Just MAJ experiment part

[kahina_paper1.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 91bdab7af02177a97b9e40a6974566ae2ed4f0e8..7907494f1d577ecf3be95f67c24ca4a0c32b5034 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -321,6 +321,7 @@ Applying this solution for the Aberth method we obtain the
 iteration function with 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}
 iteration function with 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}

 H_{i}(z)=z_{i}^{k}-\exp \left(\ln \left(
 p(z_{k})\right)-\ln\left(p(z_{k}^{'})\right)- \ln
 \left(1-Q(z_{k})\right)\right),
 H_{i}(z)=z_{i}^{k}-\exp \left(\ln \left(
 p(z_{k})\right)-\ln\left(p(z_{k}^{'})\right)- \ln
 \left(1-Q(z_{k})\right)\right),


@@ -329,12 +330,17 @@ p(z_{k})\right)-\ln\left(p(z_{k}^{'})\right)- \ln
 where:
 
 \begin{equation}
 where:
 
 \begin{equation}

+\label{Log_H1}

 Q(z_{k})=\exp\left( \ln (p(z_{k}))-\ln(p(z_{k}^{'}))+\ln \left(
 \sum_{k\neq j}^{n}\frac{1}{z_{k}-z_{j}}\right)\right).
 \end{equation}
 
 This solution is applied when the root except the circle unit, represented by the radius $R$ evaluated as:
 Q(z_{k})=\exp\left( \ln (p(z_{k}))-\ln(p(z_{k}^{'}))+\ln \left(
 \sum_{k\neq j}^{n}\frac{1}{z_{k}-z_{j}}\right)\right).
 \end{equation}
 
 This solution is applied when the root except the circle unit, represented by the radius $R$ evaluated as:

-$$R = \exp( \log(DBL\_MAX) / (2*n) )$$ where $DBL\_MAX$ stands for the maximum representable double value.
+\begin{equation}
+\label{R}
+R = \exp( \log(DBL\_MAX) / (2*n) )
+\end{equation}
+ where $DBL\_MAX$ stands for the maximum representable double value.

 
 \section{The implementation of simultaneous methods in a parallel computer}
 \label{secStateofArt}   
 
 \section{The implementation of simultaneous methods in a parallel computer}
 \label{secStateofArt}   


@@ -642,7 +648,7 @@ We report the execution times of the Ehrlisch-Aberth method implemented on one c
 
 
 \subsubsection{Influence of the number of threads on the execution times of different polynomials (sparse and full)}
 
 
 \subsubsection{Influence of the number of threads on the execution times of different polynomials (sparse and full)}

-It is also interesting to see the influence of the number of threads per block on the execution time. For that, we notice that the maximum number of threads per block for the Nvidia Tesla K40c GPU is 1024, so we varied the number of threads per block from 8 to 1024.we took into account the execution time for both sparse and full polynomials of size 50000 and 500000 degrees.
+It is also interesting to see the influence of the number of threads per block on the execution time. For that, we notice that the maximum number of threads per block for the Nvidia Tesla K40c GPU is 1024, so we varied the number of threads per block from 8 to 1024. We took into account the execution time for both sparse and full polynomials of size 50000 and 500000 degrees.

 
 \begin{figure}[H]
 \centering
 
 \begin{figure}[H]
 \centering


@@ -651,16 +657,26 @@ It is also interesting to see the influence of the number of threads per block o
 \label{fig:01}
 \end{figure}
 
 \label{fig:01}
 \end{figure}
 

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

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

 \begin{figure}[H]
 \centering
   \includegraphics[width=0.8\textwidth]{figures/log_exp}
 \caption{The impact of exp-log solution to compute very high degrees of  polynomial.}
 \label{fig:01}
 \end{figure}
 \begin{figure}[H]
 \centering
   \includegraphics[width=0.8\textwidth]{figures/log_exp}
 \caption{The impact of exp-log solution to compute very high degrees of  polynomial.}
 \label{fig:01}
 \end{figure}

+
+The figure 3, show a comparison between the execution time of the Ehrlisch-Aberth algorithm applying log-exp solution and the execution time of the Ehrlisch-Aberth algorithm without applying log-exp solution, with full polynomials degrees. We can see that the execution time for the both algorithms are the same while the polynomials degrees are less than 4500. After,we show clearly that the classical version of Ehrlisch-Aberth algorithm (without applying log.exp) stop to converge and can not solving polynomial exceed 4500, in counterpart, the new version of Ehrlisch-Aberth algorithm (applying log.exp solution) can solve very high and large full polynomial exceed 100,000 degrees.
+
+in fact, when the modulus of the roots are up than 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 Ehrlisch-Aberth algorithm. However, applying log.exp solution given in ~\ref{sec2} took into account the limit of floating using the iterative function in~\ref{Log_H1} ~\ref{Log_H2}.
+
+
+
+%we report the performances of the exp.log for the Ehrlisch-Aberth algorithm for solving very high degree of polynomial. 
+

  
 \subsubsection{A comparative study between Aberth and Durand-kerner algorithm}
 
  
 \subsubsection{A comparative study between Aberth and Durand-kerner algorithm}