Using the logarithm (eq.~\ref{deflncomplex}) and the exponential (eq.~\ref{defexpcomplex}) operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations
manipulate lower absolute values and the roots for large polynomial's degrees can be looked for successfully~\cite{Karimall98}.
-Applying this solution for the Ehrlich-Aberth method we obtain the
-iteration function with exponential and logarithm:
+Applying this solution for the iteration function Eq.~\ref{Eq:Hi} of Ehrlich-Aberth method, we obtain the iteration function with exponential and logarithm:
%%$$ \exp \bigl( \ln(p(z)_{k})-ln(\ln(p(z)_{k}^{'}))- \ln(1- \exp(\ln(p(z)_{k})-ln(\ln(p(z)_{k}^{'})+\ln\sum_{i\neq j}^{n}\frac{1}{z_{k}-z_{j}})$$
\begin{equation}
\label{Log_H2}
EA.EL: z^{k+1}_{i}=z_{i}^{k}-\exp \left(\ln \left(
-p(z_{i}^{k})\right)-\ln\left(p'(z^{k}_{i})\right)- \ln
-\left(1-Q(z^{k}_{i})\right)\right),
+p(z_{i}^{k})\right)-\ln\left(p'(z^{k}_{i})\right)- \ln\left(1-Q(z^{k}_{i})\right)\right),
\end{equation}
where:
This solution is applied when the root except the circle unit, represented by the radius $R$ evaluated in C language as :
\begin{equation}
\label{R.EL}
-R = exp(log(DBL_MAX)/(2*n) );
+R = exp(log(DBL\_MAX)/(2*n) );
\end{equation}
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 Ehrlich-Aberth method on a model of
-Optoelectronic Transpose Interconnection System (OTIS).The
+Optoelectronic Transpose Interconnection System (OTIS). 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.
\subsection{Parallel implementation with CUDA }
In order to implement the Ehrlich-Aberth method in CUDA, it is
-possible to use the Jacobi scheme or the Gauss Seidel one. With the
+possible to use the Jacobi scheme or the Gauss-Seidel one. With the
Jacobi iteration, at iteration $k+1$ we need all the previous values
$z^{k}_{i}$ to compute the new values $z^{k+1}_{i}$, that is :
\begin{equation}
\label{eq:Aberth-H-GS}
EAGS: z^{k+1}_{i}=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}
-{1-\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}(\sum^{i-1}_{j=1}\frac{1}{z^{k}_{i}-z^{k+1}_{j}}+\sum_{j=1,j\neq i}^{j=n}{\frac{1}{(z_{i}^{k}-z_{j}^{k})}})}, i=1,. . . .,n
+{1-\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}(\sum^{i-1}_{j=1}\frac{1}{z^{k}_{i}-z^{k+1}_{j}}+\sum_{j=i+1}^{j=n}{\frac{1}{(z_{i}^{k}-z_{j}^{k})}})}, i=1,. . . .,n
\end{equation}
Using Eq.~\ref{eq:Aberth-H-GS} to update the vector solution
%In CUDA programming, all the instructions of the \verb=for= loop are executed by the GPU as a kernel. A kernel is a function written in CUDA and defined by the \verb=__global__= qualifier added before a usual \verb=C= function, which instructs the compiler to generate appropriate code to pass it to the CUDA runtime in order to be executed on the GPU.
-Algorithm~\ref{alg2-cuda} shows sketch of the Ehrlich-Aberth algorithm using CUDA.
+Algorithm~\ref{alg2-cuda} shows sketch of the Ehrlich-Aberth method using CUDA.
\begin{enumerate}
\begin{algorithm}[H]
%\LinesNumbered
\caption{CUDA Algorithm to find roots with the Ehrlich-Aberth method}
-\KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (error tolerance
+\KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (Error tolerance
threshold), P (Polynomial to solve), Pu (Derivative of P), $n$ (Polynomial's degrees), $\Delta z_{max}$ (Maximum value of stop condition)}
\KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}
