MAJ des figures

[kahina_paper1.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index c8a669875ef7f21354062b4e62411a0c98328977..9897244a20b19a6506dc8eeb096f446bec2fbf4d 100644 (file)
--- 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}
 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}
 
 \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}
 \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}
 \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.
 
 \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}
 %%\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} 
 \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
 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
 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}
 
 
 \subsection{Comparative study}


@@ -678,7 +672,12 @@ We initially carried out the convergence of Aberth algorithm with various sizes
 \label{fig:01}
 \end{figure}
 
 \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]
 
 \subsubsection{A comparative study between Aberth and Durand-kerner algorithm}
 \begin{table}[htbp]