new

diff --git a/paper.tex b/paper.tex
index 493bb3996d87560220f76b26a00b3ee6446416cc..8f03506423130357b631377cc4fd94c433c7937e 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -363,11 +363,13 @@ Q(z^{k}_{i})=\exp\left( \ln (p(z^{k}_{i}))-\ln(p'(z^{k}_{i}))+\ln \left(
 \sum_{i\neq j}^{n}\frac{1}{z^{k}_{i}-z^{k}_{j}}\right)\right)i=1,...,n,
 \end{equation}
 
 \sum_{i\neq j}^{n}\frac{1}{z^{k}_{i}-z^{k}_{j}}\right)\right)i=1,...,n,
 \end{equation}
 

-This solution is applied when the root except the circle unit, represented by the radius $R$ evaluated in C language as:
-
+This solution is applied when the root except the circle unit, represented by the radius $R$ evaluated in C language as :
+\label{R.EL}
+\begin{center}

 \begin{verbatim}
 R = exp(log(DBL_MAX)/(2*n) );
 \begin{verbatim}
 R = exp(log(DBL_MAX)/(2*n) );

-\end{verbatim} 
+\end{verbatim}
+\end{center} 

 
 %\begin{equation}
 
 
 %\begin{equation}
 


@@ -581,8 +583,9 @@ quickly because, just as any Jacobi algorithm (for solving linear systems of equ
 
 %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. 
 
 
 %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 a sketch of the Ehrlich-Aberth algorithm using CUDA.
+Algorithm~\ref{alg2-cuda} shows steps of the Ehrlich-Aberth algorithm using CUDA.

 
 

+\begin{enumerate}

 \begin{algorithm}[H]
 \label{alg2-cuda}
 %\LinesNumbered
 \begin{algorithm}[H]
 \label{alg2-cuda}
 %\LinesNumbered


@@ -595,21 +598,22 @@ Algorithm~\ref{alg2-cuda} shows a sketch of the Ehrlich-Aberth algorithm using C
 
 \BlankLine
 
 
 \BlankLine
 

-Initialization of the of P\;
-Initialization of the of Pu\;
-Initialization of the solution vector $Z^{0}$\;
-Allocate and copy initial data to the GPU global memory\;
-k=0\;
+\item Initialization of the of P\;
+\item Initialization of the of Pu\;
+\item Initialization of the solution vector $Z^{0}$\;
+\item Allocate and copy initial data to the GPU global memory\;
+\item k=0\;

 \While {$\Delta z_{max} > \epsilon$}{
 \While {$\Delta z_{max} > \epsilon$}{

- Let $\Delta z_{max}=0$\;
-$ kernel\_save(ZPrec,Z)$\;
-k=k+1\;
-$ kernel\_update(Z,P,Pu)$\;
-$kernel\_testConverge(\Delta z_{max},Z,ZPrec)$\;
+\item Let $\Delta z_{max}=0$\;
+\item $ kernel\_save(ZPrec,Z)$\;
+\item  k=k+1\;
+\item $ kernel\_update(Z,P,Pu)$\;
+\item $kernel\_testConverge(\Delta z_{max},Z,ZPrec)$\;

 
 }
 
 }

-Copy results from GPU memory to CPU memory\;
+\item Copy results from GPU memory to CPU memory\;

 \end{algorithm}
 \end{algorithm}

+\end{enumerate}

 ~\\ 
 
 After the initialization step, all data of the root finding problem
 ~\\ 
 
 After the initialization step, all data of the root finding problem


@@ -622,8 +626,8 @@ polynomial's root found at the previous time-step in GPU memory, in
 order to check the convergence of the roots after each iteration (line
 8, Algorithm~\ref{alg2-cuda}).
 
 order to check the convergence of the roots after each iteration (line
 8, Algorithm~\ref{alg2-cuda}).
 

-The second kernel executes the iterative function $H$ and updates
-Z, according to Algorithm~\ref{alg3-update}. We notice that the
+The second kernel executes the iterative function and updates
+$Z$, according to Algorithm~\ref{alg3-update}. We notice that the

 update kernel is called in two forms, according to the value
 \emph{R} which determines the radius beyond which we apply the
 exponential logarithm algorithm. 
 update kernel is called in two forms, according to the value
 \emph{R} which determines the radius beyond which we apply the
 exponential logarithm algorithm. 


@@ -644,7 +648,7 @@ The first form executes formula the EA function Eq.~\ref{Eq:Hi} if the modulus
 of the current complex is less than the a certain value called the
 radius i.e. ($ |z^{k}_{i}|<= R$), else the kernel executes the EA.EL
 function Eq.~\ref{Log_H2}
 of the current complex is less than the a certain value called the
 radius i.e. ($ |z^{k}_{i}|<= R$), else the kernel executes the EA.EL
 function Eq.~\ref{Log_H2}

-(with Eq.~\ref{deflncomplex}, Eq.~\ref{defexpcomplex}). The radius $R$ is evaluated as :
+(with Eq.~\ref{deflncomplex}, Eq.~\ref{defexpcomplex}). The radius $R$ is evaluated as in ~\ref{R.EL} :

 
 $$R = \exp( \log(DBL\_MAX) / (2*n) )$$ where $DBL\_MAX$ stands for the maximum representable double value.
 
 
 $$R = \exp( \log(DBL\_MAX) / (2*n) )$$ where $DBL\_MAX$ stands for the maximum representable double value.