set ylabel "execution times (in s)"
set xlabel "polynomial's degree"
-set logscale x
-set logscale y
+#set logscale x
+#set logscale y
#set key on outside left bmargin
set style line 1 lc rgb '#0060ad' lt 1 lw 2 pt 1 ps 1.5 # --- blue
-set style line 3 lc rgb '#dd181f' lt 1 lw 2 pt 1 ps 1.5 # --- red
+set style line 3 lc rgb '#dd181f' lt 1 lw 2 pt 2 ps 1.5 # --- red
set style line 2 lc rgb '#dd181f' lt 1 lw 2 pt 5 ps 1.5 # --- red
plot 'EA_DK.txt'index 0 using 1:2 t "EA with sparse polynomials" with linespoints ls 1,\
# First data block (index 0)
-#EA sparse full
+#EA sparse full
#Taille_Poly times nb iter times nb iter
5000 0.40 17 0.748784 25
-50000 3.92 17 139.87 195
-100000 12.45 16 1459.35 31
-150000 28.67 17 754.24 27
-200000 40 23 718.623 27
-250000 93.76 20 715.554 27
-300000 138.94 21 1089.61 27
+50000 3.92 17 25.9504 40
+100000 12.45 16 54.5215 30
+150000 28.67 17 156.63 33
+200000 40 23 330.456 43
+250000 93.76 20 518.342 47
+300000 138.94 21 912.078 50
350000 159.65 18 1746.53 22
400000 258.91 22 3112 20
450000 339.47 23
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 \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 Ehrlisch-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 Ehrlisch-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 .
\subsubsection{A comparative study between Ehrlisch-Aberth algorithm and Durand-kerner algorithm}
-In this part, we are interesting to compare the simultaneous methods, Ehrlisch-Aberth and Durand-Kerner in parallel computer using GPU. We took into account the execution time, the number of iteration and the polynomials size
+In this part, we are interesting to compare the simultaneous methods, Ehrlisch-Aberth and Durand-Kerner in parallel computer using GPU. We took into account the execution time, the number of iteration and the polynomial's size. for the both sparse and full polynomials.
\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{figures/EA_DK}
-\caption{The execution time of Ehrlisch-Aberth versus Durand-Kerner algorithm}
+\caption{The execution time of Ehrlisch-Aberth versus Durand-Kerner algorithm on GPU}
\label{fig:01}
\end{figure}
+This figure show the execution time of the both algorithm EA and DK with sparse polynomial degrees ranging from 1000 to 1000000. We can see that the Ehrlisch-Aberth algorithm are faster than Durand-Kerner algorithm, with an average of 25 times as fast. Then, when degrees of polynomial exceed 500000 the execution time with EA is of the order 100 whereas DK passes in the order 1000. %with double precision not exceed $10^{-5}$.
+
\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{figures/EA_DK_nbr}
