corrections

diff --git a/paper.tex b/paper.tex
index b3e3076c085c43d6067814158e10073a3f180407..d9c332420842844c03ce34d20615eaa0123df349 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -185,22 +185,21 @@ time.
 
 Many authors have dealt with the parallelization of
 simultaneous methods, i.e. that find all the zeros simultaneously. 
 
 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
 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.
 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 parallelization 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
 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
 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
 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
 
 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}
 
 \section{The Sequential Aberth method}
 \label{sec1}