}-\exp \left(\ln \left(
-p(z_{k})\right)-\ln\left(p(z_{k}^{'})\right)- \ln
-\left(1-Q(z_{k})\right)\right),
+H_{i}(z^{k+1})=z_{i}^{k}-\exp \left(\ln \left(
+p(z_{i}^{k})\right)-\ln\left(p'(z^{k}_{i})\right)- \ln
+\left(1-Q(z^{k}_{i})\right)\right),
\end{equation}
where:
\begin{equation}
\label{Log_H1}
-Q(z_{k})=\exp\left( \ln (p(z_{k}))-\ln(p(z_{k}^{'}))+\ln \left(
-\sum_{k\neq j}^{n}\frac{1}{z_{k}-z_{j}}\right)\right).
+Q(z^{k}_{i})=\exp\left( \ln (p(z^{k}_{i}))-\ln(p'(z^{k}^{i}))+\ln \left(
+\sum_{k\neq j}^{n}\frac{1}{z^{k}_{i}-z^{k}_{j}}\right)\right).
\end{equation}
-This solution is applied when the root except the circle unit, represented by the radius $R$ evaluated as:
-\begin{equation}
-\label{R}
-R = \exp( \log(DBL\_MAX) / (2*n) )
-\end{equation}
- where $DBL\_MAX$ stands for the maximum representable double value.
+This solution is applied when the root except the circle unit, represented by the radius $R$ evaluated in C language as:
+\begin{verbatim}
+R = exp(log(DBL_MAX)/(2*n) );
+\end{verbatim}
+
+%\begin{equation}
+
+%R = \exp( \log(DBL\_MAX) / (2*n) )
+%\end{equation}
+ where \verb=DBL_MAX= stands for the maximum representable \verb=double= value.
\section{The implementation of simultaneous methods in a parallel computer}
\label{secStateofArt}
texture space, which reside in external DRAM, and are accessed via
read-only caches.
-\subsection{ The implementation of Aberth method on GPU}
+\section{ The implementation of Aberth method on GPU}
+\label{sec5}
%%\subsection{A CUDA implementation of the Aberth's method }
%%\subsection{A GPU implementation of the Aberth's method }
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.
+z^{k+1}_{i}=\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.
\end{equation}
-With the Gauss-seidel iteration, we have:
+With the Gauss-Seidel iteration, we have:
\begin{equation}
\label{eq:Aberth-H-GS}
-H(i,z^{k+1})=\frac{p(z^{(k)}_{i})}{p'(z^{(k)}_{i})-p(z^{(k)}_{i})(\sum^{i-1}_{j=1}\frac{1}{z^{(k)}_{i}-z^{(k+1)}_{j}}+\sum^{n}_{j=i+1}\frac{1}{z^{(k)}_{i}-z^{(k)}_{j}})}, i=1,...,n.
+z^{k+1}_{i}=\frac{p(z^{k}_{i})}{p'(z^{k}_{i})-p(z^{k}_{i})(\sum^{i-1}_{j=1}\frac{1}{z^{k}_{i}-z^{k+1}_{j}}+\sum^{n}_{j=i+1}\frac{1}{z^{k}_{i}-z^{k}_{j}})}, i=1,...,n.
\end{equation}
-
+%%Here a finiched my revision %%
Using Equation.~\ref{eq:Aberth-H-GS} for the update sub-step of $H(i,z^{k+1})$, we expect the Gauss-Seidel iteration to converge more quickly because, just as its ancestor (for solving linear systems of equations), it uses the most fresh computed roots $z^{k+1}_{i}$.
The $4^{th}$ step of the algorithm checks the convergence condition using Equation.~\ref{eq:Aberth-Conv-Cond}.
\label{eq:T-global}
T=\left[n\left(T_{i}(n)+T_{j}\right)+O(n)\right].K
\end{equation}
-The execution time increases with the increasing of the polynomial degree, which justifies to parallelise these steps in order to reduce the global execution time. In the following, we explain how we did parrallelize these steps on a GPU architecture using the CUDA platform.
+The execution time increases with the increasing of the polynomial degree, which justifies to parallelize these steps in order to reduce the global execution time. In the following, we explain how we did parallelize these steps on a GPU architecture using the CUDA platform.
\subsubsection{A Parallel implementation with CUDA }
On the CPU, both steps 3 and 4 contain the loop \verb=for= and a single thread executes all the instructions in the loop $n$ times. In this subsection, we explain how the GPU architecture can compute this loop and reduce the execution time.
or from GPU memory to CPU memory \verb=(cudaMemcpyDeviceToHost))=.
%%HIER END MY REVISIONS (SIDER)
\section{Experimental study}
-
+\label{sec6}
\subsection{Definition of the used polynomials }
We study two categories of polynomials : the sparse polynomials and the full polynomials.
\paragraph{A sparse polynomial}: is a polynomial for which only some coefficients are not null. We use in the following polonymial for which the roots are distributed on 2 distinct circles :
\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}
+\subsubsection{The execution time in seconds of Ehrlich-Aberth algorithm on CPU OpenMP (1 core, 4 cores) vs. on a Tesla GPU}
+%\begin{figure}[H]
+%\centering
+ % \includegraphics[width=0.8\textwidth]{figures/Compar_EA_algorithm_CPU_GPU}
+%\caption{The execution time in seconds of Ehrlich-Aberth algorithm on CPU core vs. on a Tesla GPU}
+%\label{fig:01}
+%\end{figure}
+
\begin{figure}[H]
\centering
- \includegraphics[width=0.8\textwidth]{figures/Compar_EA_algorithm_CPU_GPU}
-\caption{The execution time in seconds of Ehrlich-Aberth algorithm on CPU core vs. on a Tesla GPU}
+ \includegraphics[width=0.8\textwidth]{figures/openMP-GPU}
+\caption{The execution time in seconds of Ehrlich-Aberth algorithm on CPU OpenMP (1 core, 4 cores) vs. on a Tesla GPU}
\label{fig:01}
\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 respectively the execution time of the Ehrlich-Aberth method implemented initially on one core of the Quad-Core Xeon E5620 CPU than on four cores of the same machine with \textit{OpenMP} platform and the execution time of the same method 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 (4 cores). 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(4 cores) 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 an execution time under to 2500 s CPU (4 cores) implementation can resolve polynomials degrees of only 200,000, 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(4 cores). However the CPU(4 cores) implementation are almost 4 faster then his implementation on CPU (1 core). Furthermore, we verify that the number of iterations and the convergence precision is the same for the both CPU and GPU implementation. %This reduction of time allows us to compute roots of polynomial of more important degree at the same time than with a CPU.
-
+ %We notice that the convergence precision is a round $10^{-7}$ for the both implementation on CPU and GPU. Consequently, we can conclude that Ehrlich-Aberth on GPU are faster and accurately then CPU implementation.
\subsubsection{Influence of the number of threads on the execution times of different polynomials (sparse and full)}
-It is also interesting to see the influence of the number of threads per block on the execution time. For that, we notice that the maximum number of threads per block for the Nvidia Tesla K40c GPU is 1024, so we varied the number of threads per block from 8 to 1024. We took into account the execution time for both sparse and full polynomials of size 50000 and 500000 degrees.
+To optimize the performances of an algorithm on a GPU, it is necessary to maximize the use of cores GPU (maximize the number of threads executed in parallel) and to optimize the use of the various memoirs GPU. In fact, it is interesting to see the influence of the number of threads per block on the execution time of Ehrlich-Aberth algorithm.
+For that, we notice that the maximum number of threads per block for the Nvidia Tesla K40 GPU is 1024, so we varied the number of threads per block from 8 to 1024. We took into account the execution time for both sparse and full of 10 different polynomials of size 50000 and 10 different polynomials of size 500000 degrees.
\begin{figure}[H]
\centering
\label{fig:01}
\end{figure}
-The figure 2 show that, the best execution time for both sparse and full polynomial are given while the threads number varies between 64 and 256 threads per bloc. We notice that with small polynomials the number of threads per block is 64, Whereas, the large polynomials the number of threads per block is 256. However,In the following experiments we specify that the number of thread by block is 256.
+The figure 2 show that, the best execution time for both sparse and full polynomial are given when the threads number varies between 64 and 256 threads per bloc. We notice that with small polynomials the best number of threads per block is 64, Whereas, the large polynomials the best number of threads per block is 256. However,In the following experiments we specify that the number of thread by block is 256.
\subsubsection{The impact of exp-log solution to compute very high degrees of polynomial}
\label{fig:01}
\end{figure}
-The figure 3, show a comparison between the execution time of the Ehrlich-Aberth algorithm applying log-exp solution and the execution time of the Ehrlich-Aberth algorithm without applying log-exp solution, with full polynomials degrees. We can see that the execution time for the both algorithms are the same while the polynomials degrees are less than 4500. After,we show clearly that the classical version of Ehrlich-Aberth algorithm (without applying log.exp) stop to converge and can not solving polynomial exceed 4500, in counterpart, the new version of Ehrlich-Aberth algorithm (applying log.exp solution) can solve very high and large full polynomial exceed 100,000 degrees.
+The figure 3, show a comparison between the execution time of the Ehrlich-Aberth algorithm applying log-exp solution and the execution time of the Ehrlich-Aberth algorithm without applying log-exp solution, with full and sparse polynomials degrees. We can see that the execution time for the both algorithms are the same while the full polynomials degrees are less than 4000 and full polynomials are less than 150,000. After,we show clearly that the classical version of Ehrlich-Aberth algorithm (without applying log.exp) stop to converge and can not solving any polynomial sparse or full. In counterpart, the new version of Ehrlich-Aberth algorithm (applying log.exp solution) can solve very high and large full polynomial exceed 100,000 degrees.
-in fact, when the modulus of the roots are up than \textit{R} given in ~\ref{R},this exceed the limited number in the mantissa of floating points representations and can not compute the iterative function given in ~\ref{eq:Aberth-H-GS} to obtain the root solution, who justify the divergence of the classical Ehrlich-Aberth algorithm. However, applying log.exp solution given in ~\ref{sec2} took into account the limit of floating using the iterative function in(Eq.~\ref{Log_H1},Eq.~\ref{Log_H2}and allows to solve a very large polynomials degrees .
+in fact, when the modulus of the roots are up than \textit{R} given in ~\ref{R},this exceed the limited number in the mantissa of floating points representations and can not compute the iterative function given in ~\ref{eq:Aberth-H-GS} to obtain the root solution, who justify the divergence of the classical Ehrlich-Aberth algorithm. However, applying log.exp solution given in ~\ref{sec2} took into account the limit of floating using the iterative function in(Eq.~\ref{Log_H1},Eq.~\ref{Log_H2} and allows to solve a very large polynomials degrees .
%\begin{figure}[H]
\label{fig:01}
\end{figure}
+%\subsubsection{The execution time of Ehrlich-Aberth algorithm on OpenMP(1 core, 4 cores) vs. on a Tesla GPU}
+
+
+
+
+
+
+\section{Conclusion and perspective}
+\label{sec7}
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\end{document}
