X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/kahina_paper1.git/blobdiff_plain/e6206cbc48d80bf6ff86c59ba812f861c2b1cb17..572aa90d1d2b8a3c1220ab9ad1e1f4c4477583f2:/paper.tex?ds=sidebyside

diff --git a/paper.tex b/paper.tex
index c49a705..d9c3324 100644
--- a/paper.tex
+++ b/paper.tex
@@ -136,7 +136,7 @@ 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
 century showed that polynomials of degree five or more could not
-be solved by  direct methods. Since then, mathmathicians have
+be solved by  direct methods. Since then, mathematicians have
 focussed on numerical (iterative) methods such as the famous
 Newton method, the Bernoulli method of the 18th, and the Graeffe method.
 
@@ -183,24 +183,23 @@ drastically increases like the degrees of high polynomials. It is expected that
 parallelization of these algorithms will improve the convergence
 time.
 
-Many authors have dealt with the parallelisation of
+Many authors have dealt with the parallelization of
 simultaneous methods, i.e. that find all the zeros simultaneously. 
-Freeman~\cite{Freeman89} implemeted and compared DK, EA and another method of the fourth order proposed
-by Farmer and Loizou~\cite{Loizon83}, on a 8- processor linear
+Freeman~\cite{Freeman89} implemented and compared DK, EA and another method of the fourth order proposed
+by Farmer and Loizou~\cite{Loizon83}, on a 8-processor linear
 chain, for polynomials of degree up to 8. The third method often
-diverges, but the first two methods have speed-up 5.5
-(speed-up=(Time on one processor)/(Time on p processors)). Later,
+diverges, but the first two methods have speed-up equal to 5.5. Later,
 Freeman and Bane~\cite{Freemanall90}  considered asynchronous
 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 and al~\cite{Raphaelall01} proposed two methods of parallelisation for
+Couturier and al.~\cite{Raphaelall01} proposed two methods of parallelization 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
+compute the roots of sparse polynomials of degree 10000 in 430 seconds with only 8
 personal computers and 2 communications per iteration. Comparing to the sequential implementation
-where it takes up to 3300 seconds to obtain the same results, the authors show an interesting speedup, indeed.
+where it takes up to 3300 seconds to obtain the same results, the authors show an interesting speedup.
 
-Very few works had been since this last work until the appearing of
+Very few works had been performed 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 CPUs. However, CUDA adopts a totally new computing architecture to use the
@@ -210,14 +209,25 @@ computing ability to the massive data computing.
 
 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 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.  
+result showed that a parallel CUDA implementation is about 10 times faster than
+the sequential implementation on a single CPU for  sparse
+polynomials of degree 48000. 
+
+
+In this paper, we focus on the implementation of the Ehrlich-Aberth
+method for high degree polynomials on GPU. We propose an adaptation of
+the exponential logarithm in order to be able to solve sparse and full
+polynomial of degree up to $1,000,000$. The paper is organized as
+follows. 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}
@@ -346,7 +356,7 @@ where:
 
 \begin{equation}
 \label{Log_H1}
-Q(z^{k}_{i})=\exp\left( \ln (p(z^{k}_{i}))-\ln(p'(z^{k}^{i}))+\ln \left(
+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}
 
@@ -365,7 +375,7 @@ R = exp(log(DBL_MAX)/(2*n) );
 \label{secStateofArt}   
 The main problem of simultaneous methods is that the necessary
 time needed for convergence is increased when we increase
-the degree of the polynomial. The parallelisation of these
+the degree of the polynomial. The parallelization of these
 algorithms is expected to improve the convergence time.
 Authors usually adopt one of the two following approaches to parallelize root
 finding algorithms. The first approach aims at reducing the total number of