\begin{equation}
\label{eq:Aberth-Conv-Cond}
- \forall i \in
- [1,n];\frac{z_{i}^{k}-z_{i}^{k-1}}{z_{i}^{k}}<\xi
+ \forall i \in [1,n];\vert\frac{z_{i}^{k}-z_{i}^{k-1}}{z_{i}^{k}}\vert<\xi
\end{equation}
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-Aberth methods are the most practical choices among
them~\cite{Bini04}. These two methods have been extensively
-studied for parallelization due to their intrinsics, i.e. the
+studied for parallelization due to their intrinsics parallelism, 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
algorithms are mapped on an OTIS-2D torus using N processors. This
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.
+%Until very recently, the literature did 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
