MAJ du commentaire de la figure log.exp with sparse and full poly....

[kahina_paper1.git] / paper.tex

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@@ -662,16 +662,22 @@ The figure 2 show that, the best execution time for both sparse and full polynom
 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
 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}
+  \includegraphics[width=0.8\textwidth]{figures/sparse_full_explog}

 \caption{The impact of exp-log solution to compute very high degrees of  polynomial.}
 \label{fig:01}
 \end{figure}
 
 \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 Ehrlich-Aberth algorithm applying log-exp solution and the execution time of the Ehrlich-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 Ehrlich-Aberth algorithm (without applying log.exp) stop to converge and can not solving polynomial exceed 4500, 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 log-exp solution and the execution time of the Ehrlich-Aberth algorithm without applying log-exp 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.

 
 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 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 . 
 
 

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

 
 %we report the performances of the exp.log for the Ehrlich-Aberth algorithm for solving very high degree of polynomial. 
 
 
 %we report the performances of the exp.log for the Ehrlich-Aberth algorithm for solving very high degree of polynomial.