test

[kahina_paper1.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index d0913cca27ecbbd96ca8786a92908ef6c5dd80a6..cf26c41ab112cea5701731e0132e6d4e21ab2fc4 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -365,6 +365,7 @@ Q(z^{k}_{i})=\exp\left( \ln (p(z^{k}_{i}))-\ln(p'(z^{k}_{i}))+\ln \left(
 \end{equation}
 
 This solution is applied when the root except the circle unit, represented by the radius $R$ evaluated in C language as:
 \end{equation}
 
 This solution is applied when the root except the circle unit, represented by the radius $R$ evaluated in C language as:

+

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


@@ -582,8 +583,7 @@ Algorithm~\ref{alg2-cuda} shows a sketch of the Ehrlich-Aberth algorithm using C
 \caption{CUDA Algorithm to find roots with the Ehrlich-Aberth method}
 
 \KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (error tolerance
 \caption{CUDA Algorithm to find roots with the Ehrlich-Aberth method}
 
 \KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (error tolerance

-  threshold), P(Polynomial to solve), Pu (the derivative of P), $n$
-  (Polynomial's degrees), $\Delta z_{max}$ (maximum value of stop condition)}
+  threshold), P(Polynomial to solve), Pu (the derivative of P), $n$ (Polynomial's degrees),$\Delta z_{max}$ (maximum value of stop condition)}

 
 \KwOut {$Z$ (The solution root's vector), $ZPrec$ (the previous solution root's vector)}
 
 
 \KwOut {$Z$ (The solution root's vector), $ZPrec$ (the previous solution root's vector)}
 


@@ -606,7 +606,7 @@ Copy results from GPU memory to CPU memory\;
 \end{algorithm}
 ~\\ 
 
 \end{algorithm}
 ~\\ 
 

-After the initialization step, all data of the root finding problem to be solved must be copied from the CPU memory to the GPU global memory, because the GPUs only access data already present in their memories. Next, all the data-parallel arithmetic operations inside the main loop \verb=(while(...))= are executed as kernels by the GPU. The first kernel named \textit{save} in line 6 of Algorithm~\ref{alg2-cuda} consists in saving the vector of 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}).
+After the initialization step, all data of the root finding problem to be solved must be copied from the CPU memory to the GPU global memory, because the GPUs only access data already present in their memories. Next, all the data-parallel arithmetic operations inside the main loop \verb=(do ... while(...))= are executed as kernels by the GPU. The first kernel named \textit{save} in line 6 of Algorithm~\ref{alg2-cuda} consists in saving the vector of 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}).

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


@@ -617,16 +617,16 @@ exponential logarithm algorithm.
 \begin{algorithm}[H]
 \label{alg3-update}
 %\LinesNumbered
 \begin{algorithm}[H]
 \label{alg3-update}
 %\LinesNumbered

-\caption{Kernel\_update}
+\caption{Kernel update}

 
 \eIf{$(\left|d\_Z\right|<= R)$}{
 
 \eIf{$(\left|d\_Z\right|<= R)$}{

-$kernel\_update((d\_Z,d\_Pcoef,d\_Pdegres,d\_Pucoef,d\_Pudegres)$\;}
+$kernel\_update((d\_Z,d\_P,d\_Pu)$\;}

 {
 {

-$kernel\_update\_ExpoLog((d\_Z,d\_Pcoef,d\_Pdegres,d\_Pucoef,d\_Pudegres))$\;
+$kernel\_update\_ExpoLog((d\_Z,d\_P,\_Pu))$\;

 }
 \end{algorithm}
 
 }
 \end{algorithm}
 

-The first form executes formula \ref{eq:SimplePolynome} if the modulus
+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}


@@ -732,19 +732,19 @@ For that, we notice that the maximum number of threads per block for the Nvidia
 
 The figure 2 show that, the best execution time for both sparse and full polynomial are given when the threads number varies between 64 and 256 threads per bloc. We notice that with small polynomials the best number of threads per block is 64, Whereas, the large polynomials the best number of threads per block is 256. However,In the following experiments we specify that the number of thread by block is 256.
 
 
 The figure 2 show that, the best execution time for both sparse and full polynomial are given when the threads number varies between 64 and 256 threads per bloc. We notice that with small polynomials the best number of threads per block is 64, Whereas, the large polynomials the best number of threads per block is 256. However,In the following experiments we specify that the number of thread by block is 256.
 

-\subsection{The impact of exp-log solution to compute very high degrees of  polynomial}
+\subsection{The impact of exp.log solution to compute very high degrees of  polynomial}

 
 

-In this experiment we report the performance of log.exp solution describe in ~\ref{sec2} to compute very high degrees polynomials.   
+In this experiment we report the performance of exp.log solution describe in ~\ref{sec2} to compute very high degrees polynomials.   

 \begin{figure}[htbp]
 \centering
   \includegraphics[width=0.8\textwidth]{figures/sparse_full_explog}
 \begin{figure}[htbp]
 \centering
   \includegraphics[width=0.8\textwidth]{figures/sparse_full_explog}

-\caption{The impact of exp-log solution to compute very high degrees of  polynomial.}
+\caption{The impact of exp.log solution to compute very high degrees of  polynomial.}

 \label{fig:03}
 \end{figure}
 
 \label{fig:03}
 \end{figure}
 

-The figure 3, show a comparison between the execution time of the Ehrlich-Aberth algorithm applying exp.log solution and the execution time of the Ehrlich-Aberth algorithm without applying exp.log solution, with full and sparse polynomials degrees. We can see that the execution time for the both algorithms are the same while the full polynomials degrees are less than 4000 and full polynomials are less than 150,000. After,we show clearly that the classical version of Ehrlich-Aberth algorithm (without applying log.exp) stop to converge and can not solving any polynomial sparse or full. In counterpart, the new version of Ehrlich-Aberth algorithm (applying log.exp solution) can solve very high and large full polynomial exceed 100,000 degrees.
+The figure 3, show a comparison between the execution time of the Ehrlich-Aberth algorithm applying exp.log solution and the execution time of the Ehrlich-Aberth algorithm without applying exp.log solution, with full and sparse polynomials degrees. We can see that the execution time for the both algorithms are the same while the full polynomials degrees are less than 4000 and full polynomials are less than 150,000. After,we show clearly that the classical version of Ehrlich-Aberth algorithm (without applying exp.log) stop to converge and can not solving any polynomial sparse or full. In counterpart, the new version of Ehrlich-Aberth algorithm (applying exp.log solution) can solve very high and large full polynomial exceed 100,000 degrees.

 
 

-in fact, when the modulus of the roots are up than \textit{R} given in ~\ref{R},this exceed the limited number in the mantissa of floating points representations and can not compute the iterative function given in ~\ref{eq:Aberth-H-GS} to obtain the root solution, who justify the divergence of the classical Ehrlich-Aberth algorithm. However, applying log.exp solution given in ~\ref{sec2} took into account the limit of floating using the iterative function in(Eq.~\ref{Log_H1},Eq.~\ref{Log_H2} and allows to solve a very large polynomials degrees . 
+in fact, when the modulus of the roots are up than \textit{R} given in ~\ref{R},this exceed the limited number in the mantissa of floating points representations and can not compute the iterative function given in ~\ref{eq:Aberth-H-GS} to obtain the root solution, who justify the divergence of the classical Ehrlich-Aberth algorithm. However, applying exp.log solution given in ~\ref{sec2} took into account the limit of floating using the iterative function in(Eq.~\ref{Log_H1},Eq.~\ref{Log_H2} and allows to solve a very large polynomials degrees .