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.
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
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}
\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}
\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
