X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/kahina_paper1.git/blobdiff_plain/4bba0524dc08c15ba13863012d6bc90480c18721..6f76482b569516829ba1ec3f0358dec6c479c001:/paper.tex

diff --git a/paper.tex b/paper.tex
index c8a6698..9897244 100644
--- a/paper.tex
+++ b/paper.tex
@@ -96,7 +96,7 @@ root finding of polynomials, high degree, iterative methods, Durant-Kerner, GPU,
 Polynomials are algebraic structures used in mathematics that capture physical phenomenons and that express the outcome in the form of a function of some unknown variable. Formally speaking,  a polynomial $p(x)$ of degree \textit{n} having $n$ coefficients in the complex plane \textit{C} and zeros $\alpha_{i},\textit{i=1,...,n}$ 
 %%\begin{center}
 \begin{equation}
-     {\Large p(x)=\sum{a_{i}x^{i}}=a_{n}\prod(x-\alpha_{i}),a_{0} a_{n}\neq 0}.
+     {\Large p(x)=\sum_{i=0}^{n}{a_{i}x^{i}}=a_{n}\prod_{i=1}^{n}(x-\alpha_{i}), a_{0} a_{n}\neq 0}.
 \end{equation}
 %%\end{center}
 
@@ -583,33 +583,27 @@ or from GPU memory to CPU memory \verb=(cudaMemcpyDeviceToHost))=.
 \section{Experimental study}
 
 \subsection{Definition of the polynomial used}
-We use two forms of  polynomials:
-\paragraph{sparse polynomial}:
-in this following form, the roots are distributed on 2 distinct circles:
+We study two forms of  polynomials the sparse polynomials and the full polynomials:
+\paragraph{Sparse polynomial}: in this following form, the roots are distributed on 2 distinct circles:
 \begin{equation}
-	\forall \alpha_{1} \alpha_{2} \in C,\forall n_{1},n_{2} \in N^{*}; P(z)= (z^{n^{1}}-\alpha_{1})(z^{n^{2}}-\alpha_{2})
+	\forall \alpha_{1} \alpha_{2} \in C,\forall n_{1},n_{2} \in N^{*}; P(z)= (z^{n_{1}}-\alpha_{1})(z^{n_{2}}-\alpha_{2})
 \end{equation}
 
 This form makes it possible to associate roots having two
 different modules and thus to work on a polynomial constitute
 of four non zero terms.
 
-\paragraph{Full polynomial}:
- the second form used to obtain a full polynomial is:
+\paragraph{Full polynomial}: the second form used to obtain a full polynomial is:
 %%\begin{equation}
 	%%\forall \alpha_{i} \in C,\forall n_{i}\in N^{*}; P(z)= \sum^{n}_{i=1}(z^{n^{i}}.a_{i})
 %%\end{equation}
 
 \begin{equation}
-     {\Large \forall a_{i} \in C, i\in N; p(x)=\sum^{n-1}_{i=1} a_{i}.x^{i}} 
+     {\Large \forall a_{i} \in C, i\in N;  p(x)=\sum^{n}_{i=0} a_{i}.x^{i}} 
 \end{equation}
 with this form, we can have until \textit{n} non zero terms.
 
 \subsection{The study condition} 
-In order to have representative average values, for each
-point of our curves we measured the roots finding of 10
-different polynomials.
-
 The our experiences results concern two parameters which are
 the polynomial degree and the execution time of our program
 to converge on the solution. The polynomial degree allows us
@@ -617,7 +611,7 @@ to validate that our algorithm is powerful with high degree
 polynomials. The execution time remains the
 element-key which justifies our work of parallelization.
 	For our tests we used a CPU Intel(R) Xeon(R) CPU
-E5620@2.40GHz and a GPU K40 (with 6 Go of ram)
+E5620@2.40GHz and a GPU K40 (with 6 Go of ram).
 
 
 \subsection{Comparative study}
@@ -678,7 +672,12 @@ We initially carried out the convergence of Aberth algorithm with various sizes
 \label{fig:01}
 \end{figure}
 
-
+\begin{figure}[htbp]
+\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}
 
 \subsubsection{A comparative study between Aberth and Durand-kerner algorithm}
 \begin{table}[htbp]