section 1 2, et 3

diff --git a/paper.tex b/paper.tex
index c213794d35699bf2504b82e3ecaaa3681f8c45b0..a6e1cf8792fbcf69e33a09596d00cf25387669d6 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -161,7 +161,7 @@ drastically increases like the degrees of high polynomials. It is expected that
 parallelization of these algorithms will improve the convergence
 time.
 
 parallelization of these algorithms will improve the convergence
 time.
 

-Many authors have dealt with parallelisation of
+Many authors have dealt with the parallelisation 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
 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


@@ -341,48 +341,39 @@ Winogard~\cite{Winogard72}. The second approach aims at reducing the
 computation time per iteration, as reported
 in~\cite{Benall68,Jana06,Janall99,Riceall06}. 
 
 computation time per iteration, as reported
 in~\cite{Benall68,Jana06,Janall99,Riceall06}. 
 

-There are many
-schemes for simultaneous approximations of all roots of a given
+There are many schemes for the simultaneous approximation of all roots of a given

 polynomial. Several works on different methods and issues of root
 polynomial. Several works on different methods and issues of root

-finding have been reported in~\cite{Azad07, Gemignani07, Kalantari08, Skachek08, Zhancall08, Zhuall08}. However, Durand-Kerner and Ehrlich methods are the most practical choices among
+finding have been reported in~\cite{Azad07, Gemignani07, Kalantari08, Skachek08, Zhancall08, Zhuall08}. However, Durand-Kerner and Ehrlisch-Aberth methods are the most practical choices among

 them~\cite{Bini04}. These two methods have been extensively
 them~\cite{Bini04}. These two methods have been extensively

-studied for parallelization due to their following advantages. The
-computation involved in these methods has some inherent
+studied for parallelization due to their intrinsics, i.e. the
+computations involved in both methods has some inherent

 parallelism that can be suitably exploited by SIMD machines.
 Moreover, they have fast rate of convergence (quadratic for the
 parallelism that can be suitably exploited by SIMD machines.
 Moreover, they have fast rate of convergence (quadratic for the

-Durand-Kerner method and cubic for the Ehrlich). Various parallel
+Durand-Kerner and cubic for the Ehrlisch-Aberth). Various parallel

 algorithms reported for these methods can be found
 in~\cite{Cosnard90, Freeman89,Freemanall90,,Jana99,Janall99}.
 Freeman and Bane~\cite{Freemanall90} presented two parallel
 algorithms on a local memory MIMD computer with the compute-to
 communication time ratio O(n). However, their algorithms require
 each processor to communicate its current approximation to all
 algorithms reported for these methods can be found
 in~\cite{Cosnard90, Freeman89,Freemanall90,,Jana99,Janall99}.
 Freeman and Bane~\cite{Freemanall90} presented two parallel
 algorithms on a local memory MIMD computer with the compute-to
 communication time ratio O(n). However, their algorithms require
 each processor to communicate its current approximation to all

-other processors at the end of each iteration. Therefore they
+other processors at the end of each iteration (synchronous). Therefore they

 cause a high degree of memory conflict. Recently the author
 in~\cite{Mirankar71} proposed two versions of parallel algorithm
 cause a high degree of memory conflict. Recently the author
 in~\cite{Mirankar71} proposed two versions of parallel algorithm

-for the Durand-Kerner method, and Aberth method on model of
+for the Durand-Kerner method, and Ehrlisch-Aberth method on a model of

 Optoelectronic Transpose Interconnection System (OTIS).The
 algorithms are mapped on an OTIS-2D torus using N processors. This
 Optoelectronic Transpose Interconnection System (OTIS).The
 algorithms are mapped on an OTIS-2D torus using N processors. This

-solution need N processors to compute N roots, that it is not
-practical (is not suitable to compute large polynomial's degrees).
-Until then, the related works are not able to compute the root of
-the large polynomial's degrees (higher then 1000) and with small
-time.
-
-    Finding polynomial roots rapidly and accurately it is our
-objective, with the apparition of the CUDA(Compute Unified Device
-Architecture), finding the roots of polynomials becomes rewarding
-and very interesting, 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. In~\cite{Kahinall14} we proposed the first implantation
-of the root finding polynomials method on GPU (Graphics Processing
-Unit),which is the Durand-Kerner method. The main result prove
-that a parallel implementation is 10 times as fast as the
+solution needs N processors to compute N roots, which is not
+practical for solving polynomials with large degrees.
+Until very recently, the literature doen not mention implementations able to compute the roots of
+large degree polynomials (higher then 1000) and within small or at least tractable times. Finding polynomial roots rapidly and accurately is the main objective of our work. 
+With the advent of CUDA (Compute Unified Device
+Architecture), finding the roots of polynomials receives a new attention because of the new possibilities to solve higher degree polynomials in less time. 
+In~\cite{Kahinall14} we already proposed the first implementation
+of a root finding method on GPUs, that of the Durand-Kerner method. The main result showed
+that a parallel CUDA implementation is 10 times as fast as the

 sequential implementation on a single CPU for high degree
 sequential implementation on a single CPU for high degree

-polynomials that is greater than about 48000. Indeed, in this
-paper we present a parallel implementation of Aberth method on
-GPU, more details are discussed in the following of this paper.
+polynomials of 48000. In this paper we present a parallel implementation of Ehlisch-Aberth method on
+GPUs, which details are discussed in the sequel.

 
 
 \section {A parallel implementation of Aberth method}
 
 
 \section {A parallel implementation of Aberth method}